Flexible inner-product free Krylov methods for inverse problems

TL;DR

This paper introduces novel flexible and inner-product free Krylov methods, including randomized variants, to enhance inverse problem solving with variational regularization, improving computational efficiency and result quality.

Contribution

It develops new flexible and inner-product free Krylov methods, including a generalized Hessenberg method and randomized versions, with theoretical analysis and numerical validation.

Findings

New methods outperform existing approaches in inverse problems.

Randomized variants improve computational speed and memory efficiency.

Numerical experiments demonstrate effectiveness across different regularization terms.

Abstract

Flexible Krylov methods are a common standpoint for inverse problems. In particular, they are used to address the challenges associated with explicit variational regularization when it goes beyond the two-norm, for example involving an $\ell_p$ norm for $0 < p \leq 1$. Moreover, inner-product free Krylov methods have been revisited in the context of ill-posed problems, to speed up computations and improve memory requirements by means of using low precision arithmetics. However, these are effectively quasi-minimal residual methods, and can be used in combination with tools from randomized numerical linear algebra to improve the quality of the results. This work presents new flexible and inner-product free Krylov methods, including a new flexible generalized Hessenberg method for iteration-dependent preconditioning. Moreover, it introduces new randomized versions of the methods, based on the sketch-and-solve framework. Theoretical considerations are given, and numerical experiments are provided for different variational regularization terms to show the performance of the new methods.