diff --git "a/59E3T4oBgHgl3EQfpgot/content/tmp_files/load_file.txt" "b/59E3T4oBgHgl3EQfpgot/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/59E3T4oBgHgl3EQfpgot/content/tmp_files/load_file.txt" @@ -0,0 +1,515 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf,len=514 +page_content='Recursive Fermi-operator expansion strategies to accelerate subspace diagonalization for large eigenvalue problems in density functional theory Sameer Khadatkar1 and Phani Motamarri1 Indian Institute of Science, Bengaluru, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (*Electronic mail: phanim@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='in) Quantum mechanical calculations for material modeling using density functional theory (DFT) involves solving a large-scale nonlinear eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' These calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons in the material system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The efficient computational strategies used to solve these large nonlinear DFT eigenvalue problems rely on iterative orthogonal projection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The Rayleigh-Ritz projection step and the subspace diagonalization incur the dominant computational cost in these projec- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In this work, we explore scalable polynomial expansion based on recursive Fermi-operator expansion approaches using mixed-precision arithmetic as an alternative to subspace diagonalization of the projected Hamiltonian to reduce the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The performance and accuracy of these approaches have been thoroughly assessed by comparing them with the explicit diagonalization approach using the state-of-the-art ELPA library on both multinode CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' INTRODUCTION Eigenvalue problems are frequently encountered in many scientific disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For instance, the accurate and efficient computation of eigenvectors and eigenvalues is critical in the study of resonance, understanding the stability of fluid flows subjected to small perturbations, obtaining insights into vibra- tional modes of lattices, dimensionality reduction, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Another well-known and challenging application of eigenvalue problems is in the area of quantum modeling of materials using Kohn-Sham density functional theory (DFT)1, which has been immensely successful in providing critical in- sights into various ground-state material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To com- pute the ground-state electronic structure in DFT, one is con- fronted with solving a large-scale nonlinear eigenvalue prob- lem using a self-consistent field iteration procedure (SCF) for N smallest eigenvalue/eigenvector pairs, with N being pro- portional to the number of electrons in the material system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' This results in asymptotic cubic complexity O(N3) with the number of electrons for DFT, making these calculations com- putationally demanding and often restrictive in terms of sys- tem sizes that can be handled using widely used DFT codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Many of these codes employ plane-wave basis sets, which re- strict simulation domains to periodic or atomic-orbital type basis sets, which are not systematically convergent, and these basis sets are not amenable for massive parallelization on par- allel computing architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To extend the range of system sizes to be studied, numerous past efforts have focused on de- veloping systematically convergent real-space computational methodologies 2–6 that have focused on reducing the prefac- tor associated with the cubic computational complexity along- side improving the parallel scalability, thereby enabling large- scale DFT calculations up to 100,000 electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' These real- space DFT discretization approaches result in large sparse Hermitian eigenvalue problems of the form Hψψψi = εh i ψψψi to be solved for N smallest eigenvalue/eigenvector pairs, with N being proportional to M, the dimension of the sparse Hamil- tonian matrix H (M ≈ 105 −107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' We note that N depends on the number of electrons in the material system and is usually 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='5% of M, the degrees of freedom (DoFs) used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In the electronic structure community, the most popular eigensolver strategies employed to solve these large DFT eigenvalue problems include the Davidson approach, Lo- cally Optimal Block Pre-conditioned Conjugate Gradient (LOBPCG) method, or the Chebyshev filtered subspace it- eration (ChFSI) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' These eigensolvers belong to the category of iterative orthogonal projection methods (IOP) wherein the matrix H is orthogonally projected onto a care- fully constructed subspace rich in the wanted eigenvectors (Rayleigh-Ritz step), and subsequently, the resulting smaller dense projected Hamiltonian Hp is explicitly diagonalized (subspace diagonalization) to approximate the desired eigen- value/eigenvector pairs of the H matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The cubic scal- ing computational cost of this subspace diagonalization step dominates for medium to large-scale material systems (N > 20,000) in comparison to the costs associated with subspace