Numerical considerations of block GMRES methods when applied to linear discrete ill-posed problems

Abstract

Linear systems of equations with a matrix whose singular values decay to zero with increasing index number without a significant gap are commonly referred to as linear discrete ill-posed problems. We are interested in solving large systems of this kind when the right-hand side has k>1 column vectors. The systems may be regarded as one system of equations with a block vector (with k columns) as the right-hand side and then solved by a block iterative method, or as k linear systems of equations (one for each right-hand side vector) that can be solved independently. Thus, the solution is a block vector with k columns. In many applications, including the restoration of color images, the right-hand side represents measurements that are contaminated by errors. Block iterative methods compute all columns of the solution block vector simultaneously. We will illustrate the performance of standard block GMRES methods and global GMRES methods, which also are block methods, and show that they may determine computed solutions of lower quality than when each column of the solution block vector is computed independently by a “standard” iterative method. We introduce a new local block GMRES method that can overcome the problems associated with block GMRES methods applied to linear discrete ill-posed problems.