Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.