diff --git "a/F9FLT4oBgHgl3EQfGi9H/content/tmp_files/load_file.txt" "b/F9FLT4oBgHgl3EQfGi9H/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/F9FLT4oBgHgl3EQfGi9H/content/tmp_files/load_file.txt" @@ -0,0 +1,1502 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf,len=1501 +page_content='DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D¨OLZ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We consider the H2-formatted compression and computational estimation of co- variance functions on a compact set in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The classical sample covariance or Monte Carlo estimator is prohibitively expensive for many practically relevant problems, where often ap- proximation spaces with many degrees of freedom and many samples for the estimator are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In this article, we propose and analyze a data sparse multilevel sample covariance es- timator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', a multilevel Monte Carlo estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For this purpose, we generalize the notion of asymptotically smooth kernel functions to a Gevrey type class of kernels for which we derive new variable-order H2-approximation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' These variable-order H2-approximations can be considered as a variant of hp-approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Our multilevel sample covariance estimator then uses an approximate multilevel hierarchy of variable-order H2-approximations to compress the sample covariances on each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The non-nestedness of the different levels makes the reduction to the final estimator nontrivial and we present a suitable algorithm which can handle this task in linear complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This allows for a data sparse multilevel estimator of Gevrey covariance ker- nel functions in the best possible complexity for Monte Carlo type multilevel estimators, which is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Numerical examples which estimate covariance matrices with tens of billions of entries are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Covariance functions or kernel functions g: D × D → R, on a compact set D ⊂ Rd arise in many fields of application such as Gaussian process computations [44], machine learning [33, 49], and uncertainty quantification [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' However, in many cases these functions are not available in closed form, but must be suitably estimated from samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The canonical estimator for this purpose is the sample covariance estimator or Monte Carlo estimator g ≈ 1 M M � k=1 z(k) ⊗ z(k), see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', [34], where the sample functions z(k), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , M, are assumed to be independent, identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=') elements of a Hilbert space and ⊗ is understood as the Hilbertian tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The challenge with the above estimator is that the covariance function and the samples are often infinite-dimensional objects which in practice need to be discretized for computational purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' After discretization, the sample functions themselves are represented as elements of Rn and the covariance function as a covariance matrix in Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Assuming that the samples are approximated to an accuracy of ε = n−α, roughly M = ε−2 = n2α samples need to be drawn to reach an overall error of O(ε) of the sample covariance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, the computational effort of the sample covariance estimator is O(Mn2) = O(ε−2−2/α) = O(n2α+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This is prohibitive for large n, as it is often required for sufficient accuracy in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This article presents an algorithm with rigorous error bounds for approximating the covariance function in optimal complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Here, optimal complexity is understood such that estimating the covariance has asymptotically the same complexity as estimating the mean, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', as good as O(ε−2) = O(n2α) to reach an accuracy of O(ε) under certain assumptions on the underlying approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='11992v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='NA] 27 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The challenges of large covariance matrices are commonly overcome by using data sparse approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Here, the main difference between methods is how the data sparse format is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Purely algebraic methods operate in a black-box fashion on the samples of the sample covariance estimator to estimate suitable compression parameters for previously chosen data sparse formats such as banded matrices [3] or sparse matrices [2, 3, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' See also [11] for recent literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' However, a simultaneous estimate on approximation quality and computational complexity is not available without additional assumptions on the algebraic properties of the samples and/or covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' These properties are usually inferred from assumed analytical properties of the underlying statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Here, an often considered analog to some of the matrix approximation classes considered in [2] are asymptotically smooth covariance functions, which assume a certain decrease of the covariance with increasing spatial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' These kinds of functions are also considered in the fast multipole method [25] and its and abstract counterparts H- and H2-matrices [4, 27], as well as in wavelet compression [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The first have been applied in machine learning [5] and uncertainty quantification [18, 30, 35, 48] where complexity and approximation estimates have been derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The available machinery was also applied to estimate hyperparameters of covariance functions [12, 22, 36, 39, 41], but we stress that the objective of this article is to estimate the full covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Finally, wavelet based approaches have been used in [28, 29, 30, 32, 46] for compression and estimation of covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Similar to wavelet based approaches, sparse grid approaches are also based on a multilevel hierarchy and provide a sparse representation of the covariance matrix, but assume some global smoothness of the covariance [1, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' All of the mentioned methods operating on assumed analytical properties of covariance functions are capable to reduce the storage requirements of corresponding covariance matrices in Rn×n from O(n2) to O(n) or O(n logβ n), β > 0, with a negligible approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, the n2 part of the computational cost of the sample covariance estimator can significantly be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Reducing the computational cost of the sampling process can essentially achieved by two ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The first approach is to see the sample covariance estimator as a Monte Carlo quadrature for a stochastic integral and to replace that quadrature rule by a more efficient method such as quasi-Monte Carlo methods [16] and sparse grid approaches [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' However, bare strong assump- tions, further measures to reduce the number of samples are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The second approach to reduce computational cost during sampling are variance reduction techniques and in particular the multilevel Monte Carlo method, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', [24, 31] for a general overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The basic idea is to exploit a multi-level hierarchy in the approximation spaces for the covariance discretization to obtain covariance matrices of decreasing size and to combine many smaller and only a few larger matrices to a covariance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' It was applied to smaller and dense covariance matrices in [42] for the estimation of Sobol indices and to larger covariance matrices combined with a sparse grid approximation in [1, 14] and combined with a wavelet approximation in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Gδ-asymptotical smoothness and Gevrey kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' As we will show in a moment, there is a large class of covariance functions which is not asymptotically smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The first objective of this paper is to generalize some of the available H2-compression techniques, which can be seen as a special variant of hp-approximation, to a more general class of covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' However, we stress that all of the presented algorithms also apply to the classical, asymptotically smooth kernel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To this end, we assume that D is equipped with a measure µ, write L2 µ(D) = L2(D), and assume that we are given a probability space (Ω, Σ, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Following the stochastic partial differential equation approach to Gaussian random fields [38, 51], we note that realizations Z ∈ L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Hθ(D)) of any Gaussian random field with positive definite covariance function g have a representation as the solution to the equation AZ = W, where W is white noise on L2(D) and A = C−1/2 with (Cϕ)(x) = � D g(x, y)ϕ(y) dµ(x), DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 3 see [28, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3] for an explicit derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Vice versa, any self-adjoint and positive definite operator A: Hθ(D) → L2(D) yields a covariance operator C = A−2 with covariance function g given as the Schwartz kernel of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For example, the well known Mat´ern covariance kernels [40] are given through D = Rd and A = (κ2 − ∆)θ/2 with κ > 0, θ > d/2, and are asymptotically smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' More generally, we may consider any self-adjoint and positive definite pseudo-differential operator A ∈ OPSθ cl,δ(D) of order θ > d/2 with symbol of Gevrey class δ ≥ 1 in the sense of [8, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This implies C = A−2 ∈ OPS−2θ cl,δ (D) as a consequence of the pseudo-differential operator calculus for Gevrey classes developed in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In analogy to [47, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2] we obtain that the covariance kernel g (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', the Schwartz kernel) of C is smooth away from the diagonal and satisfies |∂α x ∂β y g(x, y)| ≤ CGA|α|+|β|(α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' )δ∥x − y∥2θ−d−|α|−|β| 2 , x, y ∈ D, x ̸= y, (1) for all α, β ∈ Nd and kernel dependent constants CG, A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We note that the special case δ = 1 corresponds to the classical asymptotical smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For δ ≥ 1 we will refer to Gδ-asymptotical smoothness and call the kernel function a Gevrey kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' These considerations make clear that a unified treatment of asymptotically smooth and more generally Gδ-asymptotically smooth covariance functions as presented in this article is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The objective of this article is to present an algorithm with rigorous error bounds and complexity estimates for estimating Gevrey kernels and covariance functions in op- timal complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This will be achieved by using a multilevel sample covariance estimator on an approximate multilevel