Inner Product Free Krylov Methods for Large-Scale Inverse Problems

TL;DR

This paper introduces two new inner-product free Krylov subspace methods, LSLU and Hybrid LSLU, designed for large-scale linear inverse problems, emphasizing efficiency and suitability for high-performance computing.

Contribution

The paper presents two novel Krylov methods, LSLU and Hybrid LSLU, that are inner-product free and tailored for large-scale inverse problems, with theoretical analysis and numerical validation.

Findings

Hybrid LSLU is effective for large-scale inverse problems.

Hybrid LSLU has comparable performance to existing methods.

Methods are suitable for high-performance computing environments.

Abstract

In this study, we introduce two new Krylov subspace methods for solving rectangular large-scale linear inverse problems. The first approach is a modification of the Hessenberg iterative algorithm that is based off an LU factorization and is therefore referred to as the least squares LU (LSLU) method. The second approach incorporates Tikhonov regularization in an efficient manner; we call this the Hybrid LSLU method. Both methods are inner-product free, making them advantageous for high performance computing and mixed precision arithmetic. Theoretical findings and numerical results show that Hybrid LSLU can be effective in solving large-scale inverse problems and has comparable performance with existing iterative projection methods.