Convergence Rates of Multilevel and Sparse Tensor Approximations for a Random Elliptic PDE

Abstract

Partial differential equations with random coefficients can be cast as parametric problems with a potentially infinite-dimensional parameter domain. For a class of elliptic equations, we derive convergence rates of approximations based on a tensorized polynomial basis on the parameter domain and a sequence of spatial discretizations. We prove that a sparse tensor product construction achieves essentially the same convergence rate as a multilevel approximation in which the spatial discretization level is chosen separately for each coefficient of the solution with respect to the polynomial basis. In some cases, the same rate is attained also by a single level approximation, using just one spatial discretization for all coefficients. We suggest an adaptive algorithm that reaches the optimal convergence rate with respect to the total number of degrees of freedom in a multilevel setting, without prior knowledge of this rate. Numerical computations confirm theoretical results.