A maximal projection solution of ill-posed linear system in a column subspace, better than the least squares solution
A better method than the least squares solution is proposed in this paper to solve ann n -dimensional ill-posed linear equations system A x = b in an m m -dimensional column subspace Cm C m , which is selected in such a way that each column in Cm C m is in a closer proximity to b . We maximize the orthogonal projection of b onto y ? A x to find an approximate solution x ? 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 ? b ? A x MP ? < ? b ? A x LS ? . Numerical examples of inverse problems under a large noise maybe up to 30% 30 % are discussed which confirm the efficiency of presently developed MP algorithms: MPA and MPA(m).
maximal projection solution column subspace least squares solution linear equations system inverse problems maximal projection algorithm