hierarchy of H2-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' More precisely we generalize the variable-order H2-approximation theory, see [4, 7, 6], to Gδ-asymptotically smooth kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The basis for this generalization is a new approximation result for Gevrey regular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' we develop a multilevel algorithm which allows to evaluate the sample covariance estima- tor in variable-order H2-compressed form with negligible approximation error in optimal complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' we provide numerical examples which estimate covariance matrices with tens of billions of entries, underlying the feasibility of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' One of the major implications of these contributions is that Gδ-asymptotically smooth covariance functions of a Gaussian processes can now be asymptotically estimated with the same complexity as the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We also note that variable-order results imply fixed order results as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' First, in Section 2, we provide a new ap- proximation result for Gevrey-regular functions and use this result for establishing the required variable-order H2-approximation rates for Gevrey kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' These results are then used in Section 3 for establishing approximation rates of a single-level H2-formatted sample covariance estimator and its computational realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Section 4 is concerned with the construction and analysis of the H2-formatted multilevel sample covariance estimator, whereas Section 5 considers its algorith- mic implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Finally, in Section 6, we provide the numerical experiments underlining our theoretical considerations before we draw our conclusions in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' H2-approximation of Gevrey kernels 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Interpolation of Gevrey functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We start our considerations by recalling the definition of functions of Gevrey class and some basic facts on polynomial interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let D ⊂ Rd and f ∈ C∞(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' f is of Gevrey class δ ≥ 1 with CG, A > 0, f ∈ Gδ(D, CG, A), if for every K ⋐ D and α ∈ Nd it holds |∂αf(x)| ≤ CGA|α|(α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' )δ for all x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' A function is analytic, if it is of Gevrey class δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 1We refrain from making this notion more explicit as we will not need it for the remainder of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The polynomial interpolation I[a,b] m : C([a, b]) → Pm on m + 1 distinct points in [a, b] is stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', ��I[a,b] m [f] �� C([a,b]) ≤ Λm∥f∥C([a,b]), for all m ∈ N, with stability constant Λm ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' An example satisfying this assumption is the interpolation on Chebychev points, which is stable with stability constant Λm ≤ 2 π ln(m) + 1, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', [45, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3 ([4, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For m ∈ N and f ∈ C([a, b]) it holds ��f − I[a,b] m [f] �� C([a,b]) ≤ (Λm + 1) min p∈Pm ∥f − p∥C([a,b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The following theorem is the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In comparison to other approx- imation results in the literature, we note that the dependence of the contraction factor on A is explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This is an essential ingredient for establishing the H2-approximation rates later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let f ∈ Gδ([−1, 1], CG, A), ρ(r) = r + √ 1 + r2, and m ∈ N, m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then it holds min p∈Pm ∥f − p∥C([−1,1]) ≤ C(A, δ)CGρ(1/A)−m1/δ/e2, where C(A, δ) is monotonically increasing in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The proof is inspired by the one of [43, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Denote by I3 : H2([−1, 1]) → P3 the Hermite interpolation operator given by I3f(±1) = f(±1), (I3f)′(±1) = f ′(±1) and, for m ∈ N, m ≥ 3, denote by πm−2,0 : L2([−1, 1]) → Pm−2 the L2-orthogonal projection onto the first m − 1 Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then, the projector H2([−1, 1]) → Pm defined by (πm,2f)(x) = (I3f)(x) + � x −1 � y −1 � πm−2,0 � (f − I3f)′′�� (z) dz dy satisfies the error estimate, see [15, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1], ∥f − πm,2f∥2 H2([−1,1]) ≤ C (m − 1 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (m − 1 + k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' ��f (k+2)��2 L2([−1,1]), 2 ≤ k ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Now, fix α = (2ρ(1/A)ρ(A)A)−1/δ, k = ⌊αγm1/δ⌋ with γ = min{max{ 2 αm1/δ , 1}, m−1 αm1/δ }, and note that 2 ≤ k ≤ m − 1, k ≤ αγm1/δ ≤ k + 1, and ρ(1/A)ρ(A) ≤ (2/A + 1)2A+1 =: Ξ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Gevrey regularity f ∈ Gδ([−1, 1], CG, A) and Stirling’s formula √ 2πn(n/e)n ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' ≤ e√n(n/e)n, n ∈ N imply ∥f − πm,2f∥2 H2([−1,1]) ≤ CC2 GA2k+4 (m − 1 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (m − 1 + k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' � (k + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' �2δ ≤ CC2 GA2k+4 e1+2k √ 2π (m − 1 − k)m−1−k+1/2 (m − 1 + k)m−1+k+1/2 � k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (k + 2)2�2δ ≤ CC2 GA2k+4 e1+2k √ 2π (m − 1 − k)m−1−k+1/2 (m − 1 + k)m−1+k+1/2 e2δ(1−k)k2kδkδ(k + 2)4δ ≤ CC2 GA2k+4 e1+2δ+2(1−δ)k √ 2π �m − 1 − k m − 1 + k �m−1−k+1/2 m−2kk2kδkδ(k + 1)4δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Since 1 − δ ≤ 0, m − 1 − k + 1/2 ≥ 0 for k ≤ m − 1, m−k ≤ (αγ/k)δk, and kδ(k + 2)4δ ≤ C(δ)22k for k ≥ 2 this implies ∥f − πm,2f∥H2([−1,1]) ≤ C(δ)CGAk+2γδkαδk2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We next remark that γδk ≤ 1 for 2 ≤ αm1/δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For for 2 > αm1/δ, we remark that γδk ≤ γ2δ ≤ C(δ)(Ξ(A)A)2, where Ξ(A)A is continuous and monotonically increasing on (0, ∞) with limt→0 Ξ(t)t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, γδk ≤ χ(A, δ) is monotonically increasing in A with χ(A, δ) ≥ 4C(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The continuous embedding H2([−1, 1]) �→ L∞([−1, 1]) and the definition of α then yield ∥f − πm,2f∥C([−1,1]) ≤ C(A, δ)CGA2ρ(1/A)ρ(A)ρ(1/A)−ρ(A)(k+1) ≤ C(A, δ)CGρ(1/A)−ρ(A)αγm1/δ, DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 5 where C(A, δ) is monotonically increasing in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To obtain the desired exponent, we consider that ρ(A)α(A, δ) is monotonically increasing in δ, and that it is bounded from below by e−2 for δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For αm1/δ < m − 1 this yields ρ(A)αγm1/δ ≥ m1/δ/e2 due to γ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For αm1/δ ≥ m − 1 we observe that ρ(A)αγm1/δ ≥ m − 1 ≥ m1/δ/e2, which yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For any f ∈ Gδ([a, b], CG, A), B = A(b − a)/2, and m ∈ N, m ≥ 3, it holds that ��f − I[a,b] m [f] �� C([a,b]) ≤ C(B, δ)CG(Λm + 1)ρ(1/B)−m1/δ/e2, where C(B, δ) is monotonically increasing in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Denoting Φ[a,b] : [−1, 1] → [a, b] with Φ[a,b](t) = (b + a)/2 + t(b − a)/2, one easily verifies that f ∈ Gδ([a, b], CG, A) implies f ◦ Φ[a,b] ∈ Gδ([−1, 1], CG, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4 yield the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ We close the subsection by generalizing the result to tensor product domains in higher dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For Q = × d i=1[ai, bi], f ∈ C(Q), and m ∈ N, we define the tensor product interpolation operator IQ m = �d i=1 IQ m,i, with IQ m,i denoting the action of Im in coordinate direction i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , d of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Q = × d i=1[ai, bi], f ∈ Gδ(Q, CG, A), and m ∈ N, m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then it holds ∥f − IQ m[f]∥C(Q) ≤ C(A diam∞(Q)/2, δ)CGd(Λm + 1)dρ � 2 A diam∞(Q) �−m1/δ/e2 , where C(A, δ) is monotonically increasing in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In complete analogy to the proof of [4, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='21], using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 and Assump- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Interpolation of Gevrey kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' As outlined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3, it is desirable to generalize the approximation theory of the widely known class of asymptotically smooth kernel functions to kernels satisfying the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Dx, Dy ⊂ Rd and g ∈ C∞({(x, y) ∈ Dx × Dy : x ̸= y}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For δ ≥ 1, g is called Gδ(CG, A)-asymptotically smooth on Dx × Dy if there exist CG, A > 0 and q ∈ R such that it holds |∂α x ∂β y g(x, y)| ≤ CGA|α|+|β|(α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' )δ∥x − y∥−2q−d−|α|−|β| 2 , x ∈ Dx, y ∈ Dy, x ̸= y, (2) for all α, β ∈ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For δ = 1 we obtain the classical asymptotical smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The following theorem generalizes the very similar result for asymptotically smooth kernels proven in [4, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Qt = × d i=1[ai, bi] and Qs = × 2d i=d+1[ai, bi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let η > 0 and Qt and Qs be admissible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', max � diam∞ Qt, diam∞ Qs} = diam∞(Qt × Qs) ≤ 2η dist2(Qt, Qs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (3) Let g be Gδ(CG, A)-asymptotically smooth on Qt × Qs and ˜g = IQt×Qs m [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then it holds for m ∈ N, m ≥ 3, ∥g − ˜g∥C(Qt×Qs) ≤ C(Aη, δ)CG 2d(Λm + 1)2d dist2(Qt, Qs)2q+d ρ � 1 Aη �−m1/δ/e2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In complete analogy to the proof of [4, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ To improve readability we may note that limt→∞ p(t)˜ρt1/δ = 0 for any polynomial p and ˜ρ ∈ (0, 1) to follow [4, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='23] and reformulate Equation (4) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9 as ∥g − ˜g∥C(Qt×Qs) ≤ Cin dist2(Qt, Qs)2q+d ˜ρm1/δ, ˜ρ : = min � Aη Aη + 1, Aη 2 �1/e2 > ρ � 1 Aη �−1/e2 , (5) for some fixed Cin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' All further results from the classical theory for asymptotically smooth kernels are generalized with only minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In the following subsection we highlight a result going back to [6] which allows to choose the polynomial degree of the interpolation according to the spatial size of the clusters, yielding linear storage complexity for the compression of Gevrey kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The classical results for asymptotically smooth kernel functions depend on the analyticity of the kernel function in admissible clusters since these estimates are based on analytic continuations into Bernstein ellipses in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In contrast, the arguments of our generalizations to Gδ(CG, A)-asymptotically smooth kernels only require finite smoothness in each direction and do not require extensions into the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Cluster trees and block-cluster trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Cluster trees and block-cluster trees are the basis for H2-approximations of kernel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We recall the basic notions along the lines of [27, Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2] and [4, Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let I ⊂ N be a finite index set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The cluster tree TI is a tree whose vertices correspond to non-empty subsets of I and are referred to as clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We require that the root of TI corresponds to I and that it holds ˙∪s∈children(t)s = t for all non-leaf clusters t ∈ TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The leafs of TI are denoted by LI and the distance of a cluster t ∈ TI to the root is denoted by level(t) ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The depth of the cluster tree is the maximal level of its clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let D ⊂ Rd be bounded and {Di}i∈I a decomposition of D into simply connected sets indexed by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We say that Qt = × d i=1[ai, bi] is a bounding box of t if Dt = ∪i∈tDi ⊂ Qt, for all t ∈ TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We remark that the definition implies that LI provides a decomposition of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Further, for computational reasons, we make the following assumptions on the considered cluster trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a cluster tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We assume that (1) the cluster tree is built on a decomposition {Di}i∈I of D ⊂ Rd bounded into simply con- nected sets, (2) the number of children for non-leaf clusters bounded from below and above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', 2 ≤ | children(t)| ≤ Cab, t ∈ TI \\ LI, (6) for some Cab > 0, (3) the cardinality of the leaf clusters is bounded from below and above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', nmin/Cab ≤ |t| ≤ nmin, t ∈ LI, (7) for some nmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Most standard algorithms for constructing cluster trees result in cluster trees satisfying these conditions, see also [4, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Given a cluster tree TI, the block-cluster tree TI×I is a tree with vertices corresponding to cluster pairs, referred to as block-clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Starting with t × s = I × I the block-cluster tree is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (1) Check whether t × s has admissible bounding boxes in the sense of Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (2) (a) If t × s has admissible bounding boxes, add it to L+ I×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (b) Otherwise, perform Item 1 for all t′ × s′, t′ ∈ children(t), s′ ∈ children(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' If t or s have no children, add t × s to L− I×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The algorithm induces a tree structure TI×I whose set of leafs is given as LI×I = L+ I×I ∪ L− I×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='0 Interpolation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='0 Reinterpolation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='0 Second reinterpolation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='00 Third reinterpolation Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Illustration of iterated interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The continuous polynomial (upper left) is replaced by a piecewise polynomial of lower degree (lower right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We remark that the definition implies that LI×I provides a partition of I × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Moreover, if t × s ∈ TI×I, then also s × t ∈ TI×I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', the block-cluster tree is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The following constant allows to quantify the sparsity of a block-cluster tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Given a block-cluster tree TI×I, its sparsity constant Csp is defined as Csp = max t∈TI ��� s ∈ TI : t × s ∈ TI×I ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Variable-order H2-approximation spaces of Gevrey kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The following definitions aim at defining H2-approximation spaces of kernel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a cluster tree and LI its leafs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For all t, s ∈ TI we define Lt = {t0 ∈ LI : ∃ cluster chain t0 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' ⊆ tn = t with ti−1 ∈ children(ti), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , n}, and Lt×s = {t0 × s0 : t0 ∈ Lt, s0 ∈ Ls}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let q ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The family of bounding boxes (Qt)t∈TI is called q-regular if all cluster chains t0 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' ⊆ tn = t, t ∈ TI, t0 ∈ Lt, yield families of bounding boxes (Qi)n i=0, Qi = × d j=1 Ji j bounding box to ti, satisfying |Ji−1 j | ≤ q|Ji j| for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , n, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a cluster tree and (Qt)t∈TI a q-regular family of bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let α ∈ N0, β ∈ N and kδ i = ⌈(β + αi)δ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let t, s ∈ TI, t0 ∈ Lt, s0 ∈ Ls and t0 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' ⊆ tn = t and s0 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' ⊆ sm = s cluster chains in TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We define the interpolation operators It t0 = It0 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' ◦ Itn, with Iti = IQi kδ p−level(ti) for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , n, and It×s t0×s0 = It t0 ⊗ Is s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' An illustration of the iterated interpolation process can be found in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We asume that TI is a cluster tree of depth p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In accordance with [4, 6] we assume that 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ (1) there are constants CΛ, λ ≥ 1 such that the stability constant Λm of the interpolation operator I[a,b] m , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2, satisfies Λm ≤ CΛ(m + 1)λ for all m ∈ N0, (2) (Qt)t∈TI is a q-regular family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' [4, 6] also assume that TI×I is locally homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This condition is automati- cally satisfied for all block-clusters as constructed in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We are now in the position to define H2-spaces of kernel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a cluster tree of depth p with a q-regular family of bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let α ∈ N0, β ∈ N, kδ i = ⌈(β + αi)δ⌉ and TI×I be a block-cluster tree constructed from TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We define Pt×s = � Pkδ p−level(t) ⊗ Pkδ p−level(s) ��� t×s for all t, s ∈ TI, Ppw t×s = {f : t × s → R: f = It×s t0×s0p, t0 × s0 ∈ Lt×s, p ∈ Pt×s} for all t × s ∈ L+ I×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We define the H2-space of kernel functions as V H = � g: D × D → R: k �� t×s ∈ Ppw t×s for all t × s ∈ L+ I×I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We remark that the definition implies that each cluster t ∈ TI contains Kt = � kδ p−level(t) �d = � (β + α(p − ℓ))δ�d (8) interpolation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' All further results from the variable-order H2-theory for asymptotically smooth kernels are generalized with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In the following we use the common assumptions and state a slightly modified error estimate in the L2-norm, rather than the maximums norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' L2-error of variable-order H2-approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For Gevrey-regular kernels, the approx- imation error in each block-cluster can be estimated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='17 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let 2q ∈ [−d, 0), let the kernel function g: Rd×Rd → R be Gδ(CG, A)-asymptotically smooth, and let α ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there are constants Cin ∈ R>0 and β0 ∈ N0 such that ��g − It×s t0×s0g �� C(Qt0×Qs0) ≤ Cin � ˜ρβ+α(p−level(t)) diam∞(Qt)2q+d �1/2� ˜ρβ+α(p−level(s)) diam∞(Qs)2q+d �1/2 holds with ˜ρ as in Equation (5) for all β ≥ β0, all blocks t × s ∈ L+ I×I satisfiyng Equation (3), and all t0 ∈ Lt, s0 ∈ Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The proof follows the arguments of [6] and [4, Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7] with only minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The restriction on 2q can be lifted to 2q ∈ R<0, if t × s ∈ L+ I×I, t ∈ children(t′), s ∈ children(s′), t′, s′ ∈ TI, and t′ × s′ does not satisfy Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This is the case for most block-cluster trees, in particular for the ones constructed as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Although the results from the literature can be generalized to Gevrey kernels, most of the analysis in the literature is based on an C(Qt0 × Qs0)-type estimate, which is not compatible with the L2-setting of the Monte Carlo type error analysis, for which an L2-estimate is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let µ be a measure on D with a suitable σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We write L2(D) = L2 µ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Moreover, to shorten notation, we assume that D × D is equipped with the product measure ˜µ and write L2(s × t) = L2 ˜µ(Ds × Dt) for any t × s ∈ TI×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We remark that the assumptions on D and its measure are quite general, covering manifolds, graphs, and multi-screens as well as point measures, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 9 Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In addition to Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='17 we assume that there are constants Ccu ∈ R>0, hH ∈ R>0, Cgr ∈ R>0, and ζ ∈ R≥1 such that µ(Dt) ≤ Ccu diam∞(Qt)d, for all t ∈ TI, C−1 gr hH ≤ diam∞(Qt) ≤ CgrhH for all t ∈ LI, and diam∞(Qt) ≤ ζ diam∞(Qt′) for all t′ ∈ children(t), t ∈ TI, see also [4, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='58) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='59)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='23, and the assumptions of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='20 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then it holds ����g − � t0×s0∈Lt×s It×s t0×s0g ���� L2(t×s) ≤ Clch−2q H ˜ρβ(ζ−2q ˜ρα)p−level(t)/2−level(s)/2, where Clc = CinCcuC−2q gr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='23 implies diam∞(Qt) ≤ ζlevel(t′)−level(t) diam∞(Qt′) ≤ CgrhHζp−level(t) for all t′ ∈ Lt, t ∈ TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, µ(Dt) diam∞(Qt)2q+d ≤ Ccu diam∞(Qt)2q ≤ CcuC−2q gr h−2q H ζ−2q(p−level(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The assertion follows from H¨olders inequality and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='20 due to ����g − � t0×s0∈Lt×s It×s t0×s0g ���� L2(t×s) ≤ max t0×s0∈Lt×s ∥g − It×s t0×s0g∥C(Qt0×Qs0)µ(Dt)1/2µ(Ds)1/2 ≤ Cin �µ(Dt)˜ρβ+α(p−level(t)) diam∞(Qt)2q+d �1/2�µ(Ds)˜ρβ+α(p−level(s)) diam∞(Qs)2q+d �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Storage requirements of H2-farfield approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The following estimate on the storage requirements of the farfield of variable-order H2-approximations follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='23 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let g ∈ V H with α ∈ N0 and β ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then the storage requirements for the coefficients of all leafs t ∈ L+ I×I are bounded by CH2((α + β)δd|I|), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', they are linear with respect to the cardinality of the underlying index set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The constant CH2 is independent of the depth of TI×I and depends only on δ, d, Csp, and the shape of TI (see Appendix A for a precise statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We use the framework provided in [4, Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2 shows that the rank as given by Equation (8) yields a (1, α, β, δd, Cab)-bounded rank distribution in the sense of [4, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='44], see also Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4 yields that TI is a (Crc, α, β, δd, Cab)-regular cluster tree in the