A Range Condition for Polyconvex Variational Regularization. (27th July 2018)

• Record Type: Journal Article
• Authors: Kirisits, Clemens; Scherzer, Otmar
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 10(2018)
• Page Start: 1064
• Page End: 1076
• Publication Date: 2018-07-27
• Subjects: Convergence rates -- inverse problems -- polyconvex functions -- regularization theory -- source conditions; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1467447
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In the context of convex variational regularization, it is a known result that, under suitable differentiability assumptions, source conditions in the form of variational inequalities imply range conditions, while the converse implication only holds under an additional restriction on the operator. In this article, we prove the analogous result for polyconvex regularization. More precisely, we show that the variational inequality derived by the authors in 2017 implies that the derivative of the regularization functional must lie in the range of the dual-adjoint of the derivative of the operator. In addition, we show how to adapt the restriction on the operator in order to obtain the converse implication.