MiS Preprint Repository

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 $\mathcal{O}(N^{-\alpha /d})$ are proven to have the storage complexity $\mathcal{O}(d{N}^{1/d}{\log }^{q}N)$ 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 $\frac{1}{|x-y|}$, $x,y\in {\mathbb{R}}^d$, $d\ge 3$. Numerical illustrations are given in the case of $d=3$.