A new choice rule for regularization parameters in Tikhonov regularization

Abstract

This article proposes and analyses a novel heuristic rule for choosing regularization parameters in Tikhonov regularization for inverse problems. The existence of solutions to the regularization parameter equation is shown, and a variational characterization of the inverse solution is provided. Some a posteriori error estimates of the approximate solutions to the inverse problem are also established. An iterative algorithm is suggested for the efficient numerical realization of the new choice rule, which is shown to have a practically desirable monotone convergence. Numerical experiments for both mildly and severely ill-posed benchmark inverse problems with various regularizing functionals of Tikhonov type, e.g. L ²–L ² with constraints, L ²–ℓ¹ and L ¹–TV, are presented to demonstrate its effectiveness, and compared with the quasi-optimality criterion.

Keywords:

• parameter choice
• a posteriori error estimate
• Tikhonov regularization

AMS Subject Classifications:

• 65J22
• 65J20
• 49M15
• 49M29

Acknowledgements

The authors would like to thank Prof. Per Christian Hansen for his package Regularization Tool with which some of our numerical experiments were conducted. The work of the third author was substantially supported by Hong Kong RGC grants (Projects 405110 and 404407) and partially supported by CUHK Focused Investment Scheme 2008/2010.