We describe the analysis and the implementation of two Finite Element (FE) algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients. They are based on separation of deterministic and stochastic parts of the input data by a Karhunen-Lo`eve expansion, truncated after M terms. With a change of measure we convert the problem to a sequence of M-dimensional, parametric deterministic problems. Two sparse, high order polynomial approximations of the random solution’s joint pdf’s, parametrized in the input data’s Karhunen-Lo`eve expansion coordinates, are analyzed: a sparse stochastic Galerkin FEM (sparse sGFEM) and a sparse stochastic Collocation FEM (sparse sCFEM). A-priori and a-posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution’s joint pdf’s to the stochastic regularity of the input data. sCFEM and sGFEM yield deterministic approximations of the random solutions joint pdf’s that converge spectrally in the number of deterministic problems to be solved. Numerical examples with random inputs of small correlation length in diffusion problems are presented. High order gPC approximations of solutions with stochastic parameter spaces of dimension up to M = 80 are computed on workstations. Show more
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Journal / seriesResearch Report
PublisherSeminar für Angewandte Mathematik, ETH
Organisational unit03435 - Schwab, Christoph / Schwab, Christoph
03217 - Künsch, Hans Rudolf
NotesShorted version published: Computer Methods in Applied Mechanics and Engineering, Volume 198, Issues 13-14, March 2009, Pages 1149-1170, NEBIS Systemnr. 000045255, DOI: 10.1016/j.cma.2008.08.019.
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