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Stochastic Galerkin approximation of operator equations with infinite dimensional noise

by C. J. Gittelson

(Report number 2011-10)

Abstract
It is common practice in the study of stochastic Galerkin methods for boundary value problems depending on random fields to truncate a series representation of this field prior to the Galerkin discretization. We show that this is unnecessary; the projection onto a finite dimensional subspace automatically replaces the infinite series expansion by a suitable partial sum. We construct tensor product polynomial bases on infinite dimensional parameter domains, and use these to recast a random boundary value problem as a countably infinite system of deterministic equations. The stochastic Galerkin method can be interpreted as a standard finite element discretization of a finite section of this infinite system.

Keywords:

@Techreport{G11_118,
  author = {C. J. Gittelson},
  title = {Stochastic Galerkin approximation of operator equations with infinite dimensional noise},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-10},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-10.pdf },
  year = {2011}
}

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