Existence and Computation of a Low Kronecker-Rank Approximant to the Solution of a Tensor System with Tensor Right-Hand Side

In this paper we construct an approximation to the solution x of a linear system of equations Ax = b of tensor product structure as it typically arises for finite element and finite difference discretisations of partial differential operators on tensor grids. For a right-hand side b of tensor product structure we can prove that the solution x can be approximated by a sum of O(log(ε)2) tensor product vectors where ε is the relative approximation error. Numerical examples for systems of size 1024 indicate that this method is suitable for high-dimensional problems.