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A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs

by M. Bieri

(Report number 2009-08)

Abstract
This work presents a stochastic collocation method for solving elliptic PDEs with random coefficients and forcing term which are assumed to depend on a finite number of random variables. The method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution's statistics. The selection of collocation points is based on a Smolyak construction of zeros of orthogonal polynomials w.r.t the probability density function of each random input variable. A sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed, resulting in a substantial reduction of overall degrees of freedom. Like in the Monte Carlo approach, the algorithm results in solving a number of uncoupled, purely deterministic elliptic problems, which allows the integration of existing fast solvers for elliptic PDEs. Numerical examples on two-dimensional domains will then demonstrate the superiority of this sparse composite collocation FEM compared to the `full composite' collocation FEM and the Monte Carlo method.

Keywords: Stochastic partial differential equations, stochastic collocation methods, Smolyak approximation, multilevel approximations, sparse tensor products

@Techreport{B09_122,
  author = {M. Bieri},
  title = {A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-08},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-08.pdf },
  year = {2009}
}

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