Incomplete alternating projection method for large inconsistent linear systems

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∈Rm. 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) xk+1=PPPQxk which was introduced by von Neumann. We take in the modification an approximative projection instead of an exact projection PP 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 and the convergence of xk to an element of is preserved. We also present preliminary numerical results for the method and compare these results with the APM.