Incomplete alternating projection method for large inconsistent linear systems

Abstract

We consider in the paper the problem of finding an approximat solution of a large scale inconsistent linear system A ⊤ x = b , where A is an n × m real matrix and b ∈ R m . The problem is a special case of the following problem. Let P , Q be nonempty and affine subspaces; find an element of the intersection P ∩ Q or find points p ∈ P and q ∈ Q which realize the distance between these two subspaces. Problems of this kind appear in many applications, e.g. in the image reconstruction or in the intensity modulated radiation therapy (see, e.g. [Y. Censor, S.A. Zenios, Parallel Optimization, Theory, Algorithms and Applications, Oxford University Press, New York, 1997; H. Stark, Y. Yang, Vector Space Projections. A Numerical Approach to Signal and Image Processing, Neural Nets and Optics, John Wiley & Sons, Inc., New York, 1998; H.W. Hamacher, K.-H. Küfer, Inverse radiation therapy planning – a multiple objective optimization approach, Discrete Appl. Math. 118 (2002) 145–161]). In order to solve the problem we deal with a modification of the so-called alternating projection method (APM) x k + 1 = P P P Q x k which was introduced by von Neumann. We take in the modification an approximative projection P ∼ P instead of an exact projection P P with appropriate stopping criteria. A similar idea was considered by Scolnik et al. [H.D. Scolnik, N. Echebest, M.T. Guardarucci, M.C. Vacchino, Incomplete oblique projections for solving large inconsistent linear systems, Math. Program. Ser. B 111 (2008) 273–300]. We modify the APM in such a way that the Fejér monotonicity with respect to Fix P P P Q and the convergence of x k to an element of Fix P P P Q is preserved. We also present preliminary numerical results for the method and compare these results with the APM.