Krylov Subspace Methods for Linear Systems with Tensor Product Structure

• Authors: Daniel Kressner;Christine Tobler
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2010

The numerical solution of linear systems with certain tensor product structures is considered. Such structures arise, for example, from the finite element discretization of a linear PDE on a $d$-dimensional hypercube. Linear systems with tensor product structure can be regarded as linear matrix equations for $d=2$ and appear to be their most natural extension for $d2$. A standard Krylov subspace method applied to such a linear system suffers from the curse of dimensionality and has a computational cost that grows exponentially with $d$. The key to breaking the curse is to note that the solution can often be very well approximated by a vector of low tensor rank. We propose and analyze a new class of methods, so-called tensor Krylov subspace methods, which exploit this fact and attain a computational cost that grows linearly with $d$.