
Journal of Convex Analysis 25 (2018), No. 2, 545569 Copyright Heldermann Verlag 2018 Identification in Variational and QuasiVariational Inequalities Joachim Gwinner Institute of Mathematics, Department of Aerospace Engineering, Universität der Bundeswehr, WernerHeisenbergWeg 39, 85577 Neubiberg  Munich, Germany joachim.gwinner@unibw.de Baasansuren Jadamba Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, U.S.A. bxjsma@rit.edu Akhtar A. Khan Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, U.S.A. aaksma@rit.edu Miguel Sama Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, 28040 Madrid, Spain Our objective is to investigate the inverse problem of identifying variable parameters in certain variational and quasivariational inequalities. To this end we extend a trilinear form based optimization framework that has been used quite effectively for parameter identification in variational equations emerging from partial differential equations. An abstract nonsmooth regularization approach is developed that encompasses the total variation regularization and permits the identification of discontinuous parameters. We investigate the inverse problem in an optimization setting using the outputleast squares formulation. We give existence and convergence results for the optimization problem. We also penalize the variational inequality and arrive at an optimization problem for which the constraint variational inequality is replaced by the penalized equation. For this case, the smoothness of the parametertosolution map is studied and convergence analysis and optimality conditions are given. We also discretize the identification problem for quasivariational inequalities and give the convergence analysis for the discrete problems. Examples are given to justify the theoretical framework. Keywords: Inverse problems, illposed problems, regularization, total variation, parameter identification, output leastsquares, variational inequalities, quasivariational inequalities, penalization, finite elements. MSC: 49J40, 49N45, 90C26 [ Fulltextpdf (163 KB)] for subscribers only. 