Adaptive wavelet methods for the stochastic Poisson equation
type: Item
instance of: scholarly article
title: Adaptive wavelet methods for the stochastic Poisson equation
zbMATH Open document ID: 1260.65003
author: Petru A. Cioica
author: Nicolas Döhring
author: Stefan Kinzel
author: Felix Lindner
author: Thorsten Raasch
author: Klaus Ritter
author: Stephan Dahlke
author: R. L. Schilling
published in: BIT
publication date: 2012-11-21T00:00:00Z
review text: This paper proposes numerical algorithms for the Poisson equation \(-\Delta U=X\) in a bounded Lipschitz domain \(D\subset \mathbb{R}^d\) with Dirichlet's boundary condition \(U=0\) on \(\partial D\) and \(X\) a random function with values in \(L^2(D)\). The field \(X\) is defined in terms of a stochastic wavelet expansion and its smoothness along Sobolev and Besov spaces is controlled by two parameters \(\alpha\) and \(\beta\), where \(\beta\) is a sparsity parameter. Efficient algorithms for the nonlinear approximation of the random functions \(X\) and \(U\) are constructed. Suitable adaptive wavelet algorithms achieve the nonlinear approximation of \(U\) at a computational cost that is proportional to the degrees of freedom. Numerical experiments are presented to complement the asymptotic error analysis.
reviewed by: Dominique Lépingle
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 65T60
zbMATH DE Number: 6109256
zbMATH Keywords: elliptic stochastic partial differential equation
zbMATH Keywords: Besov regularity
zbMATH Keywords: approximation rates
zbMATH Keywords: nonlinear approximation
zbMATH Keywords: adaptive methods
zbMATH Keywords: Poisson equation
zbMATH Keywords: stochastic wavelet expansion
zbMATH Keywords: algorithms
zbMATH Keywords: numerical experiments
describes a project that uses: Wavelet and Multiscale Library
MaRDI profile type: Publication
full work available at URL: s10543-011-0368-7
OpenAlex ID: W2133112166
cites work: Wavelet Thresholding via A Bayesian Approach
cites work: Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations
cites work: Q3580928
cites work: The wavelet element method. I: Construction and analysis
cites work: Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains
cites work: Q4434436
cites work: Nonlinear Approximation of Random Functions
cites work: Biorthogonal bases of compactly supported wavelets
cites work: Adaptive wavelet methods for elliptic operator equations: Convergence rates
cites work: Adaptive wavelet methods. II: Beyond the elliptic case
cites work: On the importance of combining wavelet-based nonlinear approximation with coding strategies
cites work: Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic SPDEs
cites work: Free-knot spline approximation of stochastic processes
cites work: Besov regularity for elliptic boundary value problems in polygonal domains
cites work: Besov regularity for elliptic boundary value problems
cites work: Optimal approximation of elliptic problems by linear and nonlinear mappings. I
cites work: Optimal approximation of elliptic problems by linear and nonlinear mappings. II
cites work: Adaptive frame methods for elliptic operator equations
cites work: Optimal approximation of elliptic problems by linear and nonlinear mappings. IV: Errors in \(L_{2}\) and other norms
cites work: Wavelets with complementary boundary conditions -- functions spaces on the cube
cites work: Composite wavelet bases for operator equations
cites work: Wavelets on Manifolds I: Construction and Domain Decomposition
cites work: Q4215356
cites work: Compression of Wavelet Decompositions
cites work: On the convergence of generalized polynomial chaos expansions
cites work: Q3470840
cites work: Q4023365
cites work: Q4023463
cites work: The inhomogeneous Dirichlet problem in Lipschitz domains
cites work: Information-based nonlinear approximation: an average case setting
cites work: Q3135829
cites work: An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
cites work: On the average complexity of multivariate problems
cites work: New stable biorthogonal spline-wavelets on the interval
cites work: Average-case analysis of numerical problems
cites work: Q4717960
cites work: A Milstein-based free knot spline approximation for stochastic differential equations
cites work: Adaptive Solution of Operator Equations Using Wavelet Frames
cites work: A multiplicative Schwarz adaptive wavelet method for elliptic boundary value problems
cites work: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients
cites work: On besov - hardy - sobolev spaces in domains and regular elliptic boundary value problems. the case
cites work: Wavelet Analysis of Refinement Equations
cites work: Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
cites work: Integration and approximation of multivariate functions: Average case complexity with isotropic Wiener measure
cites work: Average case complexity of linear multivariate problems. I: Theory

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cites work: The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations

Adaptive wavelet methods for the stochastic Poisson equation (DOI)
DOI: 10.1007/S10543-011-0368-7