An extrapolated TSVD method for linear discrete ill-posed problems with Kronecker structure

Abstract

This paper describes a new numerical method for the solution of large linear discrete ill-posed problems, whose matrix is a Kronecker product. Problems of this kind arise, for instance, from the discretization of Fredholm integral equations of the first kind in two space-dimensions with a separable kernel. The available data (right-hand side) of many linear discrete ill-posed problems that arise in applications is contaminated by measurement errors. Straightforward solution of these problems generally is not meaningful because of severe error propagation. We discuss how to combine the truncated singular value decomposition (TSVD) with reduced rank vector extrapolation to determine computed approximate solutions that are fairly insensitive to the error in the data. Exploitation of the structure of the problem keeps the computational effort quite small.