sense of [4, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='47], see also Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3, with Crc given as in Equation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The assertion follows from [4, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='49], see also Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' H2-sample covariance estimation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Approximation of Gaussian random field samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We consider finite dimensional approximation spaces Vh ⊂ L2(D), h > 0, and denote the L2-projection onto Vh by Πh : L2(D) → Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The approximation spaces are assumed to satisfy the approximation estimate ∥u − Πh∥L2(D) ≤ CL2hγ∥u∥Hγ(D), for all u ∈ Hγ(D), (9) for all 0 ≤ γ ≤ m for some m ∈ N with the Hilbert spaces Hγ(D) ⊂ L2(D) appropriately chosen such that Hγ(D) ⊂ Hγ′(D) ⊂ L2(D), 0 ≤ γ′ ≤ γ ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' These approximation estimates hold in scattered data approximation [50] and for the standard piecewise polynomial finite element spaces of polynomial degree m on quasi uniform meshes on manifolds or graphs [9] with Hm(D) being the standard Sobolev spaces, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Denoting by ⊗ the Hilbertian tensor product, we identify L2(D × D) ≃ L2(D) ⊗ L2(D) and write Πmix h = Πh ⊗ Πh for the L2-projection Πmix h : L2(D × D) → Vh ⊗ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We further introduce the spaces of mixed regularity Hθ mix(D × D) = Hθ(D) ⊗ Hθ(D) for θ > 0 and note that for any given centered Gaussian random field Z ∈ L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Hθ(D)) it holds g = E[Z ⊗ Z] ∈ Hθ mix(D × D) for its covariance function g due to ∥g∥Hθ mix(D×D) = ��E[Z ⊗ Z] �� Hθ mix(D×D) ≤ ∥Z ⊗ Z∥L1 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hθ mix(D×D)) ≤ ∥Z∥2 L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hθ(D)), (10) see also [14, Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10)], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Z ∈ L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Hθ(D)), θ > 0, be a Gaussian random field and g ∈ Hθ mix(D) its covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Vh be an approximation space such that Equation (9) holds for γ = min{θ, m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is a constant C⊗ L2 ∈ R>0 depending on CL2 such that it holds ∥g − Πmix h g∥L2(D×D) ≤ C⊗ L2hγ∥g∥Hγ mix(D×D) ≤ C⊗ L2hγ∥Z∥2 L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hγ(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The first estimate is standard, the second follows from Equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' L2-projection onto H2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Given the discrete approximation in a tensor product ap- proximation space Vh ⊗ Vh ⊂ L2(D × D) to a Gδ(CG, A)-asymptotically smooth kernel, we would like to convert this approximation into a variable-order H2-approximation of the kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This is accomplished by L2-projection into the vector space of H2-approximated kernel functions V H from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We denote the L2-projection of k ∈ L2(D × D) onto V H by ΠHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Due to Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='23, computing ΠHk is equivalent to computing the L2(t × s) projections ΠH t×sk of k|t×s onto Ppw t×s and setting ΠHk = � t×s∈L+ I×I ΠH t×sk + � t×s∈L− I×I k|t×s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' for k ∈ L2(D × D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We extend ΠH t×sk and k|t×s by zero outside of t × s to simplify notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The assumptions of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='24 together with Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3 imply ��g − ΠHg �� L2(t×s) = ��g − ΠH t×sg �� L2(t×s) ≤ Clch−2q H ˜ρβ(ζ−2q ˜ρα)p−level(t)/2−level(s)/2 for all blocks t × s ∈ L+ I×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Follows immediately from C´ea’s lemma and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let the assumptions of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='24 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Choose α ∈ N such that ζ−2q ˜ρα < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is β0 ∈ N such that ��g − ΠHg �� L2(D×D) ≤ ClcCsph−2q H ˜ρβ 1 − ζ−2q ˜ρα for all β ≥ β0 with ˜ρ as in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Due to L2(D × D) ≃ L2(D) ⊗ L2(D) ≃ L2(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' L2(D)) we may write ��g − ΠHg �� L2(D×D) = sup u,v∈L2(D) u,v̸=0 � D � D � g(x, y) − ΠHg(x, y) � u(x)v(y) dµ(x) dµ(y) ∥u∥L2(D)∥v∥L2(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' the Cauchy-Schwartz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' and sparsity of TI×I the numerator is estimated by � D � D � g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' y) − ΠHg(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' y) � u(x)v(y) dµ(x) dµ(y) ≤ � t×s∈L+ I×I ��g − ΠHg �� L2(t×s)∥u∥L2(t)∥v∥L2(s) ≤ Clch−2q H ˜ρβ � t×s∈L+ I×I (ζ−2q ˜ρα)(p−level(t))/2∥u∥L2(t)(ζ−2q ˜ρα)(p−level(s))/2∥v∥L2(s) ≤ ClcCsph−2q H ˜ρβ � � t∈TI (ζ−2q ˜ρα)p−level(t)∥u∥2 L2(t) �1/2� � s∈TI (ζ−2q ˜ρα)p−level(s)∥v∥2 L2(t) �1/2 ≤ ClcCsph−2q H ˜ρ�� � p � ℓ=0 (ζ−2q ˜ρα)p−ℓ � t∈TI level(t)=ℓ ∥u∥2 L2(t) �1/2� p � ℓ=0 (ζ−2q ˜ρα)p−ℓ � s∈TI level(s)=ℓ ∥v∥2 L2(t) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Finally, ζ−2q ˜ρα < 1 implies p � ℓ=0 (ζ−2q ˜ρα)p−ℓ � t∈TI level(t)=ℓ ∥u∥2 L2(t) ≤ ∥u∥2 L2(D) p � ℓ=0 (ζ−2q ˜ρα)ℓ ≤ ∥u∥2 L2(D) 1 − ζ−2q ˜ρα , which yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let the assumptions of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='24 hold and let Vh be an approximation space such that Equation (9) holds for γ = min{θ, m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Choose α ∈ N such that ζ−2q ˜ρα < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is β0 ∈ N such that ��g − ΠHΠmix h g �� L2(D×D) ≤ ClcCsph−2q H ˜ρβ 1 − ζ−2q ˜ρα + C⊗ L2hγ∥Z∥2 L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hγ(D)) for all β ≥ β0 with ˜ρ as in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Follows from stability of the L2-projection, ��g − ΠHΠmix h g �� L2(D×D) ≤ ��g − ΠHg �� L2(D×D) + ��g − Πmix h g �� L2(D×D), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ In the next subsection we discuss how we can apply ΠH to simple tensors with elements in Vh in linear complexity in dim(Vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Algorithmic realization of ΠH applied to simple tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' As we will see below, com- puting ΠH(zh⊗zh), zh ∈ Vh, efficiently is one of the central operations in the H2-formatted (single- and multi-level) estimation of covariance functions and thus deserves some discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3 implies that for any zh ∈ Vh we have ΠH(zh ⊗ zh) = � t×s∈L+ I×I ΠH t×s(zh|t ⊗ zh|s) + � t×s∈L− I×I zh|t ⊗ zh|s, where ΠH t×s(zh|t ⊗ zh|s) = upw t×s ∈ Ppw t×s are the solutions of the local variational problems Find upw t×s ∈ Ppw t×s s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (upw t×s, ppw t×s)L2(t×s) = (zh|t ⊗ zh|s, ppw t×s)L2(t×s) for all ppw t×s ∈ Ppw t×s, (11) for all t × s ∈ L+ t×s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ Crucially, Ppw t×s inherits the tensor product structure of Pt×s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', it holds Ppw t×s = Ppw t ⊗ Ppw s , for all t × s ∈ L+ I×I, where Ppw t = {f ∈ L2(t): f = It t0p, t0 ∈ Lt, p ∈ Pkp−level(t) �� t}, for all t ∈ TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, Equation (11) is equivalent to solving the finite dimensional variational problems Find upw r ∈ Ppw t s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (upw r , ppw r )L2(r) = (zh|r, ppw r )L2(r) for all ppw r ∈ Ppw r , for r ∈ {t, s} and setting upw t×s = upw t ⊗ upw s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Fixing appropriate nodal bases Ppw r = span{ψr i }m i=1 with m as in Equation (8) this is equivalent to solving the systems of linear equations Qrur = qh r (12) with Qr = � (ψr i , ψr j)L2(r) �m i,j=1, qh r = � (zh|r, ψr i )L2(r) �m i=1, ur = � ψr i �m i=1, (13) for r ∈ {t, s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The expression for qh r can be further simplified to qh r = Mrzh r, where Mr = � (ψr i , φr j)L2(r) � i,j, ψr i ∈ Ppw r , φr j ∈ Vj|r, is the moment matrix on r and zh r is the coefficient vector of zh|r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We note that Ppw t = Pt for all t ∈ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We will now show that, for a given sample zh ∈ Vh, computing ΠH(zh⊗zh) can be accomplished in O(dim Vh) complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To avoid technicalities, we make the following simplifying assumption, which is satisfied if Vh is suitably build on refinements of the decomposition {Di}i∈I, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We assume that dim(Vh|s) ≤ Cminnmin for all s ∈ LI and some constant Cmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let t ∈ TI \\LI, t′ ∈ children(t), and Et′ be the matrix representation of Et′ : Pt → Pt′ defined by p �→ It′p with respect to the bases {ψt i}m i=1 and {ψt′ i }m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We refer to {Et}t∈TI\\{I} as the transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For the constant order case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', for α = 0, we denote the family of transfer matrices by {Ft}t∈TI\\{I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7 hold and let zh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then we can compute {qh t }t∈TI defined as in Equation (13) in at most CH2(α + β)δd|I| operations with the H2-forward transformation, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', [4], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e, as follows: (1) Compute qh t = Mtzh t for all t ∈ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (2) Recursively compute qh t = � t′∈children(t) E⊺ t′qh t′ for all t ∈ TI \\ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This is a classical result from the literature, see [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='45 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='48], using the same constants as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We can compute {Qt}t∈TI as defined in Equation (13) in in at most 2CH2(α + β)2δd|I| operations as follows: (1) Compute Qt for all t ∈ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Keep in mind that Ppw t = Pt in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (2) Recursively compute Qt = � t′∈children(t) E⊺ t′Qt′Et′ for all t ∈ TI \\ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In complete analogy to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9, see also Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25 and [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='45 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ We remark that actual implementations would compute and factorize {Qt}t∈TI once and use it for all samples, whereas {qt}t∈TI needs to be recomputed for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' However, we will not further exploit this fact in the following estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7 hold and let zh ∈ Vh and TI be a cluster tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then we can compute ΠH(zh ⊗ zh) in at most 7CH2(α + β)2δd|I| operations as follows: (1) Compute {qh t }t∈TI and {Qt}t∈TI as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (2) Solve the local systems Qtut = qh t , see Equation (12), for all t ∈ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 13 (3) Compute ut ⊗ us to obtain ΠH t×s(zh|t ⊗ zh|s) = upw t×s ∈ Ppw t×s for all t × s ∈ L+ I×I and zh|t ⊗ zh|s for all t × s ∈ L− I×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Computing {qh t }t∈TI and {Qt}t∈TI is achivable in a combined 3CH2(α + β)2δd|I|, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Solving the local systems t ∈ TI is achievable in at most 3K3 t complexity if a dense solver is used, with Kt given as in Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='45 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='48] with the same constants as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25 yields