construction and Rayleigh-Ritz steps in IOP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For in- stance, the authors6 employing the ChFSI approach have re- ported that the subspace diagonalization constitutes roughly 30% of the total ChFSI cost for N ≈ 30,000, whereas it ac- counts for around 56% of the total cost for N ≈ 50,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, the current work explores recursive polynomial ex- pansion approaches based on Fermi-operator expansion as an alternative to the subspace diagonalization procedure to im- prove the computational efficiency, thereby reducing the com- putational prefactor associated with the cubic complexity of the subspace diagonalization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Furthermore, the en- ergy efficiency and parallel scaling efficiency of these ap- proaches is examined on both multinode CPUs and multinode GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Recursive polynomial expansion approaches (RPE) rely on the key idea that constructing a density matrix (projec- tor matrix corresponding to N smallest eigenvectors) suffices to compute ground-state electronic structure in DFT at zero- temperature without the necessity of knowing explicit eigen- values and eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' These RPE approaches 7–12 have been explored in the past for conducting ground-state DFT calculations using atomic-orbital basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' However, the compu- tational efficiency, scaling and energy efficiency of these ap- proaches have not been explored in comparison to subspace arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='04642v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='comp-ph] 11 Jan 2023 2 diagonalization procedures for their use in iteration orthogo- nal projection methods on multinode CPU and GPU architec- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The evolving computing architectures in today’s ex- ascale era place a heavy emphasis on scalable methodolo- gies with a focus on reduced data movement and increased arithmetic intensity, with an equal emphasis on using energy- efficient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The current work assumes significance in this regard and is useful for solving large-scale eigenvalue problems arising from the discretization of DFT using sys- tematically convergent basis sets employing IOP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' the key contributions of our current work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' as de- scribed in the subsequent sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' include – (a) efficient im- plementation strategies of various recursive polynomial ex- pansion (RPE) techniques based on Fermi-operator expansion on both multinode CPU and GPU architectures for both zero- temperature case and the finite-temperature case of Fermi- dirac smearing of the occupancy function (b) design of mixed precision strategies in conjunction with RPE to reduce com- pute and data access costs (c) assessing accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' scaling effi- ciency and energy efficiency of the proposed implementation procedures by comparing it with explicit diagonalization al- gorithms provided by state-of-the-art ELPA library13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' RELATED WORK AND BACKGROUND This section discusses key ideas central to recursive poly- nomial expansion approaches which are used to approximate the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Density matrix At zero electronic temperature, the density matrix (D) can be defined as a projector matrix corresponding to the lowest occupied (Nocc ≤ N) eigenvectors of the Kohn-Sham Hamil- tonian H matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Mathematically, it takes the form of a shifted Heaviside step function, θ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' ), given by D = θ[µI−H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The density matrix (D) in the case of finite-temperature is a smeared version of zero-temperature density matrix and math- ematically represented by a Fermi-operator matrix function given by D = [eβ(H−µI) + I]−1, where, I denotes the identity matrix, β = 1/(kBTe) is the inverse electronic temperature, µ is the Fermi-energy, and H is the Hamiltonian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Note that the eigenvalues fi of D are referred to as occupancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' fi is either 0 or 1 for a zero-temperature case whereas for the case of a finite-temperature case, fi ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Recursive polynomial expansion techniques to approximate the density matrix Two types of polynomial expansion schemes can be used to approximate the density matrix – (a) Serial Fermi-operator expansion schemes (Chebyshev Fermi-operator expansion scheme14, Green’s function expansion scheme15, etc), (b) Recursive Fermi-operator expansion schemes 8–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In this work, we employ the latter approach i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=', the recursive Fermi- operator expansion schemes as they are shown to be more ef- ficient and can be used to approximate the density matrix for both zero-temperature, and finite-temperature cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Recursive Fermi-operator expansion for zero-temperature density matrix (Z-RFOE) The recursive Fermi-operator expansion8 involves succes- sive projections of a matrix Xn, where X0 = H and Xn+1 = F(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The functions F(Xn) are chosen to project the eigen- value spectrum of Xn to eigenvalues closer either to 1 or to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Mathematically this can be represented as D = θ(µI−H) ≈ Fm(Fm−1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='F0(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=')) (1) One of the most efficient techniques in Z-RFOE is to use the second-order projection polynomials (SP2) 9 given by Xn+1 = Fn(Xn) = Xn ± (Xn − X2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The SP2 here are continuously increasing and decreasing functions in the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The ± sign is chosen to adjust the trace of Xn+1 in each projection such that it converges to Nocc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Accelerated recursive polynomial expansion for zero-temperature density matrix (A-Z-RFOE) This technique works on the concept of shifting and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In Z-RFOE, we used SP2 polynomials, which either took the form F(X) = X2 or F(X) = 2X−X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In A-Z-RFOE, instead of restricting ourselves to these fixed expansion functions, we give it some freedom to choose the expansion functions such that it moves the eigenvalues closer to either 1 or 0 faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To optimize the convergence, we chose the polynomial such that each iteration gives the highest slope of projection around the eigenvalues, which are rescaled values of the HOMO (Highest Occupied Molecular Orbital) and LUMO (Lowest Unoccu- pied Molecular Orbital) eigenvalues and done such that there is no risk of eigenvalues switching the places between the oc- cupied and the unoccupied states10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Recursive Fermi-operator expansion scheme for finite-temperature cases (T-RFOE) Finite-temperature density matrix has occupancies fi ∈ [0,1] and the SP2 recursion scheme discussed above is not well suited for approximating density matrix with fractional occupancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, an intermediate function gener- ated in Z-RFOE that is obtained before the convergence of the algorithm to the projector matrix is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' This serves as a smeared function to zero-temperature density matrix (Heavi- side step function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, the truncated expansion for computing, the density matrix D can be given by the expres- sion in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Gm(H) = Fm(Fm−1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='F0(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=')) (2) Lower the electronic temperature Te higher will be the β value (refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' IIA), and more recursion steps m will be re- quired to approximate the density matrix11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Accuracy of the polynomial expansion procedures The accuracy of the aforementioned polynomial expansion procedures is given in terms of the degree npl of the polyno- mial needed to approximate the density matrix and is given by npl ∝ (εN −ε1) with ε1,εN being spectral bound estimates 3 of H16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' It is well known that DFT discretized Hamiltonian H using real-space approaches has large spectral width εN − ε1 resulting in higher npl for approximating the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Often this leads to an inefficient computational procedure to approximate the density matrix since the dimension of H can be of O(105 −107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' COMPUTATIONAL METHODOLOGY AND IMPLEMENTATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Proposed methodology Due to the aforementioned limitations of employing the re- cursive polynomial expansion procedures on the real-space discretized DFT Hamiltonian (H), we resort to iterative or- thogonal projection (IOP) methods of solving a large sparse eigenvalue problem and choose to work with the smaller dense projected Hamiltonian Hp in the subspace rich with eigenvec- tors of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, we employ the recursive polynomial expansion procedures on Hp to approximate the density ma- trix in the subspace as an alternative to explicit subspace diag- onalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Since the spectral width of Hp is commensurate with spectral width corresponding to occupied eigenstates, it is small and the proposed approach is computationally effi- cient as demonstrated subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Algorithmic details Using Hp, Z-RFOE and A-Z-RFOE schemes employing SP2 polynomials for approximating zero-temperature density matrix and the T-RFOE scheme for the finite-temperature den- sity matrix have been implemented in a distributed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Figure 1 shows the schematic of the RFOE algorithm imple- mented in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Furthermore, we also explored mixed-precision strategies in conjunction with the RFOE schemes implemented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, we rely on the fact that far away from RFOE convergence to the appropriate density matrix, the floating point operations involved in the initial RFOE itera- tions can be performed in single precision (FP32) and switch- ing to FP64 operations thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The