Existence and Computation of Low Kronecker-Rank Approximations for Large Linear Systems of Tensor Product Structure

Abstract

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 (log(ɛ)²) tensor product vectors where ɛ is the relative approximation error. Numerical examples for systems of size 1024²⁵⁶ indicate that this method is suitable for high-dimensional problems.