Embedded techniques for choosing the parameter in Tikhonov regularization

TL;DR

This paper proposes a new parameter selection strategy for Tikhonov regularization in large-scale ill-posed problems, utilizing the Arnoldi method and discrepancy principle without prior error norm knowledge.

Contribution

It introduces an adaptive, discrepancy-based parameter choice rule integrated with Arnoldi-Tikhonov, improving regularization without requiring initial error estimates.

Findings

Effective parameter estimation demonstrated on test problems

Improved regularization accuracy in image deblurring

Theoretical support for the discrepancy-based approach

Abstract

This paper introduces a new strategy for setting the regularization parameter when solving large-scale discrete ill-posed linear problems by means of the Arnoldi-Tikhonov method. This new rule is essentially based on the discrepancy principle, although no initial knowledge of the norm of the error that affects the right-hand side is assumed; an increasingly more accurate approximation of this quantity is recovered during the Arnoldi algorithm. Some theoretical estimates are derived in order to motivate our approach. Many numerical experiments, performed on classical test problems as well as image deblurring are presented.