criteria to decide the number of initial FP32 iterations is linked to relative trace change of Xn (εtr) of two successive RFOE iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' An es- timate of εtr could be obtained by examining the dependence of εtr on the relative change in occupied eigensubspace be- tween the starting matrix X0 = Hp and the intermediate ma- trices Xn generated during the course of RFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Our numerical studies on smaller size representative Hp arising in DFT show that εtr ≈ O(10−4) gives an acceptable error of O(10−7) with respect to fully double precision (FP64) computation of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Implementation details The multinode parallel implementation of RFOE codes was done in C++ employing Message Passing Interface (MPI) library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Software for Linear Algebra Targeting Exascale (SLATE) library17 was used for storing the parallel matrices encountered during the course of RFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' SLATE stores the matrix in a 2-D block-cyclic manner on both CPUs and GPUs within a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The tile size is the most basic parameter that can affect the SLATE routines’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Numerical ex- periments were conducted by varying the tile size to decide the optimal tile size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Some of the key aspects of the implementation are high- lighted below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Trace Calculations Traces of matrix squares are required during the course of RFOE iterations and are computed by evaluating the square of the Frobenius norm of the given symmetric matrix (Tr(A2) = ||A||2 F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, Frobenius norm function available in the SLATE library was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Further, the computations of matrix traces which was required only in the beginning and end of RFOE involved a traversal through the diagonal elements of the global matrix and the use of an MPI collective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Matrix-matrix multiplication The computationally dominant step in all the RFOE algo- rithms implemented is the matrix-matrix multiplication step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' We used the SLATE library functions to perform this step in parallel across multinode CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The perfor- mance of the Communication-optimal Matrix Multiplication (COSMA)18 and cuBLASMg (made available from CUDA Math Library Early Access Program19) library was also ex- plored to compute parallel matrix-matrix multiplications on CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Our studies indicates that COSMA was slower in terms of computational times compared to the SLATE library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And the cuBLASMg library is restricted to multi-GPUs within a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Metrics for accuracy benchmarking of RFOE For accuracy benchmarking of the RFOE methods imple- mented, we computed two errors: (a) Relative error between the exact and approximated density matrix (f(H)) using the Frobenius norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='e ε1 = (||D − Dref ||F)/||Dref ||F, (b) Rela- tive error between the trace of actual and the approximated f(H)H, ε2 = (tr(DH) − tr(Dref H))/tr(Dref H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Dref was computed by explicit diagonalization using ELPA library13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' RESULTS AND DISCUSSION In order to assess the accuracy and performance of the pro- posed methods, we employ synthetic matrices representative of the subspace projected Hamiltonians (Hp) arising in DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, the matrix Hp is constructed in such a way that the spectral width is smaller and remains con- stant with increase in matrix size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' We choose H p ij = H p ji = e−d∗|i−j| ∗ sin(i + 1), and the matrix sizes used were 8192 × 8192, 16384 × 16384, 32768 × 32768, and 65536 × 65536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The multinode CPU study was done on PARAM Pravega hav- ing Intel Xeon Cascade Lake 8268 CPU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='9 GHz) with 48 cores (96 threads) on each node, while multinode GPU study was done on a local lab cluster having 16x (8 on each node) NVIDIA Tesla V100 GPUs with 32 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 1: General implementation details flowchart for all the RFOE codes The performance metrics used for comparisons are: Node-hrs ⇒ Execution time (in hours) × the number of nodes taken in the best scaling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' It gives a measure of computational efficiency on CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' GPU-hrs ⇒ Execution time (in hours) × the number of GPUs taken in the best scaling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' It gives a measure of computational efficiency on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Minimum walltime ⇒ Least possible time for the job execution using as many resources as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' It is a measure of scaling efficiency of the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Energy consumption ⇒ Upper bound of the energy re- quired by the