Multilevel frames for sparse tensor product spaces

For Au = f with an elliptic differential operator A : H → H′ and stochastic data f , the m-point correlation function Mmu of the random solution u satisfies a deterministic, hypoelliptic equation with the m-fold tensor product operator A of A. Sparse tensor products of hierarchic FE-spaces in H are known to allow for approximations to Mmu which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases [28]. If wavelet bases are not available, we show here how to achieve log-linear complexity computation of sparse approximations of Mmu for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method. Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany Institut für Informatik, Christian–Albrechts–Universität zu Kiel, Olshausenstr. 40, 24098 Kiel, Germany