Analytic regularity and collocation approximation for elliptic PDEs with random domain deformations

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• 作者：Julio E. Castrilló ; n-Candá ; s^a ; ¹ ; ^{uvel@alum.mit.edu" class="auth_mail" title="E-mail the corresponding author} ; Fabio Nobile^b ; ^{fabio.nobile@epfl.ch" class="auth_mail" title="E-mail the corresponding author} ; Raú ; l F. Tempone^c ; ^a ; ^{raul.tempone@kaust.edu.sa" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Uncertainty quantification ; Stochastic collocation ; Stochastic PDEs ; Finite elements ; Complex analysis ; Smolyak sparse grids
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：71
• 期：6
• 页码：1173-1197
• 全文大小：893 K

文摘

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116000080&_mathId=si19.gif&_user=111111111&_pii=S0898122116000080&_rdoc=1&_issn=08981221&md5=35f2963a8341a25b8f9292a4ff83b4ed" title="Click to view the MathML source">Nclass="mathContainer hidden">class="mathCode">$N$ random variables. The elliptic problem is remapped onto a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116000080&_mathId=si20.gif&_user=111111111&_pii=S0898122116000080&_rdoc=1&_issn=08981221&md5=dcc36f8bef58eba8ecb9af40ea3ae38a" title="Click to view the MathML source">C^Nclass="mathContainer hidden">class="mathCode">${\mathbb{C}}^N$ with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and variance of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments.