Incomplete oblique projections for solving large inconsistent linear systems

Abstract

The authors introduced in previously published papers acceleration schemes for Projected Aggregation Methods (PAM), aiming at solving consistent linear systems of equalities and inequalities. They have used the basic idea of forcing each iterate to belong to the aggregate hyperplane generated in the previous iteration. That scheme has been applied to a variety of projection algorithms for solving systems of linear equalities or inequalities, proving that the acceleration technique can be successfully used for consistent problems. The aim of this paper is to extend the applicability of those schemes to the inconsistent case, employing incomplete projections onto the set of solutions of the augmented system Ax − r = b. These extended algorithms converge to the least squares solution. For that purpose, oblique projections are used and, in particular, variable oblique incomplete projections are introduced. They are defined by means of matrices that penalize the norm of the residuals very strongly in the first iterations, decreasing their influence with the iteration counter in order to fulfill the convergence conditions. The theoretical properties of the new algorithms are analyzed, and numerical experiences are presented comparing their performance with several well-known projection methods.