Mixed precision iterative refinement for linear inverse problems

TL;DR

This paper explores a mixed precision iterative refinement approach for solving severely ill-posed linear inverse problems, demonstrating comparable accuracy to double precision methods through filtering properties and numerical experiments.

Contribution

It introduces a novel mixed precision iterative refinement method formulated as a filtered solution using the preconditioned Landweber method with a Tikhonov preconditioner for ill-posed problems.

Findings

Achieves accuracy comparable to double precision solutions

Demonstrates filtering properties in mixed precision iterative refinement

Shows effectiveness through numerical experiments

Abstract

This study investigates the iterative refinement method applied to the solution of linear discrete inverse problems by considering its application to the Tikhonov problem in mixed precision. Previous works on mixed precision iterative refinement methods for the solution of symmetric positive definite linear systems and least-squares problems have shown regularization to be a key requirement when computing low precision factorizations. For problems that are naturally severely ill-posed, we formulate the iterates of iterative refinement in mixed precision as a filtered solution using the preconditioned Landweber method with a Tikhonov-type preconditioner. Through numerical examples simulating various mixed precision choices, we showcase the filtering properties of the method and the achievement of comparable working accuracy of discrete inverse problems (i.e., to within a few decimal places in relative error) compared to results computed in double precision as well as another approximate iterative refinement method which we use as a benchmark.