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 an $n$ ‐dimensional ill‐posed linear equations system $Ax=b$ in an $m$ ‐dimensional column subspace ${\mathcal{C}}_m$ , which is selected in such a way that each column in ${\mathcal{C}}_m$ is in a closer proximity to $b$ . We maximize the orthogonal projection of $b$ onto $y≔Ax$ to find an approximate solution $x\in \text{span}\{a,{\mathcal{C}}_m\}$ , 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 $\parallel b‐A{x}_{\text{MP}}\parallel <\parallel b‐A{x}_{\text{LS}}\parallel$ . 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).