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Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs
(2022)SAM Research ReportWe establish summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions for countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments ...Report -
ReLU Neural Network Galerkin BEM
(2022)SAM Research ReportWe introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation based approach, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built ...Report -
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Multilevel Domain Uncertainty Quantification in Computational Electromagnetics
(2022)SAM Research ReportWe continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), {\bf 8}:301--341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping the physical domains to a nominal polygonal domain with piecewise smooth maps. The regularity of the pullback ...Report -
Multiresolution Kernel Matrix Algebra
(2022)SAM Research ReportWe propose a sparse arithmetic for kernel matrices, enabling efficient scattered data analysis. The compression of kernel matrices by means of samplets yields sparse matrices such that assembly, addition, and multiplication of these matrices can be performed with essentially linear cost. Since the inverse of a kernel matrix is compressible, too, we have also fast access to the inverse kernel matrix by employing exact sparse selected ...Report -
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Neural and gpc operator surrogates: construction and expression rate bounds
(2022)SAM Research ReportApproximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for Deep Neural Operator and Generalized Polynomial Chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs ...Report -
Multilevel Monte Carlo FEM for Elliptic PDEs with Besov Random Tree Priors
(2022)SAM Research ReportReport -
Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D
(2022)SAM Research ReportWe prove weighted analytic regularity for the solution of the integral frac tional Poisson problem on bounded intervals with analytic right-hand side. Based on this regularity result, we prove exponential convergence of the hp FEM on geometric boundary-refined meshes.Report -
Exponential Convergence of hp-Time-Stepping in Space-Time Discretizations of Parabolic PDEs
(2022)SAM Research ReportFor linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without ...Report