Inverse problems for quasi-variational inequalities

Abstract

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.

Inverse problems for nonlinear quasi-variational inequalities with an application to implicit obstacle problems ofp -Laplacian type

“…However, in this paper, we deal with inequality (MQVI) in which the mapping T is not necessarily strongly monotone and linear, and also a convex and lower semicontinuous function appears in the inequality. This results in the additional difficulty that the variational selection is not a single-valued map, and some important properties obtained in [18] are not available.…”

“…Remark 2.3. Hypotheses (H 1 ) and (H 3 ) have been used in [18]. However, assumption (H 2 ) is weaker than the following one required in [18]: for any sequence {b k } ⊂ B with b k → 0 in L, any bounded sequence {u k } ⊂ V , and fixed v ∈ V , we have…”

Inverse problems for vector variational and vector quasi-variational inequalities

“…The direct problem in this setting is to solve the variational or quasi-variational inequality. In contrast, the inverse problem seeks the coefficient from an estimate of the solution of the associated variational problem, see [4,9,17]. Recently, vector variational and vector quasi-variational inequalities appeared as natural and essential generalizations of variational and quasi-variational inequalities, see [7,12,14,18,20,21,22,24,25].…”

“…In the following, we describe the general setting of the inverse problem which will be studied in this paper. A similar framework has been used for scalar quasi-variational inequalities in [17]. The parameter space in this work is denoted by B which is assumed to be a Banach space.…”

“…where κ > 0 is the regularization parameter and R is the regularizing mapping. The above method is motivated by the work [17]. To the best of our knowledge, this is the first paper where the underlying problem of (1.3) is not scalar but vector-valued.…”