A maximal projection solution of ill-posed linear system in a column subspace, better than the least squares solution

作者：Liu Chein Shan^*

来源：Computers & Mathematics with Applications, 2014, 67(10): 1998-2014.

DOI：10.1016/j.camwa.2014.04.011

• Linear equations system
• Column subspace
• Maximal projection solution
• Maximal projection algorithm
• Least squares solution
• Inverse problems

A better method than the least squares solution is proposed in this paper to solve an n-dimensional ill-posed linear equations system Ax = b in an m-dimensional column subspace C-m, which is selected in such a way that each column in C-m is in a closer proximity to b. We maximize the orthogonal projection of b onto y := Ax to find an approximate solution x is an element of span{a, C-m}, where a is a nonzero free vector. Then, we can prove that the maximal projection solution (MP) is better than the least squares solution (LS) with parallel to b - AX(MP)parallel to < parallel to b-Ax(LS)parallel to. Numerical examples of inverse problems under a large noise maybe up to 30% are discussed which confirm the efficiency of presently developed MP algorithms: MPA and MPA(m).

• 出版日期2014-6