In the talk 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 (a vector of Kronecker rank 1) we can prove that the solution x can be approximated by a sum of O(log(ε²)) tensor product vectors (each of Kronecker rank 1) where ε is the relative approximation error. The complexity for the approximation of the inverse is almost linear with respect to the meshwidth h^-1 and linear in the dimension d of the space where the partial differential equation is posed. If the right-hand side is the sum of few tensor vectors, then an approximation to the solution of the system can be computed in O(d h^-1 log(h^-1)).

The talk closes with numerical examples for systems of size 1024⁴⁰⁹⁶.