A Spectral Projection Preconditioner for Solving Ill Conditioned Linear Systems

TL;DR

This paper introduces a spectral projection preconditioner combined with a deflated Krylov subspace method that significantly reduces iterations and improves accuracy when solving ill-conditioned linear systems.

Contribution

The paper proposes a novel spectral projection preconditioner integrated with a deflated Krylov method for more efficient and accurate solutions of ill-conditioned linear systems.

Findings

Requires fewer iterations to converge

Achieves higher solution accuracy

Outperforms standard Krylov solvers on ill-conditioned problems

Abstract

We present a preconditioner based on spectral projection that is combined with a deflated Krylov subspace method for solving ill conditioned linear systems of equations. Our results show that the proposed algorithm requires many fewer iterations to achieve the convergence criterion for solving an ill conditioned problem than a Krylov subspace solver. In our numerical experiments, the solution obtained by the proposed algorithm is more accurate in terms of the norm of the distance to the exact solution of the linear system of equations.