Tensor-product approximation to operators and functions in high dimensions

In recent papers tensor-product structured Nyström and Galerkin-type approximations of certain multi-dimensional integral operators have been introduced and analysed. In the present paper, we focus on the analysis of the collocation-type schemes with respect to the tensor-product basis in a high spatial dimension d. Approximations up to an accuracy O(N-α/d) are proven to have the storage complexity O(dN1/dlogqN) with q independent of d, where N is the discrete problem size. In particular, we apply the theory to a collocation discretisation of the Newton potential with the kernel , x,y∈Rd, d⩾3. Numerical illustrations are given in the case of d=3.