that solving all local systems requires 3CH2(α+β)2δd|I| operations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Computing ut ⊗us, t×s ∈ L+ I×I, requires KtKs operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='49] yields that the third step can be achieved in CH2(α + β)2δd|I| operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' H2-sample covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Consider a centered Gaussian random field Z ∈ L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Hθ(D)), θ > 0, with unknown covariance function g ∈ Gδ(CG, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We would like to estimate g in H2- compressed form from approximations of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' samples of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Given an approximation space Vh ⊂ L2(D) we define the sample covariance estimator (SCE) as E[Πmix h g] ≈ EMC[Πmix h g] = 1 M M � k=1 Πmix h � z(k) ⊗ z(k)� = 1 M M � k=1 � Πhz(k) ⊗ Πhz(k)� , with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' samples z(k), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , M, M ∈ N, of Z ∈ L2 P(Ω, Hθ(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Z ∈ L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Hθ(D)), θ > 0, be a centered Gaussian random field with covariance function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Vh be an approximation space such that Equation (9) holds for γ = min{θ, m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then it holds ��g − EMC[Πmix h g] �� L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='L2(D×D)) ≤ � C⊗ L2hγ + 1 √ M � ∥Z∥2 L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hγ(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The estimate is derived by standard methods using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', also [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ As is meanwhile well known, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' [1] for a reference, the naive sample covariance estimator from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 is computationally inconvenient for the estimation of second moments since it yields a quadratic complexity in the dimension of Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Instead, we pursue the following alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The H2-formatted sample covariance estimator (H2-SCE) is defined as E[ΠHΠmix h g] ≈ EMC[ΠHΠmix h g] = 1 M M � k=1 ΠH� Πhz(k) ⊗ Πhz(k)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' As outlined in the previous subsection, a single sample of the estimator can be computed in linear complexity in |I| ∼ dim(Vh), if a solver with linear complexity for evaluating Πhz(k) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, the overall complexity of the H2-SCE is O(M|I|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let the assumptions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Choose α ∈ N such that ζ−2q ˜ρα < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is β0 ∈ N such that ��g − EMC[ΠHΠmix h g] �� L2 P(Ω,L2(D×D)) ≤ ClcCsph−2q H ˜ρβ 1 − ζ−2q ˜ρα + � C⊗ L2hγ + 1 √ M � ∥Z∥2 L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hγ(D)) for all β ≥ β0 with ˜ρ as in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We first note that EMC[ΠHΠmix h g] = ΠHEMC[Πmix h g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Stability of the L2-projection yields ��g − EMC[ΠHΠmix h g] �� L2 P(Ω,L2(D×D)) = ��g − ΠHEMC[Πmix h g] �� L2 P(Ω,L2(D×D)) ≤ ��g − ΠHg �� L2(D×D) + ��g − EMC[Πmix h g] �� L2 P(Ω,L2(D×D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The first term is estimated with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 and the second with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Computational H2-sample covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For computational covariance esti- mation one often aims at a discretization of the covariance function rather than the covariance itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In the following we provide error estimates for bilinear forms of type a(uh, vh) = � D � D g(x, y)uh(x)vh(y) dµ(x) dµ(y) (14) for uh, vh ∈ Wh with Wh ⊂ L2(D) being some approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The canonical applications are bilinear forms of Galerkin schemes and Nystr¨om discretizations in scattered data approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For the latter we chose the approximation space to be a set of dirac distributions on points xi ∈ D, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , N, such that Equation (14) reads a(u, v) = N � i,j=1 g(xi, xj)uivj (15) for u = [ui]N i=1, v = [vi]N i=1 ∈ RN, see also [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We first provide the error estimate and thereafter some assumptions one will usually make on the approximation space Wh in order to achieve linear complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let the assumptions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='15 hold and let Wh ⊂ L2(D) be an approxi- mation space satisfying Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Choose α ∈ N such that ζ−2q ˜ρα < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is β0 ∈ N such that���� � D � D � g(x, y) − EMC[ΠHΠmix h g(x, y)] � uh(x)vh(y) dµ(x) dµ(y) ���� L2 P(Ω) ≤ �ClcCsph−2q H ˜ρβ 1 − ζ−2q ˜ρα + � C⊗ L2hγ + 1 √ M � ∥Z∥2 L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hγ(D)) � ∥uh∥L2(D)∥vh∥L2(D), for all uh, vh ∈ Wh and β ≥ β0 with ˜ρ as in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The assertion follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='15 and the Cauchy-Schwarz inequality in L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ For computational reasons, the basis of the approximation space Wh needs to be local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Wh = span{φi}i∈I be an approximation space and TI a cluster tree constructed on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We require that all basis functions φi, i ∈ t with t ∈ LI, are supported on Dt, but not on Ds for s ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We readily check that the assumption is fulfilled for piecewise constant finite elements on the decomposition {Dt}t∈TI and refinements thereof and for Nystr¨om discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Wh = span{φi}i∈I be an approximation space satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We call A = [a(φj, φi)]i,j∈I with A as in Equation (14) an H2-matrix, if g ∈ V H and A is stored in compressed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In complete analogy to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='25 and in accordance with the literature we obtain linear storage requirements for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Under the assumptions of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='16 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='17, the matrix A can be stored with a storage requirement of CH2(α + β)δd|I|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', linear in the cardinality of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This yields the following optimal result complexity-result for the H2-SCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='11 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='17 the H2-SCE is computable in complexity CH2M(α+β)δd|I|, if the H2-matrix addition is used for the summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Follows from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='11, and the linear complexity of the H2-matrix addition, see [4, Chapter 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ We remark that methods relying on a sparse grid approximation of the covariance yield a complexity which is only linear up to a logarithmic factor, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Multilevel H2-sample covariance estimation: Construction and error analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Multilevel hierarchy and cluster trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To further improve the computational complex- ity of the H2-SCE we pursue in the following a multilevel approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Our considerations are guided by the characteristics of nested finite element spaces, but can be transferred to other ap- proximation spaces providing a suitable multilevel hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To that end, we note that on a given decomposition on D we can always define a finite element space and, by employing an appropriate clustering algorithm, a cluster tree such that the following assumption is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Vh0 ⊂ L2(D) be a piecewise polynomial finite element space generated from the decomposition Th0 = {D(0) i }i∈I0 and let TI0 be a cluster tree constructed on I0 which satisfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Under these circumstances we can generate a sequence of nested decompositions {Thℓ = {D(ℓ) i }i∈Iℓ}∞ ℓ=0 with |Iℓ| = |I0|Cℓ uni (16) for some Cuni > 1 and corresponding finite element spaces Vh0 ⊂ Vh1 ⊂ Vh2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' ⊂ L2(D) in the usual way using uniform refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We can also construct nested cluster trees {TIℓ}∞ ℓ=0 by repeated uniform refinement of Th0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Th0 = {D(0) i }i∈I0 and let TI0 and Th0 satisfy Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let {Thℓ = {D(ℓ) i }i∈Iℓ}∞ ℓ=0 be a sequence of nested decompositions generated by uniform refinement of Th0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Given a cluster tree TIℓ on Iℓ, we define a cluster tree TIℓ+1 on Iℓ+1 as follows: The vertices of TIℓ+1 \\ LIℓ+1 are defined by the one-to-one correspondence of the supports of the clusters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', t(ℓ+1) ∈ TIℓ+1 \\ LIℓ+1 ⇔ there is t(ℓ) ∈ TIℓ such that D(ℓ+1) t(ℓ+1) = D(ℓ) t(ℓ), (17) with D(k) t = ∪i∈tD(k) i , k = ℓ, ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The tree hierarchy between the vertices of TIℓ+1 \\ LIℓ+1 is naturally given by the tree structure induced by the nestedness of the cluster supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For all s ∈ LIℓ let ts ∈ TIℓ+1 \\ IIℓ+1 be the corresponding cluster satisfying Equation (17) and let Tts be a cluster tree on ts satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 constructed by a cluster- ing algorithm with fixed constant C′ ab in Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We define the children of ts as children(ts) = Lts, implying that LIℓ+1 = � s∈LIℓ Lts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We say that a sequence of cluster trees is nested if Equation (17) holds for all ℓ ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To simplify notation we write t = t(ℓ) = t(ℓ+1) whenever Equation (17) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' An illustration to Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3 is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let the assumptions from Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then the sequence of cluster trees {Thℓ = {D(ℓ) i }i∈Iℓ}∞ ℓ=0 as defined in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2 is nested and satisfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 with uniform constants for all ℓ ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The nestedness of the cluster trees follows by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Further, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2 implies nmin/C′ ab ≤ |t| ≤ nmin for all t ∈ Lts due to Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Since Equation (7) also implies that |ts| ≤ 4nmin, each cluster tree Tts has at most 4nmin nmin/Cab′ = 4Cab′ leafs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, TIℓ+1 satisfies Equation (6) with C′′ ab = max{Cab, 4C′ ab}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ The nestedness of the generated cluster trees directly implies that also the the sequence of block-cluster trees {TIℓ×Iℓ}∞ ℓ=1 constructed as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13 is nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Moreover the leaves of the generated block-cluster trees provide a nested sequence of decompositions of I × I and D × D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ I0 = {1, 2, 3} {1} {2, 3} {2} {3} I1 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , 9} {1, 2, 