Regularization of linear inverse problems by rational Krylov methods

TL;DR

This paper analyzes the regularization properties of rational Krylov methods, specifically the aggregation and RatCG algorithms, for solving ill-posed linear problems, showing they form optimal-order regularization schemes under certain conditions.

Contribution

It demonstrates that aggregation and RatCG methods can be understood as rational Krylov space methods and establishes their optimal regularization properties with the discrepancy principle.

Findings

Methods form optimal-order regularization schemes

Regularization effectiveness depends on large regularization parameters

Methods can be interpreted as rational Krylov space methods

Abstract

For approximately solving linear ill-posed problems in Hilbert spaces, we investigate the regularization properties of the aggregation method and the RatCG method. These recent algorithms use previously calculated solutions of Tikhonov regularization (respectively, Landweber iterations) to set up a new search space on which the least-squares functional is minimized. We outline how these methods can be understood as rational Krylov space methods, i.e., based on the space of rational functions of the forward operator. The main result is that these methods form an optimal-order regularization schemes when combined with the discrepancy principle as stopping rule and when the underlying regularization parameters are sufficiently large.