Some transpose-free CG-like solvers for nonsymmetric ill-posed problems

TL;DR

This paper proposes a new class of transpose-free Krylov subspace methods for nonsymmetric ill-posed problems, offering an efficient alternative to traditional CG-like solvers by using rank-deficient approximations of the transpose.

Contribution

It introduces and analyzes novel transpose-free Krylov methods that improve efficiency for nonsymmetric ill-posed problems, with theoretical insights and numerical validation.

Findings

New methods outperform classical Arnoldi-based solvers

Effective for large, ill-posed inverse problems

Theoretical analysis supports practical advantages

Abstract

This paper introduces and analyzes an original class of Krylov subspace methods that provide an efficient alternative to many well-known conjugate-gradient-like (CG-like) Krylov solvers for square nonsymmetric linear systems arising from discretizations of inverse ill-posed problems. The main idea underlying the new methods is to consider some rank-deficient approximations of the transpose of the system matrix, obtained by running the (transpose-free) Arnoldi algorithm, and then apply some Krylov solvers to a formally right-preconditioned system of equations. Theoretical insight is given, and many numerical tests show that the new solvers outperform classical Arnoldi-based or CG-like methods in a variety of situations.