Abstract
Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results.
Highlights
IntroductionWe consider one-parameter and multiparameter Tikhonov regularization problems of the form argmin Ax − b 2 + μi Li x 2 ( ≥ 1),
We consider one-parameter and multiparameter Tikhonov regularization problems of the form argmin Ax − b 2 + μi Li x 2 ( ≥ 1), (1)x i =1 where · denotes the 2-norm and the superscript i is used as an index
Multiparameter Tikhonov can be used when a satisfactory choice of the regularization operator is unknown in advance, or can be seen as an attempt to combine the strengths of different regularization operators
Summary
We consider one-parameter and multiparameter Tikhonov regularization problems of the form argmin Ax − b 2 + μi Li x 2 ( ≥ 1),. We present a new subspace method consisting of three phases; a new expansion phase, a new extraction phase, and a new truncation phase. A new method for selecting the regularization parameters μik in the extraction phase. The three phases work alongside each other: the intermediate solution obtained in the extraction phase is preserved in the truncation phase, whereas the remaining perpendicular component(s) from the expansion phase are removed. In the former, a straightforward parameter selection strategy for multiparameter regularization is given, in the latter, a justification using perturbation analysis.
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