3} {1} {2} {3} {4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , 9} {4, 5, 6} {4} {5} {6} {7, 8, 9} {7} {8} {9} D(0) 1 D(0) 2 D(0) 3 D(1) 1 D(1) 2 D(1) 3 D(1) 4 D(1) 5 D(1) 6 D(1) 7 D(1) 8 D(1) 9 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Illustration of nested cluster trees TI0 (upper left) and TI1 (upper right) in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2 to nested decompositions {D(0) i }i∈I0 (bottom left) and {D(1) i }i∈I1 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 25 25 25 25 25 25 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 49 49 49 49 49 49 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Illustration of three H2-approximation spaces on D × D = [0, 1]2 for three binary, nested, and perfectly balanced cluster trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' No approximation is performed within the red blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The blue blocks are approximated by ten- sorized iterated interpolation with the inscribed polynomial degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' β = 3, α = 2, and δ = 1 were used as parameters in Equation (8) for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The H2- approximation spaces are not nested, but have a similar structure which leads to an approximate multi-level hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The following definition identifies clusters and block clusters which are equivalent in the sense that they correspond to the same parts of D and D × D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To simplify notation we write t ∈ TIℓ for all t ∈ TIℓ+1, t × s ∈ TIℓ×Iℓ for all t × s ∈ TIℓ+1×Iℓ+1, and vice versa, whenever the involved clusters satisfy Equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We further note that the farfields and the nearfields of nested block-cluster trees do not provide nested decompositions of D × D, since only t × s ∈ L+ Iℓ×Iℓ ⇒ t × s ∈ L+ Iℓ+1×Iℓ+1 is guaranteed from the construction, see also Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='13 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, the sequence {V Hℓ}∞ ℓ=0 of H2-spaces from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='19 generated by the sequence of block-cluster trees is not nested, see also Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This holds also for the polynomials in the farfield, which depend on the depth of the specific block-cluster tree, see also Equation (8), which in turn DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 17 depends on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For clarification we write Ppw,ℓ t = Ppw t and Ppw,ℓ t×s = Ppw t×s for the polynomial spaces from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='19 whenever they are constructed from the cluster tree TIℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' As a last remark of this subsection, we use the introduced notation to localize the multilevel hierarchy in the finite element spaces by means of the nestedness of the cluster trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let {Vhℓ}∞ ℓ=0 and {TIℓ}∞ ℓ=0 be sequences of nested finite element spaces and nested cluster trees as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Jℓ : Vhℓ → Vhℓ+1 be the canonical prolongation operator between nested finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For t ∈ LIℓ+1 we write Jt for the matrix representation of Jℓ|t : Vhℓ|t → Vhℓ+1|t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Multilevel H2-sample covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' With a suitable (approximate) multilevel structure at hand, we now introduce a multilevel version of the H2-SCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To shorten notation we introduce the operator ΠH h,ℓ = ΠHℓΠmix hℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Given the above sequence of finite element spaces and H2-spaces and setting Πmix h−1g = 0, we define the H2-formatted multilevel sample covariance estimator (H2-MLSCE) recursively as E[ΠH h,Lg] ≈ EML L [ΠH h,Lg] = L � ℓ=0 ΠHLEℓ �� ΠH h,ℓ − ΠH h,ℓ−1 � g � (18) with the single level estimators Eℓ �� ΠH h,ℓ − ΠH h,ℓ−1 � g � = 1 Mℓ Mℓ � k=1 � ΠH h,ℓ − ΠH h,ℓ−1 �� z(k) ⊗ z(k)� , ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , L, given by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' samples z(k), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , Mℓ, Mℓ ∈ N, of Z ∈ L2 P(Ω, Hθ(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Z ∈ L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Hθ(D)), θ > 0, be a centered Gaussian random field with co- variance function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Th0 = {D(0) i }i∈I0 and let TI0 and Th0 satisfy Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let {Thℓ = {D(ℓ) i }i∈Iℓ}L ℓ=0 and {TIℓ}L ℓ=0 be sequences of decompositions with corresponding cluster trees as constructed in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2 and {Vhℓ}L ℓ=0 a nested sequence of piecewise polynomial ansatz spaces of order m ∈ N on {Thℓ}L ℓ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Define γ = min{θ, m} and choose α ∈ N such that ζ−2q ˜ρα < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is β0 ∈ N such that it holds ��g − EML L [ΠH h,Lg] �� L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='L2(D×D)) ≤ ClcCsp˜ρβ 1 − ζ−2q ˜ρα � h−2q H,L + (1 + 2−2q) L � ℓ=0 h−2q H,ℓ √Mℓ � + C⊗ L2 � hγ L + (1 + 2γ) L � ℓ=0 hγ ℓ √Mℓ � ∥Z∥2 L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hγ(D)) for all β ≥ β0 with ˜ρ as in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The estimate is proved in the usual way, using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='6, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', also [1], and using stability of the L2-projection on the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='8 hold, let ˜γ = min{−2q, γ} = min{−2q, θ, m}, and choose α ∈ N such that ζ−2q ˜ρα < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is β0 ∈ N and a constant 0 < CMLE = CMLE � ClcCsp˜ρβ0, ζ−2q ˜ρα, C⊗ L2, ChH, −2q, γ, ∥Z∥L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='Hγ(D)) � such that ��g − EML L [ΠH h,Lg] �� L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='L2(D×D)) ≤ CMLE � h˜γ L + L � ℓ=0 h˜γ ℓ √Mℓ � for all β ≥ β0 with ˜ρ as in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 25 25 25 25 25 25 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 49 49 49 49 49 49 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Illustration of the multilevel reduction algorithm for H2- approximation spaces on three different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The farfield is projected directly onto the finest level, whereas the nearfield is prolongated recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The specific construction of {Thℓ}L ℓ=0, {TIℓ}L ℓ=0, and {Vhℓ}L ℓ=0 using uniform refinement implies that C−1 hHhℓ ≤ hH,ℓ ≤ ChHhℓ for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ In analogy to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='16 we obtain the following bound on the bilinear form induced by the covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We recall that this also holds for bilinear forms of Nystr¨om type Equa- tion (15), if the corresponding assumptions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Under the assumptions of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9 there is β0 ∈ N such that ���� � D � D � g(x, y) − ΠHLEML L [Πmix hL g(x, y)] � uh(x)vh(y) dµ(x) dµ(y) ���� L2 P(Ω) ≤ CMLE � h˜γ L + L � ℓ=0 h˜γ ℓ √Mℓ � ∥uh∥L2(D)∥vh∥L2(D), for all β ≥ β0 with ˜ρ as in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Multilevel H2-sample covariance estimation: Algorithmic considerations In view of a computational implementation of the multilevel H2-MLSCE in Equation (18) we require an efficient way to combine the H2-approximations on different levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', an efficient implementation of the sum over the different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Reformulating this task, we seek an efficient implementation of the multilevel reduction ΠHL : � L× ℓ=0 W H ℓ � → W H L , � gHℓ�L ℓ=0 �→ ˜gHL = ΠHL L � ℓ=0 gHℓ, (19) with W H ℓ = � ΠHℓvhl : vhl ∈ Vhℓ ⊗ Vhℓ � , ℓ = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In the following, we will pursue a strategy which is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To that end, we exploit Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' that ΠHL can be represented as a sum of local L2-projections on t × s ∈ LIL×IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' It is clear that there is nothing to do if a target block-cluster of ΠHL is inadmissible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', if t × s ∈ L− IL×IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' If t × s is admissible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', if t × s ∈ L+ IL×IL, we observe that ΠHL t×s L � ℓ=0 gHℓ = L � ℓ=0 ΠHL t×sgHℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 19 Thus, we can compute ΠHL t×sgHℓ whenever t × s ∈ L+ IL×IL and t × s ∈ LIℓ×Iℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Otherwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', if t×s ∈ L− Iℓ×Iℓ and t×s /∈ LIL×IL, we split s×t into far- and nearfield according to the partitioning of TIℓ+1×Iℓ+1, project the resulting farfield blocks to level L and add the nearfield blocks to the nearfield of level ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Projecting admissible block-clusters to admissible block-clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To that end, we consider the case where t × s ∈ L+ Iℓ×Iℓ and t × s ∈ L+ IL×IL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', t × s is an admissible block-cluster in both block-cluster trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For these block-clusters, computing ΠHL|t×sgHℓ|t×s, gHℓ ∈ W H ℓ , amounts to the solution of Find gHL|t×s ∈ Ppw,L t×s s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (gHL|t×s, ppw,L t×s )L2(t×s) = (gHℓ|t×s, ppw,L t×s )L2(t×s) for all ppw,L t×s ∈ Ppw,L t×s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This is a finite dimensional variational problem which can be written as Qsu(L) s×tQ⊺ t = R(L,ℓ) s u(ℓ) s×t � R(L,ℓ) t �⊺ , (20) with Qr, r ∈ {s, t}, as in Equation (13), u(L) s×t and u(ℓ) s×t the coefficient matrices of gHL|t×s and gHℓ|t×s, and R(L,ℓ) r = �� ψ(r,L) i , ψ(r,ℓ) j � L2(r) � i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=',K(L) r , j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=',K(ℓ) r ∈ RK(L) r ×K(ℓ) r , for all ψ(r,L) i ∈ Ppw,L r and ψ(r,ℓ) i ∈ Ppw,ℓ r , r ∈ {s, t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Projecting inadmissible leaf block-clusters to admissible block-clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We con- sider the case t × s ∈ L− Iℓ×Iℓ and t × s ∈ L+ IL×IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Upon noting that it holds gHℓ|t×s ∈ Vhℓ|s ⊗ Vhℓ|t for all gHℓ ∈ W H ℓ we readily remark that Find gHL|t×s ∈ Ppw,L t×s s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (gHL|t×s, ppw,L t×s )L2(t×s) = (gHℓ|t×s, ppw,L t×s )L2(t×s) for all ppw,L t×s ∈ Ppw,L t×s , is a finite dimensional variational problem which can be rewritten as Qsu(L) s×tQ⊺ t = N(L,ℓ) s g(ℓ) s×t � N(L,ℓ) t �⊺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (21) As in the previous subsection, u(L) s×t and u(ℓ) s×t are the coefficient matrices of gHL|t×s and gHℓ|t×s, and N(L,ℓ) r = �� ψ(r,L) i , φ(r,ℓ) j � L2(r) � i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=',K(L) r , j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=',dim(Vhℓ|r) ∈ RK(L) r ×dim(Vhℓ|r), for all ψ(r,L) i ∈ Ppw,L r and φ(r,ℓ) i ∈ Vhℓ|r, r ∈ {s, t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Preliminary computational considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In view of an efficient solution of Equa- tion (20) and Equation (21), an efficient assembly of the matrices R(L,ℓ) t and N(L,ℓ) t is mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Before we state our algorithm for the multilevel reduction, we would like to make some preliminary remarks on how these matrices can be obtained efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7 and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1 hold and consider fam- ilies of finite element spaces and cluster trees as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Compute {Rt}t∈TL with (1) Rt = Qt for all t ∈ LIL, (2) Rt = � t′∈children(t) E⊺ t′,LRt′Ft′ for all t ∈ TIℓ \\ LIℓ, and {Nt}t∈TL with (1) Nt = Mt for all