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A. Bespalov, D. Praetorius, M. Ruggeri:
"Rate optimality of an adaptive multilevel stochastic Galerkin finite element method";
Talk: Congress of the Italian Society of Industrial and Applied Mathematics (SIMAI 2020+2021), Parma, Italy (invited); 2021-08-30 - 2021-09-03.

We consider an adaptive multilevel stochastic Galerkin finite element method for a class of elliptic boundary value problems with parametric or uncertain coefficients. Stochastic Galerkin approximations are based on a sparse polynomial expansion with spatial coefficients residing in finite element spaces associated with different locally refined meshes. The adaptive algorithm is steered by a reliable and efficient a posteriori error estimator, which can be decomposed into a two-level spatial estimator and a hierarchical parametric estimator. We show that, under an appropriate saturation assumption, the proposed adaptive strategy yields convergence of the approximation error to zero with optimal algebraic rates with respect to the overall dimension of the underlying multilevel approximation spaces.