Projected iterated Tikhonov regularization in low precision

TL;DR

This paper explores the effectiveness of a projected Krylov subspace method with Tikhonov regularization in low-precision arithmetic for solving severely ill-posed linear inverse problems, demonstrating comparable accuracy to double precision.

Contribution

It introduces a novel approach combining projected Krylov subspace methods with Tikhonov regularization in low precision, analyzing its regularizing properties and practical performance.

Findings

Method achieves similar accuracy to double precision in inverse problems.

Filtering properties effectively suppress noise and ill-posedness.

Numerical examples confirm robustness in low-precision computations.

Abstract

We investigate the regularizing behavior of an iterative Krylov subspace method for the solution of linear inverse problems in precisions lower than double. Recent works have considered the projection of iterated Tikhonov methods using Krylov subspaces for both computational efficiency and an additional regularizing effect. To investigate the regularizing behavior of this projected algorithm applied to problems that are naturally severely ill-posed, we formulate the iterates as a filtered solution using the preconditioned Landweber method with a Tikhonov-type preconditioner in a Krylov subspace. Through numerical examples simulating multiple low precision choices, we showcase the filtering properties of the method and the achievement of comparable working accuracy applied to discrete inverse problems (i.e., to within a few decimal places in relative error) compared to results computed in traditional double precision.