Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients
KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Stochastic Numerics Research Group
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AbstractIn this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane CN. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates. © 2013 Elsevier Ltd. All rights reserved.
SponsorsThe authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and Uncertainty Quantification for Models of Porous Media" is also acknowledged. The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems". The fourth author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.