On the incomplete oblique projections method for solving box constrained least squares problems

H Scolnik,N Echebest,M T Guardarucci

doi:10.1007/s11075-013-9721-z

Abstract

The aim of this paper is to extend the applicability of the incomplete oblique projections method (IOP) previously introduced by the authors for solving inconsistent linear systems to the box constrained case. The new algorithm employs incomplete projections onto the set of solutions of the augmented system Ax − r = b, together with the box constraints, based on a scheme similar to the one of IOP, adding the conditions for accepting an approximate solution in the box. The theoretical properties of the new algorithm are analyzed, and numerical experiences are presented comparing its performance with some well-known methods.

Highlights

Large and sparse systems of linear equations arise in many important applications [5, 24], as image reconstruction from projections, radiation therapy treatments planning, computational mechanics and optimization problems
We report for Bounded Incomplete Oblique Projections (BIOP) the number of iterations(Iter) and CPU time required for each problem p, for reaching the stopping condition using = 10−6
We report for BVLS the number of iterations(Iter) and CPU time required for each problem p, for reaching the stopping condition or a norm of the residual less or equal than the one obtained by BIOP

Summary

Introduction

Large and sparse systems of linear equations arise in many important applications [5, 24], as image reconstruction from projections, radiation therapy treatments planning, computational mechanics and optimization problems. Those systems are often inconsistent, and one usually seeks a point x∗ ∈ n, li ≤ x∗i ≤ ui, i = 1, . In [19] we have introduced, for inconsistent problems Ax = b, the IOP algorithm that converges to a weighted least squares solution of that system.

Objectives

Results

Conclusion