t ∈ LIL, (2) Nt = � t′∈children(t) E⊺ t′,LNt′J⊺ t′ for all t ∈ TIℓ \\ LIℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ Then {Rt}t∈TL can be computed in at most 2CH2(α + β)2δd|IL| operations and {Nt}t∈TL can be computed in at most 2CH2C2 minn2 min(α + β)2δd|IL| operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Estimating the effort for {Rt}t∈TL is complete analogy to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To estimate the one for {Nt}t∈TL, we note that the computational effort in each cluster t′ ∈ children(t) is bounded by Cminnmin � K(L) t K(L) t′ + CminnminK(L) t′ � ≤ 2C2 minn2 minK(L) t K(L) t′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The effort is then bounded in analogy to the one of {Rt}t∈TL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ The following lemma extends these considerations to the case when an multilevel hierarchy of H2-approximation spaces is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Given {Rt}t∈TL and {Nt}t∈TL as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1 and 0 ≤ ℓ ≤ L, compute {R(L,ℓ) t }t∈TIℓ by (1) R(L,ℓ) t = Rt for all t ∈ LIℓ, (2) R(L,ℓ) t = � t′∈children(t) E⊺ t′,LR(L,ℓ) t′ Et′,ℓ for all t ∈ TIℓ \\ LIℓ, and {N(L,ℓ) t }t∈TIℓ by (1) N(L,ℓ) t = Nt for all t ∈ LIℓ, (2) N(L,ℓ) t = � t′∈children(t) E⊺ t′,LN(L,ℓ) t′ Jt′ for all t ∈ TIℓ \\ LIℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then {R(L,ℓ) t }t∈TIℓ can be computed in at most 2CH2 (α(L − ℓ + 1) + β)3δd (α + β)δd |Iℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' operations and {N(L,ℓ) t }t∈TIℓ can be computed in at most 2CH2C2 minn2 min (α(L − ℓ + 1) + β)3δd (α + β)δd |Iℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We first note that TIℓ is a (Crc, α, β+(L−ℓ)α, δd, Cab)-bounded as well as a (Crc, α, β, δd, Cab)- regular cluster tree with Crc as in Equation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7 yields the assertion for {R(L,ℓ) t }t∈Tℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Modifying the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1 with similar arguments yields the assertion for {N(L,ℓ) t }t∈TIℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The multilevel H2-reduction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let Cab be the uniform constant satisfying Equation (6) for all elements of the family of cluster trees {TIℓ}L ℓ=0 constructed in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is a constant CML = CML(CH2, Cmin, Cab, Cuni, nmin, δ, d) such that the computational cost of Equation (19) are bounded by CML (α + β)⌈3δd⌉ (α + β)δd |IL|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', in linear complexity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' |IL|, if Equation (19) is computed as follows: (1) Set ˜gHL = gHL (2) Initialize {Qt}t∈TIL , {Rt}t∈TIL , and {Nt}t∈TIL as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1 (3) For ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , L − 1 proceed as follows: (a) Initialize {R(L,ℓ) t }t∈TIℓ and {N(L,ℓ) t }t∈TIℓ as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2 (b) Project all far- and nearfield blocks on level ℓ to level L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', set ˜gHL|t×s = ˜gHL|t×s + ΠHL|t×sgHℓ|t×s for all t × s ∈ LIℓ×Iℓ with t × s ∈ LIL×IL, by solving the local systems Equation (20) and Equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (c) For all t × s ∈ L− Iℓ×Iℓ, consider t × s as cluster in TIℓ+1×Iℓ+1 and DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 21 (i) set ˜gHL|t′×s′ = ˜gHL|t′×s′ + ΠHL|t′×s′gHℓ|t′×s′ for all t′ × s′ ∈ children(t × s) with t′ × s′ ∈ L+ Iℓ+1×Iℓ+1 by solving the local systems Equation (21), (ii) set ˜gHℓ+1|t′×s′ = ˜gHℓ+1|t′×s′ + gHℓ|t′×s′ for all t′×s′ ∈ children(t×s) with t′×s′ ∈ L− Iℓ+1×Iℓ+1 by dense matrix addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We first list the computational cost of every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Step 1: This step is without computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Step 2: The computational cost for assembling {Qt}t∈TIL are bounded in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10, the ones for {Rt}t∈TIL and {Nt}t∈TIL in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The total cost of this step are thus 6CH2C2 minn2 min(α + β)2δd|IL|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Step 3: We first list the computational cost for each substep for fixed ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Step 3a: The computational cost are bounded in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Summing up the cost for this step yields 4CH2C2 minn2 min (α(L − ℓ + 1) + β)3δd (α + β)δd |Iℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Step 3b: The computational cost for solving Equation (20) are given by � r∈{s,t} � 2 � K(L) r �3 + � K(L) r �� K(ℓ) r �2� ≤ 3 � r∈{s,t} � K(L) r �3 and arise for all t × s ∈ L+ Iℓ×Iℓ, while the efforts for Equation (21) are given by � r∈{s,t} � 2 � K(L) r �3 + � K(L) r � C2 minn2 min � ≤ 3C2 minn2 min � r∈{s,t} � K(L) r �3 and arise for all t × s ∈ L− Iℓ×Iℓ ∩ L+ IL×IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Step 3c: This substep is concerned with all t × s ∈ L− Iℓ×Iℓ \\ L+ IL×IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, a prolongation from Vhℓ|t ⊗ Vhℓ|s to Vhℓ+1|t ⊗ Vhℓ+1|s is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This can be accomplished in at most 2CuniC3 minn3 min operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Step 3(c)i: For all t′ × s′ ∈ children(t × s) ∩ L+ Iℓ+1×Iℓ+1 we need to solve Equation (21) on the level pair (L, ℓ + 1) instead of (L, ℓ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', Qs′u(L) s′×t′Q⊺ t′ = N(L,ℓ+1) s′ g(ℓ+1) s′×t′ � N(L,ℓ+1) t′ �⊺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The cost for a given t × s ∈ L− Iℓ×Iℓ \\ L+ IL×IL are thus bounded by � t′×s′∈children(t×s) � r∈{s′,t′} � 2 � K(L) r �3 + � K(L) r � C2 minn2 min � ≤ 3C2 minC2 abn2 min � r∈{s,t} � K(L) r �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Step 3(c)ii:: The computational cost for this step are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Steps 3b and 3c combined: Combining the preliminary considerations above and using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7, the combined total computational cost for fixed ℓ for Step 3b and 3c are bounded by 9CH2C2 minC2 abn2 min (α(L − ℓ + 1) + β)3δd (α + β)δd |Iℓ| + 2CuniC3 minn3 min|Iℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ Overall cost: Summing up the contributions of each step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' yields that the overall cost of the algorithm are bounded by L−1 � ℓ=0 � 19CH2C2 minC2 abn2 min (α(L − ℓ + 1) + β)3δd (α + β)δd + 2CuniC3 minn3 min � |Iℓ| ≤ |I0| L−1 � ℓ=0 � 19CH2C2 minC2 abn2 min (α(L − ℓ + 1) + β)3δd (α + β)δd + 2CuniC3 minn3 min � Cℓ uni ≤ |I0| � 19CH2C2 minC2 abn2 min L−1 � ℓ=0 (α(L − ℓ + 1) + β)3δd (α + β)δd Cℓ uni + 2CuniC3 minn3 min CL uni − 1 Cuni − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We note that L−1 � ℓ=0 (α(L − ℓ + 1) + β)3δd (α + β)δd Cℓ uni ≤ CL uni L � ℓ=0 ((α + β) + αℓ)3δd (α + β)δd C−ℓ uni ≤ CL uni (α + β)δd ∞ � ℓ=0 ((β + α) + αℓ)⌈3δd⌉C−ℓ uni where ∞ � ℓ=0 (β + αℓ)kqℓ ≤ � 1 + 1 1 − q � q 1 − q + 1 2 �k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' � (α + β)k for all q ∈ [0, 1) and k ∈ N0 due to [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='50 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The assertion follows with |I0|CL uni = |IL|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The implementation effort for the H2-MLSCE estimator is comparatively low and along the lines of the usual H2-algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In fact, given any H2-library, the H2-MLSCE estimator only requires the implementation of the three algorithms in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='11, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2, and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To that end, we remark that the initialization of {Qt}t∈TIL , {Rt}t∈TIL , {Nt}t∈TIL , {R(L,ℓ) t }t∈TIℓ , and {N(L,ℓ) t }t∈TIℓ can algorithmically all be treated by the same subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Computational work vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='20 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3 yields that the H2-MLSCE can be computed in O � �L ℓ=0 Mℓ|Iℓ| � operations, with δ entering only in the constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, it remains to choose the sample numbers such that accuracy of the finest level is achieved with minimal work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In complete analogy to various references, we mention [28, Appendix D] or [37] for example, we state the following theorem without proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let the assumptions of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='9 hold and choose ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The H2-MLSCE with L = d ˜γ ���� log(ε−1) log(Cuni) ���� and sample numbers Mℓ = M0C−2ℓ(1+˜γ/d)/3 uni , ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' , L, with M0 = � � � � � C2˜γL/d uni for 2˜γ > d, C2˜γL/d uni L2 for 2˜γ = d, C2(1+˜γ/d)L/3 uni for 2˜γ < d, achieves error estimates ��g − EML L [ΠH h,Lg] �� L2 P(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='L2(D×D)) = O(ε) DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 23 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Sample realizations of the centered Gaussian process with G3/2- asymptotically smooth covariance function taken for the numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' and sup uh,vh∈VhL ��� � D � D � g(x, y) − ΠHLEML L [Πmix hL g(x, y)] � uh(x)vh(y) dµ(x) dµ(y) ��� L2 P(Ω) ∥uh∥L2(D)∥vh∥L2(D) = O(ε) in a computational complexity of � � � � � O(ε−2) for 2˜γ > d, O � ε−2| log(ε−1)|3� for 2˜γ = d, O(ε−d/˜γ) for 2˜γ < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Thus, for 2˜γ > d, the overall error is dominated by the Monte Carlo error, whereas for 2˜γ < d the overall error is dominated by the error of the approximation spaces Vhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We note that these computational complexities are in line with the wavelet-based approach from [28], but the H2-approach does not require a hierarchical basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In contrast, wavelet-based approaches are theoretically also applicable if the smoothness of the kernel function is finite, which is, see also Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10, asymptotically not the case for the H2-approach due to the increasingly higher polynomial degrees required for interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Numerical experiments For our numerical experiments we aim at estimating the covariance of a Gaussian random field at the surface ∂D of a turbine geometry, see Figure 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', on a two-dimensional manifold embedded into R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The radius of the turbine to the end of the blades is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To that end, we prescribe a reference Gaussian random field in terms of a Karhunen-Lo´eve expansion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', Z(ω, x) = ∞ � k=0 � λkϕ(x)Yk(ω), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5e+00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5e+002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5e+00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5e+002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5e+00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5e+002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5e+00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5e+0024 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ L 0 1 2 3 4 5 6 dim Vh = dim Wh 60 240 960 3 840 15 360 61 440 245 760 dim(Wh ⊗ Wh) 3 600 57 600 921 600 ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7 · 106 ≈ 236 · 106 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='77 · 109 ≈ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4 · 109 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Dimensions of the used finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The estimated covari- ance matrices are matrices in Rdim Wh×dim Wh, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', have dim(Wh ⊗ Wh) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' L 0 1 2 3 4 5 6 M0 1 4 64 576 4 096 25 600 147 456 M1 1 16 144 1 024 6 400 36 864 M2 4 36 256 1 600 9 216 M3 9 64 400 2 304 