Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficients

Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficients

J. Beck, F. Nobile, L. Tamellini, R. Tempone, Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficients,  Computers & Mathematics with Applications. Volume 67, Issue 4, March 2014, Pages 732–751.
J. Beck, F. Nobile, L. Tamellini, R. Tempone
Uncertainty Quantification, PDEs with random data, linear elliptic equations, multivariate polynomial approximation, best M -terms polynomial approximation, Stochastic Galerkin method, sub-exponential convergence
2014
In this work we consider quasi-optimal versions of the Stochastic Galerkin Method for solving linear elliptic PDEs with stochastic coeffcients. In particular, we consider the case of a nite 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 speci c application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates.
doi:10.1016/j.camwa.2013.03.004