Regularity and generalized polynomial chaos approximation of parametric and random second-order hyperbolic partial differential equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.35197
DOI: 10.1142/S0219530512500145
author: Viet Ha Hoang
author: Christoph Schwab
publication date: 2012-08-29T00:00:00Z
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 74H30
Mathematics Subject Classification ID: 74J05
zbMATH DE Number: 6073839
zbMATH Keywords: wave equation
zbMATH Keywords: generalized polynomial chaos
zbMATH Keywords: random media
zbMATH Keywords: Gevrey regularity
MaRDI profile type: Publication
cites work: Exponential convergence of {\(hp\)} quadrature for integral operators with Gevrey kernels
cites work: Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic SPDEs
cites work: ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S
full work available at URL: s0219530512500145
OpenAlex ID: W2110065034
title: Regularity and generalized polynomial chaos approximation of parametric and random second-order hyperbolic partial differential equations
published in: Analysis and Applications (Singapore)
review text: Let \(D\) be a bounded Lipschitz domain in \(R^{d}\). In \(Q_{T}=(0,T)\times D\) the authors consider the stochastic wave equation \({\partial^2 u\over \partial t^2}-\nabla\cdot(a(x,\omega)\nabla u)=g(t,x)\), \(u|_{t=0}=g_1\), \(u_{t}|_{t=0}=g_2\). It is assumed that the coefficient \(a(x,\omega)\) is a random field on probability space \((\Omega,\Sigma,P)\) over \(L^{\infty}(D)\). The forcing \(g\) and initial data \(g_1\) and \(g_2\) are assumed to be deterministic. The authors show that the law of the random solution can be represented as a deterministic function of a countable number of coordinates. For a class of equations with regular right-hand side and compatible initial conditions it is shown that this solution is, as a function of the coordinates, smooth as mapping from the parameter domain into suitable Sobolev spaces in which deterministic wave equation is well-posed. The authors investigate the smoothness of the parametric solution in terms of Gervey regularity. Sufficient conditions for the \(p\)-summability of the generalized polynomial chaos expansion of the parametric solution in terms of the countably many input parameters are obtained and rates of convergence of best \(N\)-term polynomial chaos type approximations of the parametric solution are presented.
reviewed by: Aleksandr D. Borisenko
description: scientific article; zbMATH DE number 6073839