Adaptive wavelet methods for elliptic partial differential equations with random operators

We apply adaptive wavelet methods to boundary value problems with random coefficients, discretized by wavelets in the spatial domain and tensorized polynomials in the parameter domain. Greedy algorithms control the approximate application of the fully discretized random operator, and the construction of sparse approximations to this operator. We suggest a power iteration for estimating errors induced by sparse approximations of linear operators.

[1]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[2]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

[3]  Wolfgang Dahmen,et al.  Adaptive eigenvalue computation: complexity estimates , 2007, Numerische Mathematik.

[4]  R. DeVore,et al.  Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs , 2010 .

[5]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[6]  Albert Cohen,et al.  Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs , 2010, Found. Comput. Math..

[7]  Claude Jeffrey Gittelson,et al.  Adaptive Galerkin methods for parametric and stochastic operator equations , 2011 .

[8]  Rob P. Stevenson,et al.  Adaptive Solution of Operator Equations Using Wavelet Frames , 2003, SIAM J. Numer. Anal..

[9]  K. Ritter,et al.  Adaptive Wavelet Methods for Elliptic Stochastic Partial Differential Equations , 2022 .

[10]  Christoph Schwab,et al.  Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients , 2007 .

[11]  Rob Stevenson,et al.  Finite element wavelets with improved quantitative properties , 2009 .

[12]  D. Xiu Fast numerical methods for stochastic computations: A review , 2009 .

[13]  Rob Stevenson,et al.  On the Compressibility of Operators in Wavelet Coordinates , 2004, SIAM J. Math. Anal..

[14]  Arend Aalberthus Roeland Metselaar Handling Wavelet Expansions in numerical Methods , 2002 .

[15]  G. Karniadakis,et al.  Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures , 2006, SIAM J. Sci. Comput..

[16]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[17]  Rob Stevenson,et al.  Adaptive wavelet methods for solving operator equations: An overview , 2009 .

[18]  K. Mattbes Bauer, K.: Wahrscheinlichkeitstheorie und Grundzüge der Maßtheorie. De Gruyter Lehrbuch. Walter de Gruyter & Co., Berlin 1968. 342 S., Preis DM 32.— , 1971 .

[19]  Massimo Fornasier,et al.  Adaptive frame methods for elliptic operator equations , 2007, Adv. Comput. Math..

[20]  Claude Jeffrey Gittelson,et al.  An adaptive stochastic Galerkin method , 2011 .

[21]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

[22]  A. Cohen Numerical Analysis of Wavelet Methods , 2003 .

[23]  R. DeVore,et al.  ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S , 2011 .

[24]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

[25]  Wolfgang Dahmen,et al.  Adaptive application of operators in standard representation , 2006, Adv. Comput. Math..

[26]  Rob P. Stevenson,et al.  An optimal adaptive wavelet method without coarsening of the iterands , 2006, Math. Comput..

[27]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods II—Beyond the Elliptic Case , 2002, Found. Comput. Math..

[28]  Claude Jeffrey Gittelson,et al.  Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* , 2011, Acta Numerica.

[29]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[30]  Claude Jeffrey Gittelson,et al.  An adaptive stochastic Galerkin method for random elliptic operators , 2013, Math. Comput..

[31]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[32]  Rob Stevenson,et al.  An Adaptive Wavelet Method for Solving High-Dimensional Elliptic PDEs , 2009 .

[33]  Christoph Schwab,et al.  Sparse Tensor Discretization of Elliptic sPDEs , 2009, SIAM J. Sci. Comput..

[34]  G. Karniadakis,et al.  An adaptive multi-element generalized polynomial chaos method for stochastic differential equations , 2005 .

[35]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[36]  Claude Jeffrey Gittelson,et al.  Stochastic Galerkin approximation of operator equations with infinite dimensional noise , 2011 .

[37]  M. Fornasier,et al.  Nonlinear and adaptive frame approximation schemes for elliptic PDEs: Theory and numerical experiments , 2009 .

[38]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[39]  M. Fornasier,et al.  Adaptive Frame Methods for Elliptic Operator Equations: The Steepest Descent Approach , 2007 .

[40]  C. Schwab,et al.  Sparse high order FEM for elliptic sPDEs , 2009 .

[41]  Claude Jeffrey Gittelson,et al.  Uniformly convergent adaptive methods for a class of parametric operator equations , 2012 .

[42]  Rob P. Stevenson,et al.  Space-time adaptive wavelet methods for parabolic evolution problems , 2009, Math. Comput..