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Analytic regularity for the Navier-Stokes equations in polygons with mixed boundary conditions
(2021)SAM Research ReportWe prove weighted analytic regularity of Leray-Hopf variational solutions for the stationary, incompressible Navier-Stokes Equations (NSE) in plane polygonal domains, subject to analytic body forces. We admit mixed boundary conditions which may change type at each vertex, under the assumption that homogeneous Dirichlet (''no-slip'') boundary conditions are prescribed on at least one side at each vertex of the domain. The weighted analytic ...Report -
hp-DG-QTT solution of high-dimensional degenerate diffusion equations
(2012)SAM Research ReportWe consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hp-discontinuous Galerkin method is shown to converge exponentially. The ...Report -
Weighted analytic regularity for the integral fractional Laplacian in polyhedra
(2023)SAM Research ReportOn polytopal domains in 3D, we prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian with analytic right-hand side. Employing the Caffarelli-Silvestre extension allows to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability ...Report -
Deep Operator Network Approximation Rates for Lipschitz Operators
(2023)SAM Research ReportWe establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or Hölder) continuous maps \(\mathcal G:\mathcal X\to\mathcal Y\) between (subsets of) separable Hilbert spaces \(\mathcal X\), \(\mathcal Y\). The DON architecture considered uses linear encoders \(\mathcal E\) and decoders \(\mathcal D\) via (biorthogonal) Riesz bases of \(\mathcal X\), \(\mathcal Y\), and an ...Report -
Electromagnetic Wave Scattering by Random Surfaces: Uncertainty Quantification via Sparse Tensor Boundary Elements
(2015)SAM Research ReportFor time-harmonic scattering of electromagnetic waves from obstacles with uncertain geometry, we perform a domain perturbation analysis. Assuming as known both the scatterers’ nominal geometry and its small-amplitude random perturbations statistics, we derive a tensorized boundary integral equation which describes, to leading order, the second order statistics, i.e. the two-point correlation of the randomly scattered electromagnetic fields. ...Report -
Regularity and generalized polynomial chaos approximation of parametric and random 2nd order hyperbolic partial differential equations
(2010)SAM Research ReportInitial boundary value problems of linear second order hyperbolic partial differential equations whose coefficients depend on countably many random parameters are reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This parametric family is approximated by Galerkin projection onto finitely supported polynomial systems in the parameter space. We establish uniform ...Report -
Sparse-Grid, Reduced-Basis Bayesian Inversion: Nonaffine-Parametric Nonlinear Equations
(2015)SAM Research ReportWe extend the reduced basis accelerated Bayesian inversion methods for affine-parametric, linear operator equations which are considered in [15, 16] to non-affine, nonlinear parametric operator equations. We generalize the analysis of sparsity of parametric forward solution maps in [18] and of Bayesian inversion in [41, 42] to the fully discrete setting, including Petrov-Galerkin high-fidelity (“HiFi”) discretization of the forward maps. ...Report -
Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions
(2015)SAM Research ReportWe analyze the approximation of the solutions of second-order elliptic problems, which have point singularities but belong to a countably normed space of analytic functions, with a first-order, $h$-version finite element (FE) method based on uniform tensor-product meshes. The FE solutions are well known to converge with algebraic rate at most 1/2 in terms of the number of degrees of freedom, and even slower in the presence of singularities. ...Report -
Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems
(2015)SAM Research ReportIn this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty quantification problems in subsurface flow. We extend the convergence analysis in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite element discretizations and give a constructive proof of ...Report -
Analysis of multilevel MCMC-FEM for Bayesian inversion of log-normal diffusions
(2019)SAM Research ReportWe develop the Multilevel Markov Chain Monte Carlo Finite Element Method (MLMCMC-FEM for short) to sample from the posterior density of the Bayesian inverse problems. The unknown is the diffusion coefficient of a linear, second order divergence form elliptic equation in a bounded, polytopal subdomain of $ \mathbb{R}^d$ . We provide a convergence analysis with absolute mean convergence rate estimates for the proposed modified MLMCMC method, ...Report