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.
