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Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
(2011)Research ReportsThe numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [11, 12] that under very weak assumptions on the diffusion ...Report -
Expression Rates of Neural Operators for Linear Elliptic PDEs in Polytopes
(2024)SAM Research ReportWe study the approximation rates of a class of deep neural network approximations of operators, which arise as data-to-solution maps G † of linear elliptic partial differential equations (PDEs), and act between pairs X, Y of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators G with the structure G = R ◦ A ◦ E, with linear encoders E and decoders R. The constructive proofs are via a ...Report -
Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering
(2024)SAM Research ReportWe consider frequency-domain acoustic scattering at a homogeneous star-shaped penetrable ob stacle, whose shape is uncertain and modelled via a radial spectral parameterization with random coefficients. Using recent results on the stability of Helmholtz transmission problems with piece wise constant coefficients from [A. Moiola and E. A. Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, ...Report -
Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction
(2021)SAM Research ReportCentered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded domains in Euclidean space or smooth, compact and orientable manifolds are determined by their covariance operators. We consider centered GRFs given sample-wise as variational solutions to coloring operator equations driven by spatial white noise, with pseudodifferential coloring operator being elliptic, self-adjoint and positive from the Hörmander class. ...Report -
hp FEM for Reaction-Diffusion Equations. II: Regularity Theory
(1997)SAM Research ReportA singularly perturbed reaction-diffusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve, the boundary data are analytic, and the right hand side is analytic. We give asymptotic expansions of the solution and new error bounds that are uniform in the perturbation parameter as well as in the expansion order. Additionally, we provide growth estimates ...Report -
Mixed hp-DGFEM for incompressible flows III: Pressure stabilization
(2002)SAM Research ReportWe consider stabilized mixed hp-discontinuous Galerkin methods for the discretization of the Stokes problem in three-dimensional polyhedral domains. The methods are stabilized with a term penalizing the pressure jumps. For this approach it is shown that IQk-IQk and IQk-IQk-1 elements satisfy a generalized inf-sup condition on geometric edge and boundary layer meshes that are refined anisotropically and non quasi-uniformly towards faces, ...Report -
Mixed hp-DGFEM for incompressible flows II: Geometric edge meshes
(2002)SAM Research ReportWe consider the Stokes problem in three-dimensional polyhedral domains discretized on hexahedral meshes with hp-discontinuous Galerkin finite elements of type IQk for the velocity and IQk-1 for the pressure. We prove that these elements are inf-sup stable on geometric edge meshes that are refined anisotropically and non quasi-uniformly towards edges and corners. The discrete inf-sup constant is shown to be independent of the aspect ratio ...Report -
Mixed hp-FEM on anisotropic meshes II: Hanging nodes and tensor products of boundary layer meshes
(1997)SAM Research ReportDivergence stability of mixed hp-FEM for incompressible fluid flow for a general class of possibly highly irregular meshes is shown. The meshes may be refined anisotropically and contain hanging nodes on geometric patches. The inf-sup constant is independent of the aspect ratio of the elements and the dependence on the polynomial degree is given explicitly. Numerical estimates of inf-sup constants confirm our results.Report -
Mixed hp-FEM on anisotropic meshes
(1997)SAM Research ReportMixed hp-FEM for incompressible fluid flow on anisotropic meshes are analyzed. A discrete inf-sup condition is proved with a constant independent of the meshwidth and the aspect ratio. For each polynomial degree $k\geq 2$, velocity-pressure subspace pairs are presented which are stable on quadrilateral mesh-patches, independently of the element aspect ratio implying in particular divergence stability on the so-called Shishkin-meshes. ...Report -
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