scaomath/pde-triebel-lizorkin

There are two datasets generated by solving $p\in H^1_0$ either in

$$-\mathrm{\nabla }\cdot (a\mathrm{\nabla }p)=f-\; \backslash nabla\; \backslash cdot(a\backslash nabla\; p)\; =\; f$$

or

$$-\mathrm{\nabla }\cdot (a\mathrm{\nabla }p)+cu=f-\; \backslash nabla\; \backslash cdot(a\; \backslash nabla\; p)\; +\; c\; u=\; f$$

where $a=e^\varphi$ and $\varphi$ is generated by a Gaussian Random Field. The mean is 0 and the covariance operator (between two locations) is $C = (-\Delta + \tau^2)^{-\alpha}$ where the Laplacian has Neumann BC, basically invert a fractional Laplacian operator. $\alpha$ controls the smoothness of the $\varphi$, $\alpha \to 1$ is basically random noise, the closer $\alpha$ is to 1, the less smooth the isocurve is. $\tau$ controls the high frequency modes of the problem the bigger $\tau$ is, the more oscillatory the $\varphi$ is.