Abstract of Christine Tobler

Solution of parameter-dependent linear systems by tensor methods
Consider a parameter-dependent linear system

A(α) x(α) = b(α),

where α is a vector of p parameters. We assume that the parameter space is discretized with a regular grid and aim at computing the solution x(α) for each grid point. This becomes rather expensive for larger p as the number of grid points grows exponentially in p. We therefore propose the use of low tensor rank approximations to reduce the computational cost significantly. For this purpose, we treat the right hand side b and the solution x evaluated at all grid points as tensors of dimension p+1. Assuming that b admits a low tensor rank approximation and A is sufficiently smooth, one can show that x also admits a low tensor rank approximation. We present algorithms which exploit this fact and demonstrate their efficiency with a number of examples.


Lars Grasedyck (MPI Leipzig, Germany)
Wolfgang Hackbusch (MPI Leipzig, Germany)
Boris Khoromskij (MPI Leipzig, Germany)