A rational conjugate gradient method for linear ill-conditioned problems

TL;DR

This paper introduces a novel conjugate gradient method tailored for solving linear ill-conditioned problems, leveraging rational Krylov spaces and Tikhonov regularization to improve convergence and stability.

Contribution

It develops a new conjugate gradient algorithm that avoids explicit orthogonalization, combining rational Krylov methods with Tikhonov regularization for better handling ill-conditioned systems.

Findings

The method converges effectively on ill-conditioned problems.

Numerical examples demonstrate improved regularization properties.

Sparse pentadiagonal matrix representations facilitate computations.

Abstract

We consider linear ill-conditioned operator equations in a Hilbert space setting. Motivated by the aggregation method, we consider approximate solutions constructed from linear combinations of Tikhonov regularization, which amounts to finding solutions in a rational Krylov space. By mixing these with usual Krylov spaces, we consider least-squares problem in these mixed rational spaces. Applying the Arnoldi method leads to a sparse, pentadiagonal representation of the forward operator, and we introduce the Lanczos method for solving the least-squares problem by factorizing this matrix. Finally, we present an equivalent conjugate-gradient-type method that does not rely on explicit orthogonalization but uses short-term recursions and Tikhonov regularization in each second step. We illustrate the convergence and regularization properties by some numerical examples.