Consider a parameterdependent 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.
