Incomplete alternating projection method for large inconsistent linear systems

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• 作者：Andrzej Cegielski ; Agnieszka Suchocka
• 关键词：Linear inconsistent system ; Alternating projections ; Fej&#xe9 ; r monotonicity ; Convergence
• 刊名：Linear Algebra and its Applications
• 出版年：2008
• 出版时间：1 March 2008
• 年：2008
• 卷：428
• 期：5-6
• 页码：1313-1324
• 全文大小：171 K

文摘

We consider in the paper the problem of finding an approximat solution of a large scale inconsistent linear system SB-3&_mathId=mml1&_user=10&_cdi=5653&_rdoc=8&_acct=C000050221&_version=1&_userid=10&md5=727d36924877d37e144afbde8cf1dfd0"" title=""Click to view the MathML source"">Ax=b, where A is an n×m real matrix and e6d8c7d80d8f1"">sbottom"" border=""0"" height=13 width=49>. The problem is a special case of the following problem. Let sbottom"" border=""0"" height=16 width=33> be nonempty and affine subspaces; find an element of the intersection sbottom"" border=""0"" height=13 width=44> or find points e6e88ce356b16044fea4a658d"">sbottom"" border=""0"" height=16 width=44> and sbottom"" border=""0"" height=16 width=40> 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) sbottom"" border=""0"" height=14 width=104> which was introduced by von Neumann. We take in the modification an approximative projection sbottom"" border=""0"" height=19 width=22> instead of an exact projection sbottom"" border=""0"" height=14 width=22> 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 sbottom"" border=""0"" height=16 width=64> and the convergence of x_k to an element of sbottom"" border=""0"" height=16 width=64> is preserved. We also present preliminary numerical results for the method and compare these results with the APM.