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 annn-dimensional ill-posed linear equations system Ax=b in an mm-dimensional column subspace CmCm, which is selected in such a way that each column in CmCm is in a closer proximity to b. We maximize the orthogonal projection of b onto y≔Ax to find an approximate solution x∈span{a,Cm}, 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−AxMP‖<‖b−AxLS‖. 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).