M4 16 100 576 M5 25 144 M6 36 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Sample numbers chosen according to the case 2˜γ = d in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 for the numerical example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' with Yk ∼ U([−1, 1]) and {(λk, ϕk)}∞ k=0 the eigenpairs of the integral operator C : L2(∂D) → L2(∂D), (Cϕ)(x) = � ∂D gδ(x, y)ϕ(y) dσ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The covariance function gδ is chosen as a modified Mat´ern-9/2 kernel gδ(x, y) = ˜g(∥γδ(x) − γδ(y)∥), ˜g(r) = � 1 + 3r + 27r2 7 + 18r3 7 + 27r4 35 � e−3r, where γδ : ∂D → R3, γδ(x1, x2, x3), = � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1 + Υδ(2 ∗ x1 − 1)x1 x2 x3 � � and Υδ(t) = υδ(1 − t) υδ(1 − t) + υδ(t), υδ(t) = � 0, t ≤ 0, e−t 1 1−δ , t > 0, is a partition of Gevrey class δ ≥ 1 with Υ(t) = 1 for t < 0 and Υ(t) = 0 for t > 0, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For our numerical experiments we choose δ = 3/2, for which samples are illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This makes the covariance function gδ a G3/2-asymptotically smooth kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The H2-implementation of the numerical experiments is based on the C++-Library Bembel [17], with compression parameters α = 1, β = 2, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='8, and nmin = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We choose piecewise constant finite element spaces Vhℓ = Whℓ, ℓ = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', on uniformly refined quadrilateral meshes with Cuni = 4 and hℓ ∼ 2−ℓ, leading to dimensions of the finite element spaces and covariance matrices as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The Gaussian random field samples ΠhℓZ are generated from a Karhunen Lo´eve expansion which is truncated at 10−3hℓ and computed from a pivoted Cholesky decomposition [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' According to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 it holds ˜γ = 1 and we can expect a linear convergence rate for our H2-MLSCE, if the sample numbers are chosen proportional to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For our particular example we choose the sample numbers listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Figure 6 shows that we reach indeed the predicted rate convergence rate of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='8 and a computational work vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' accuracy as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The spectral error was computed with a power iteration up to an absolute accuracy of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The computation times are measured in wall clock time and have been carried out in parallel with 48 threads on a compute server with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3TB RAM and two Intel(R) Xeon(R) CPU E7-4850 v2 CPUs with Hyper-Threading enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 25 0 2 4 6 100 101 102 L absolute spectral error Convergence ˜γ = 1 100 101 102 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 105 106 ε wall clock time (sec) Computational work vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' accuracy ε−2| log(ε)|3 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Convergence plot of a realization of the H2-MLSCE and corresponding computational work vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' accuracy with the sample numbers as in Table 2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' also Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='10 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Conclusion In this article, we considered the multilevel estimation of covariance functions which are Gδ- asymptotically smooth, δ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This choice is motivated by the stochastic partial differential equation approach to Gaussian random fields and pseudodifferential operator theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The naive approach to estimate the covariance function from discretized samples using the single level covari- ance estimator is computationally prohibitive due to the density of the arising covariance matrices and the slow convergence of the sample covariance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' To overcome these issues, we first generalized the classical H2-approximation theory for asymptotically smooth kernels to Gevrey kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This allows to compress the arising covariance matrices by H2-matrices in linear com- plexity with respect to the underlying approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Secondly, we proposed and analyzed an H2-formatted multilevel covariance sample estimator (H2-MLCSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' This estimator exploits an approximate multilevel hierarchy in the H2-approximation spaces to estimate the covariance in the same complexity as the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The provided approximation theory is applicable to a rather general setting, covering for example domains, manifolds, graphs, and multi-screens as well as various approximation spaces such as finite element spaces and Nystr¨om discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Alternatively to the approach proposed in this paper, a wavelet based method for estimating covariance functions was proposed in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The advantage of such a wavelet method is that the wavelet-based approximation results also hold for finite smoothness of the covariance function, whereas the here presented H2-approach requires asymptotically infinite smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' In contrast, the advantage of the H2-approach in this paper is that no wavelet basis is required and that the presented algorithms can be integrated into the many readily available H2-matrix codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Acknowledgement The author would like to express his sincere gratitude to Christoph Schwab for the initial discussions on generalizing the H2-matrix approximation theory to Gevrey kernels and for critical and helpful comments during the writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Barth, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Schwab, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Zollinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Scattered Data Approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Cambridge University Press, first edition, December 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' [51] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Whittle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Stochastic processes in several dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Bulletin de l’Institut International de Statistique, 40:974–994, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Computation of H2-related constants Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='1 ([4, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a cluster tree and denote the number of interpo- lation points chosen in each cluster t ∈ TI by Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We say that {Kt}t∈TI is a rank distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We say that {Kt}t∈TI is a (Cbn, α, β, r, ξ)-bounded rank distribution, Cbn ≥ 1, α > 0, β ≥ 0, r ≥ 1, ξ ≥ 1, if ��� t ∈ TI : Kt > (α + β(ℓ − 1))r��� ≤ Cbnξ−ℓ|TI|, for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a cluster tree on the index set I satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then {Kt}t∈TI is a (1, α, β, δd, Cab)-bounded rank distribution if the number of interpolation points in (Kt)t∈TI are chosen according to Equation (8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=', Kt = � (β + α(p − level(t)))δ�d Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The proof is analogy to the example in [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let p denote the depth of TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We need to bound the number of clusters with Kt = � (β + α(p − level(t)))δ�d ≥ (β + α(p − level(t)))δd > (α + β(ℓ − 1))δd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' From this inequality we deduce that the clusters satisfying this constraint also satisfy level(t) < p + 1 − ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Due to Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12 the number of such clusters is bounded by from above by (Cp−ℓ+2 ab − 1)/(Cab − 1) and we obtain the assertion due to |TI| ≥ Cp+2 ab − 1 Cab − 1 = Cℓ ab Cp−ℓ+2 ab − C−ℓ ab Cab − 1 ≥ Cℓ ab Cp−ℓ+2 ab − 1 Cab − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ 28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' D ¨OLZ Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='3 ([4, Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='43 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a cluster tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' We say that it is (Crc, α, β, r, ξ)-bounded with Crc ≥ 1, α > 0, β ≥ 0, r ≥ 1, ξ > 1, if (22) ��� t ∈ LI : |t| > (β + α(ℓ − 1))r��� ≤ Crcξ−ℓ|TI|, for all ℓ ∈ N, and | children(t)| ≤ Crc, for all t ∈ TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (23) We say that TI is (Crc, α, β, r, ξ)-regular, if it is (Crc, α, β, r, ξ)-bounded and additionally satisfies | children(t)| ≥ 2, for all t ∈ TI \\ LI, (24) (α + β)r ≤ Crc|t|, for all t ∈ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (25) Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a cluster tree with depth p on the index set I satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then TI is (Crc, α, β, δd, Cab)-regular with (26) Crc = max � Cab, (α + β)δd nmin , C n1/(δd) min −β+α α +1 ab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Equation (6) implies 2 ≤ | children(t)| ≤ Cab, t ∈ TI \\ LI, which yields (24) and Equa- tion (23) holds with Crc ≥ Cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Inserting the upper bound from Equation (7) into Equation (25) yields (α + β)δd nmin ≤ Crc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Finally, the lower bound from Equation (7) implies that there are at most Cp+1 ab leafs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' The upper bound from Equation (7) and Equation (22) with ξ = Cab then imply that Crc must satisfy Crc ≥ � Cp+ℓ+1 ab |TI| for all ℓ with (β + α(ℓ − 1))δd < nmin 0 else Solving (β + α(ℓ − 1))δd < nmin for ℓ implies ℓ < (n1/(δd) min − β + α)/α which yields C n1/(δd) min −β+α α +1 ab ≤ Crc due to |TI| ≥ (Cp+2 ab − 1)/(Cab − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Combining all conditions on Crc yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 ([4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a (Crc, α, β, r, ξ)-bounded cluster tree and let {Kt}t∈TI be a (Cbn, α, β, r, ξ)-bounded rank distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Define kt = � max{Kt, |t|}, t ∈ LI, max{Kt, � t′∈children(t) Kt′}, t ∈ TI \\ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' (27) and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then there is a constant Ccb = Ccb(Crc, Cbn, r, ξ) ≥ 1 such that � t∈TI km t ≤ Ccb(α + β)rm|TI|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='6 ([4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be a (Crc, α, β, r, ξ)-regular cluster tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Then it holds |TI| ≤ 2Crc|I| (α + β)r Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='7 (Modification of [4, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be (Crc, α, β, r, ξ)-bounded and (Kt)t∈TI be a (Cbn, α, β, ξ)-bounded rank distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Let TI be (Crc, α′, β′, r, ξ)-regular and TI×I be a block-cluster tree with sparsity constant Csp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' For m ∈ N and {kt}t∈TI defined as in Equation (27) it holds � t∈TI km t ≤ CH2 (α + β)rm (α′ + β′)r |I| with CH2 = 2CrcCcb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' Combine Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='5 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content=' □ DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY 29 Institute for Numerical Simulation, University of Bonn, Friedrich-Hirzebruch-Allee 7, 53115 Bonn, Germany Email address: doelz@ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='uni-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9FLT4oBgHgl3EQfGi9H/content/2301.11992v1.pdf'}