Conjugate gradient for ill-posed problems: regularization by preconditioning, preconditioning by regularization

TL;DR

This paper explores the combined use of preconditioning and Tikhonov regularization in conjugate gradient methods for ill-posed problems, enabling efficient solution estimation and application to boundary data completion and optical flow estimation.

Contribution

It introduces a novel Ritz analysis approach for simultaneous preconditioning and regularization, improving solution estimation in ill-posed problems.

Findings

Effective estimation of Tikhonov's weight with negligible cost

Application to boundary data completion problem

Application as inner solver for optical flow estimation

Abstract

This paper investigates using the conjugate gradient iterative solver for ill-posed problems. We show that preconditioner and Tikhonov-regularization work in conjunction. In particular when they employ the same symmetric positive semi-definite operator, a powerful Ritz analysis allows one to estimate at negligible computational cost the solution for any Tikhonov's weight. This enhanced linear solver is applied to the boundary data completion problem and as the inner solver for the optical flow estimator.