Iterative Refinement and Flexible Iteratively Reweighed Solvers for Linear Inverse Problems with Sparse Solutions

TL;DR

This paper introduces a novel iterative refinement framework for sparse solutions in large-scale linear inverse problems, combining flexible Krylov methods with reweighted schemes for improved accuracy and efficiency.

Contribution

It proposes a new algorithmic framework that enhances sparse solution computation by integrating iterative refinement with flexible reweighted Krylov methods, outperforming existing approaches.

Findings

Outperforms other flexible Krylov methods in memory-limited scenarios

Achieves higher accuracy through suitable restarts

Demonstrates effectiveness in image deblurring and tomography

Abstract

This paper presents a new algorithmic framework for computing sparse solutions to large-scale linear discrete ill-posed problems. The approach is motivated by recent perspectives on iteratively reweighted norm schemes, viewed through the lens of iterative refinement. This framework leverages the efficiency and fast convergence of flexible Krylov methods while achieving higher accuracy through suitable restarts. Additionally, we demonstrate that the proposed methods outperform other flexible Krylov approaches in memory-limited scenarios. Relevant convergence theory is discussed, and the performance of the proposed algorithms is illustrated through a range of numerical examples, including image deblurring and computed tomography.