job in kWh to run it on the supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Indicative of the rupee cost required for the calculations on the supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For the energy consumption cal- culation we used the Thermal Design Power (TDP) rat- ings for both CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Multinode CPU comparisons Figure 2 shows that, all our RFOE implementations for zero-temperature case are better than ELPA in terms of node- hrs, which indicates that all of our implementations are com- putationally efficient compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For instance, the A- Z-RFOE results in a speedup of around 2x in comparison to ELPA for the 65536 size matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In the minimum walltime regime, we find that ELPA is slightly faster than the RFOE im- plementation for the matrix sizes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Figure 3 shows (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 2: (a) Node-hrs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot (Note: c stands for CPUs, n stands for nodes) for different implementations of RFOEs for zero-temperature case on multinode CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' that, both our T-RFOE and mixed-precision T-RFOE imple- mentations for finite-temperature case are better than ELPA in terms of node-hrs (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='2x speedup of mixed-precision T-RFOE implementation over ELPA for the 65536 size matrix), which indicates that both of our implementations are computation- ally efficient compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And, even in the minimum walltime, mixed precision T-RFOE was found to be slightly better than ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The number of CPUs on which we got min- imum walltime for different matrix sizes is also shown on the minimum walltime plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Multinode GPU comparisons Figure 4 shows that, all our implementations for zero- temperature case are better than ELPA up to 16384 size ma- trix, and beyond this size, the mixed-precision Z-RFOE and A-Z-RFOE are better than ELPA for GPU-hrs timings, which indicates that both of our implementations are computation- ally efficient compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Our A-Z-RFOE implemen- tation gave 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='5x speedup over ELPA for the 32768 size ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Due to the memory issue of ELPA, it does not work on multinode GPUs up to 16 GPUs for the 65536 size ma- trix, which indicates that the RFOE implementations use the memory efficiently compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And for minimum walltime, ELPA shows a better behaviour, suggesting that the RFOE implementations are not scaling well on multinode GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Figure 5 shows that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' our T-RFOE and mixed-precision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='T-RFOE implementations for finite-temperature case are bet- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Initializations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Nocc : Occupancy number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='So= Hp : Hamiltonian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='(Defined on Distributed System) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Xo = WoSo+bol (on Parallel architectures) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Initial scaling using spectral bound estimates of H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Ns = Tr[Xo] (Using MPl_Allreduce) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='While ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='(TrEr < Tolerance) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Final D Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Xn = WXn-12 + bXn-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Nx = IIXn-1ll-=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='FP32/FP64 GEMM for Xn-12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='( IIXn-1]l==[Tr[Xn-13]1/2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Xn and Xn-1 stored in 2-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='block-cyclic manner on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='CPUs/GPUs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Compute w = f(Ns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Nocc) and b = g(Ns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Nocc) where f() and g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=') depends on Update Ns using Nx and Ns the expansion scheme (Z-RFOE TrEr = abs(Ns - Nocc) A-Z-RFOE, T-RFOE)5 ELPA Double-PrecisionZ-RFOE 4 Mixed-Precision Z-RFOE node-hrs Double-Precision A-Z-RFOE m 1 0 8192 16384 32768 65536 Matrix Size350 ELPA (secs) 300 Double-Precision Z-RFOE 250 Mixed-PrecisionZ-RFOE (6144c128n) Double-Precision A-Z-RFOE walltime 200 150 100 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (6144c 128n) (6144c128n) （12288c256n) 50 (3072c64n) ←(12288c256n) 0 （3072c64n） (12288c256n) 8192 16384 32768 65536 Matrix Size5 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 3: (a) Node-hrs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot (Note: c stands for CPUs, n stands for nodes) for different implementations of RFOEs for finite-temperature case on multinode CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' ter than ELPA for GPU-hrs