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Homogenization and Numerical Upscaling for Spectral Fractional Diffusion
(2024)SAM Research ReportWe consider two-scale, linear spectral fractional diffusion of order \(2s\in (0,2)\) with homogeneous Dirichlet boundary condition and locally periodic, two-scale coefficients in a bounded domain \(D \subset \mathbb{R}^d\), with fundamental period \(Y=(0,1)^d \subset \mathbb{R}^d\). We derive a local limiting two-scale homogenized equation for the so-called Caffarelli-Sylvestre (CS) extension in the tensorized domain \(D\times Y \times ...Report -
Exponential expressivity of ${\rm ReLU}^k$ neural networks on Gevrey classes with point singularities
(2024)SAM Research ReportWe analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains \(\mathrm{D} \subset \mathbb{R}^d\), \(d=2,3\). We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in \(\mathrm{D}\), comprising the countably-normed ...Report -
Exponential convergence of $hp$ quadrature for integral operators with Gevrey kernels
(2009)SAM Research ReportGalerkin discretizations of integral equations in $\mathbb{R}^d$ require the evaluation of integrals $ I = \int_{S^{{(1)}}}$ $ \int_{S^{{(2)}}}$ $g(x,y)dydx$ where $S^{{(1)}}$, $S^{{(2)}}$ are $d$-simplices and $g$ has a singularity at $x=y$. We assume that g is Gevrey smooth for $x \neq y$ and satisfies bounds for the derivatives which allow algebraic singularities at $x = y$. This holds for kernel functions commonly occuring in integral ...Report -
On Kolmogorov equations for anisotropic multivariate Lévy processes
(2008)Research reportsFor d-dimensional exponential L´evy models, variational formulations of the Kolmogorov equations arising in asset pricing are derived. Well-posedness of these equations is verified. Particular attention is paid to pure jump, d-variate L´evy processes built from parametric, copula dependence models in their jump structure. The domains of the associated Dirichlet forms are shown to be certain anisotropic Sobolev spaces. Representations of ...Report -
Sparse p-version BEM for first kind boundary integral equations with random loading
(2008)Research ReportReport -
Sparse high order FEM for elliptic sPDEs
(2008)Research ReportWe describe the analysis and the implementation of two Finite Element (FE) algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients. They are based on separation of deterministic and stochastic parts of the input data by a Karhunen-Lo`eve expansion, truncated after M terms. With a change of measure we convert the problem to a sequence of M-dimensional, parametric ...Report