Contingent derivatives and regularization for noncoercive inverse problems. (3rd July 2019)

• Record Type: Journal Article
• Authors: Clason, Christian; Khan, Akhtar A.; Sama, Miguel; Tammer, Christiane
• Is Part Of: Optimization. Volume 68:Number 7(2019)
• Page Start: 1337
• Page End: 1364
• Publication Date: 2019-07-03
• Subjects: Inverse problems -- ill-posed problems -- noncoercive variational problems -- regularization -- total variation -- parameter identification -- output least-squares -- modified output least-squares -- contingent cone -- contingent sets -- contingent derivatives; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2018.1442448
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract: We study the inverse problem of parameter identification in noncoercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map using the first-order and the second-order contingent derivatives. We explore the inverse problem using the output least-squares and the modified output least-squares objectives. By regularizing the noncoercive variational problem, we obtain a single-valued regularized parameter-to-solution map and investigate its smoothness and boundedness. We also consider optimization problems using the output least-squares and the modified output least-squares objectives for the regularized variational problem. We give a complete convergence analysis showing that for the output least-squares and the modified output least-squares, the regularized minimization problems approximate the original optimization problems suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We provide discrete formulas for the gradient and the Hessian calculation and present numerical results.