On the Choice of Subspace for the Quasi-minimal Residual Method for Linear Inverse Problems

Inverse problems arise in various scientific and engineering applications, necessitating robust numerical methods for their solution. In this work, we consider the effectiveness of Krylov subspace iterative methods, including GMRES, QMR, and their range restricted variants for solving linear discrete ill-posed problems. We analyze the impact of subspace selection on solution quality. Our findings indicate that range restricted QMR can outperform standard QMR, and confirm the previously observed behavior that range restricted GMRES can be superior to conventional GMRES in terms of approximation efficacy. Notably, range restricted QMR demonstrates a key advantage over GMRES with respect to range restricted QMR's singular spectrum which can make the method less sensitive to errors that are naturally present making it particularly effective when the noise level in the problem is uncertain.