| In this dissertation, we describe LSHSS, a robust solver based on the Hermitian and skew-Hermitian splitting (HSS) method for solving the linear least squares problem corresponding to the linear system A x = f via minimizing ||f -- Ax||W, for a given symmetric positive definite matrix W. We use the restarted GMRES as the outer iteration with the HSS preconditioner, in which we use the SOR method for solving the linear systems involving the preconditioner. Theoretical analysis shows that LSHSS always converges to the unique solution of this linear least squares problem. We present numerical experiments that demonstrate the robustness of LSHSS compared to other preconditioned iterative solvers, as well as sparse direct solvers such as Pardiso. In addition, we demonstrate its parallel scalability on a cluster of multicore nodes.;Further, we propose a hybrid solver based on M-matrix splitting and block-row projection (BRP). We split A into two nonsingular M-matrices, and construct the new larger linear system which we solve using a BRP method. The robustness and parallel scalability of BRP are also compared with those of LSHSS. |
