Seminar in Numerical Analysis: Markus Bachmayr (Universität Mainz)

Optimality of adaptive stochastic Galerkin methods for affine-parametric elliptic PDEs

We consider the computational complexity of approximating elliptic PDEs with random coefficients by sparse product polynomial expansions. Except for special cases (for instance, when the spatial discretisation limits the achievable overall convergence rate), previous approaches for a posteriori selection of polynomial terms and corresponding spatial discretizations do not guarantee optimal complexity in the sense of computational costs scaling linearly in the number of degrees of freedom. We show that one can achieve optimality of an adaptive Galerkin scheme for discretizations by spline wavelets in the spatial variable when a multiscale representation of the affinely parameterized random coefficients is used. 

M. Bachmayr and I. Voulis, An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality, arXiv:2109:09136

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