Adaptive Stochastic Galerkin Methods

Claude Gittelson, ETH Zürich

Abstract: We consider stochastic Galerkin methods for elliptic PDE depending on a random field. Expanding this field into a series with independent coefficients introduces an infinite product structure on the probability space. This permits a discretization by tensor products of suitable orthonormal polynomials. The original problem can be reformulated as an infinite system of equations for the coefficients of the solution with respect to this basis.

Without any truncation of the series, restricting to a finite set of polynomial basis functions reduces this infinite system to a finite system of deterministic equations, which can be solved by standard finite element methods.

The only remaining challenge is the selection of active basis functions. We tackle this problem by iterative methods based on adaptive wavelet techniques. Our method uses adaptive local truncation of the series expansion to recursively refine the set of active indices.

These results are part of a PhD thesis under the supervision of Prof. Ch. Schwab, supported in part by the Swiss National Science Foundation under grant No. 200021-120290/1.