timings (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='5x speedup of mixed- precision T-RFOE implementation over ELPA for the 65536 size matrix), indicating that both implementations are com- putationally efficient compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And, for minimum walltime, our mixed-precision T-RFOE implementation is al- most similar to or better than ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The number of GPUs on which we got minimum walltime for different matrix sizes is shown on the minimum walltime plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Energy consumption comparisons Figure 6 shows that, in both the regimes of node-hrs/GPU- hrs and minimum walltime, we are better than ELPA in terms of energy consumption for zero-temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Figure 7 shows that, in both the regimes of node-hrs/GPU-hrs and min- imum walltime case, we are better than ELPA in terms of en- ergy consumption for finite-temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' This indicates that the rupee cost required for our calculations on the super- computer will be less than ELPA for both zero-temperature case and finite-temperature case of approximating the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Accuracy benchmarking The errors ε1 and ε2 (defined earlier) were of the O(10−10) and O(10−09) for double-precision implementation of Z- RFOE and A-Z-RFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And, were of the O(10−07) and O(10−09) for mixed-precision implementation of Z-RFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 4: (a) GPU-hrs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot for different implementations of RFOEs for zero-temperature case on multinode GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 5: (a) GPU-hrs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot for different implementations of RFOEs for finite-temperature case on multinode GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 5 ELPA Double-PrecisionT-RFOE 4 Mixed-Precision T-RFOE node-hrs m N 1 0 8192 16384 32768 65536 Matrix Size160 ELPA (12288c256n) (secs) 140 Double-Precision T-RFOE Mixed-PrecisionT-RFOE 120 (6144c128n) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime 100 80 60 (6144c128n) 40 (6144c128n) 20 (3072c 64n) (12288c256n) 0 (3072c64n) (12288c256n） 8192 16384 32768 65536 Matrix Size1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='75 ELPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='50 Double-Precision Z-RFOE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='25 Mixed-PrecisionZ-RFOE GPU-hrs Double-Precision A-Z-RFOE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='00 8192 16384 32768 65536 Matrix size700 ELPA (SDos) Double-Precision Z-RFOE 600 Mixed-Precision Z-RFOE (8 GPUs) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime 500 Double-Precision A-Z-RFOE 400 300 200 (8 GPUs) 100 (4 GPUs) (6 GPUs) (16 GPUS) 0 （6GPUS) (16GPUs) 8192 16384 32768 65536 Matrix size0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='8 ELPA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='7 Double-Precision T-RFOE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='6 Mixed-PrecisionT-RFOE GPU-hrs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='0 8192 16384 32768 65536 Matrix Size350 ELPA (secs) 300 Double-PrecisionT-RFOE Mixed-Precision T-RFOE 250 (8 GPUs) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime 200 150 (8 GPUs) 100 (6 GPUs) 50 (4 GPUS) (16GPUS) 0 (6GPUs) (16-GPUs) 8192 16384 32768 65536 Matrix size6 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 6: Energy consumption (kWh) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot in terms of (a) Node-hrs/GPU-hrs, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime for the best implementation of RFOE for zero-temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 7: Energy consumption (kWh) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot in terms of (a) Node-hrs/GPU-hrs, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime for the best implementation of RFOE for finite-temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For T-RFOE, the error ε1 was of the O(10−03) and ε2 was of O(10−06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The density matrix approximated by T-RFOE has an higher error compared to the Fermi-Dirac based den- sity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' However, in DFT, the computation of material properties relies on differences in energies and hence, the T- RFOE approach can be viewed as an alternative approach to smearing the zero-temperature density matrix, which can be practically helpful in approximating finite-temperature den- sity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' CONCLUSIONS RFOE schemes, as expected, had a lesser computational prefactor which made them computationally efficient com- pared to ELPA in the node-hrs/GPU-hrs regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In the case of minimum walltimings, ELPA timings were better as it scaled better than our RFOE implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Energy efficiency- wise, the RFOE implementations were better on both multin- ode CPUs and GPUs, which is directly proportional to the cost required for the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In terms of memory utiliza- tion, multinode GPU implementations of RFOE were better than ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 40cuBLAS Library, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 41cuSOLVER Library, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'}