Hybrid CGME and TCGME algorithms for large-scale general-form regularization

TL;DR

This paper introduces two hybrid algorithms based on Krylov subspace methods for efficiently solving large-scale general-form regularization problems, with proven convergence properties and demonstrated effectiveness through numerical experiments.

Contribution

The paper presents novel hybrid CGME and TCGME algorithms that improve convergence and accuracy in large-scale regularization, with theoretical analysis and practical validation.

Findings

Inner least squares problems become better conditioned as iterations increase.

The stopping tolerance for LSQR can be chosen to match the accuracy of the best regularized solutions.

Numerical experiments show the new algorithms outperform existing methods in efficiency and effectiveness.

Abstract

Two new hybrid algorithms are proposed for large-scale linear discrete ill-posed problems in general-form regularization. They are both based on Krylov subspace inner-outer iterative algorithms. At each iteration, they need to solve a linear least squares problem, which is the inner least squares problem. It is proved that inner linear least squares problems, solved by LSQR, become better conditioned as k increases, so LSQR converges faster. We also prove how to choose the stopping tolerance for LSQR to guarantee that the computed and exact best regularized solutions have the same accuracy. Numerical experiments are provided to demonstrate the effectiveness and efficiency of our new hybrid algorithms, along with comparisons to the existing algorithm.