A bilevel approach for parameter learning in inverse problems. (20th September 2018)

• Record Type: Journal Article
• Authors: Holler, Gernot; Kunisch, Karl; Barnard, Richard C
• Is Part Of: Inverse problems. Volume 34:Number 11(2018:Nov.)
• Publication Date: 2018-09-20
• Subjects: Tikhonov regularization -- multi-penalty regularization -- bilevel optimization -- parameter learning -- optimal control problems -- elliptic partial differential equations; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aade77
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: A learning approach for selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized in the regularization parameters. Conditions which ensure the existence of solutions to the bilevel optimization problem are derived, and these conditions are verified for two relevant examples. Difficulties arising from the possible lack of convexity of the lower level problems are discussed. Optimality conditions are given provided that a reasonable constraint qualification holds. Finally, results from numerical experiments used to test the developed theory are presented.