Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation

For initial boundary value problems of linear parabolic partial differential equations with random coefficients, we show analyticity of the solution with respect to the parameters and give an a priori error analysis for N-term generalized polynomial chaos approximations in a scale of Bochner spaces...

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Main Authors: Hoang, Viet Ha., Schwab, Christoph.
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2014
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Online Access:https://hdl.handle.net/10356/100698
http://hdl.handle.net/10220/18460
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-1006982023-02-28T19:23:03Z Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation Hoang, Viet Ha. Schwab, Christoph. School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Analytic mechanics For initial boundary value problems of linear parabolic partial differential equations with random coefficients, we show analyticity of the solution with respect to the parameters and give an a priori error analysis for N-term generalized polynomial chaos approximations in a scale of Bochner spaces. The problem is reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space by Galerkin projection onto finitely supported polynomial systems in the parameter space. Uniform stability with respect to the support of the resulting coupled parabolic systems is established. Analyticity of the solution with respect to the countably many parameters is established, and a regularity result of the parametric solution is proved for both compatible as well as incompatible initial data and source terms. The present results imply convergence rates and stability of sparse, adaptive space-time tensor product Galerkin discretizations of these infinite dimensional, parametric problems in the parameter space recently proposed in [C. Schwab and C. J. Gittelson, Acta Numer., 20 (2011), pp. 291–467; C. J. Gittelson, Adaptive Galerkin Methods for Parametric and Stochastic Operator Equations, Ph.D. thesis, ETH Z¨urich, 2011]. Published version 2014-01-13T06:18:37Z 2019-12-06T20:26:51Z 2014-01-13T06:18:37Z 2019-12-06T20:26:51Z 2013 2013 Journal Article Hoang, V. H., & Schwab, C. (2013). Sparse Tensor Galerkin Discretization of Parametric and Random Parabolic PDEs---Analytic Regularity and Generalized Polynomial Chaos Approximation. SIAM Journal on Mathematical Analysis, 45(5), 3050-3083. https://hdl.handle.net/10356/100698 http://hdl.handle.net/10220/18460 10.1137/100793682 en SIAM journal on mathematical analysis © 2013 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Mathematical Analysis and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: [http://dx.doi.org/10.1137/100793682]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. 34 p. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic DRNTU::Science::Mathematics::Analytic mechanics
spellingShingle DRNTU::Science::Mathematics::Analytic mechanics
Hoang, Viet Ha.
Schwab, Christoph.
Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation
description For initial boundary value problems of linear parabolic partial differential equations with random coefficients, we show analyticity of the solution with respect to the parameters and give an a priori error analysis for N-term generalized polynomial chaos approximations in a scale of Bochner spaces. The problem is reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space by Galerkin projection onto finitely supported polynomial systems in the parameter space. Uniform stability with respect to the support of the resulting coupled parabolic systems is established. Analyticity of the solution with respect to the countably many parameters is established, and a regularity result of the parametric solution is proved for both compatible as well as incompatible initial data and source terms. The present results imply convergence rates and stability of sparse, adaptive space-time tensor product Galerkin discretizations of these infinite dimensional, parametric problems in the parameter space recently proposed in [C. Schwab and C. J. Gittelson, Acta Numer., 20 (2011), pp. 291–467; C. J. Gittelson, Adaptive Galerkin Methods for Parametric and Stochastic Operator Equations, Ph.D. thesis, ETH Z¨urich, 2011].
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Hoang, Viet Ha.
Schwab, Christoph.
format Article
author Hoang, Viet Ha.
Schwab, Christoph.
author_sort Hoang, Viet Ha.
title Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation
title_short Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation
title_full Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation
title_fullStr Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation
title_full_unstemmed Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation
title_sort sparse tensor galerkin discretization of parametric and random parabolic pdes - analytic regularity and generalized polynomial chaos approximation
publishDate 2014
url https://hdl.handle.net/10356/100698
http://hdl.handle.net/10220/18460
_version_ 1759854772862582784