Adaptive Regularization Parameter Choice Rules for Large-Scale Problems

TL;DR

This paper introduces adaptive strategies for selecting regularization parameters in large-scale inverse problems, combining Krylov subspace methods with bilevel optimization to improve efficiency and reliability.

Contribution

It proposes a novel bilevel optimization framework for adaptive regularization parameter selection in large-scale problems, integrating standard criteria within a unified approach.

Findings

Strategies are effective and computationally efficient.

Methods outperform existing strategies in numerical tests.

Framework is adaptable to various regularization criteria.

Abstract

This paper derives a new class of adaptive regularization parameter choice strategies that can be effectively and efficiently applied when regularizing large-scale linear inverse problems by combining standard Tikhonov regularization and projection onto Krylov subspaces of increasing dimension (computed by the Golub-Kahan bidiagonalization algorithm). The success of this regularization approach heavily depends on the accurate tuning of two parameters (namely, the Tikhonov parameter and the dimension of the projection subspace): these are simultaneously set using new strategies that can be regarded as special instances of bilevel optimization methods, which are solved by using a new paradigm that interlaces the iterations performed to project the Tikhonov problem (lower-level problem) with those performed to apply a given parameter choice rule (higher-level problem). The discrepancy principle, the GCV, the quasi-optimality criterion, and Regi\'{n}ska criterion can all be adapted to work in this framework. The links between Gauss quadrature and Golub-Kahan bidiagonalization are exploited to prove convergence results for the discrepancy principle, and to give insight into the behavior of the other considered regularization parameter choice rules. Several numerical tests modeling inverse problems in imaging show that the new parameter choice strategies lead to regularization methods that are reliable, and intrinsically simpler and cheaper than other strategies already available in the literature.