Variational and Quasi-Variational Inequalities

Gwinner, Joachim; Jadamba, Baasansuren; Khan, Akhtar A.; Raciti, Fabio

doi:10.1201/9781315228969-4

Cited by 9 publications

(12 citation statements)

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“…However, in recent years, motivated by various applications, inverse problems in variational and quasi-variational inequalities have attracted a lot of attention. To mention a few of the recent contributions, we note that Gwinner-Jadamba-Khan-Sama [13] examined the inverse problem in an optimization setting using the output-least squares formulation, and obtained existence as well as convergence results for the optimization problem. Clason-Khan-Sama-Tammer [10] explored the inverse problem of parameter identification in noncoercive variational problems via the output least-squares and the modified output least-squares objectives.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasivariational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of p-Laplacian type.

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Section: Introductionmentioning

confidence: 99%

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

2019

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“…The partial derivative DeA(u, e, f ) is given by ∂e → T (δe)u. Then with the partial derivative DeL and similar to the proof of [17,Theorem 3.6] we obtain using the symmetry of t(e, •, •) (22).…”

mentioning

confidence: 84%

“…Similarly as above DwH(f, •)(w) : V → V is seen to be surjective and the differentiability follows from the implicit function theorem. Again from (17) we have…”

Section: Similarly the Differentiability Of The Mapmentioning

confidence: 99%

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An Optimization Approach to Parameter Identification in Variational Inequalities of Second Kind -- II

Gwinner¹

2020

Preprint

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This paper continues the work of [16] and is concerned with the inverse problem of parameter identification in variational inequalities of the second kind that does not only treat the parameter linked to a bilinear form, but importantly also the parameter linked to a nonlinear non-smooth function. Here we specify the abstract framework of [16] and cover frictional contact problems as well as other non-smooth problems from continuum mechanics. The optimization approach of [16] to the inverse problem using the output-least squares formulation involves the variational inequality of the second kind as constraint. Here we use regularization technics of nondifferentiable optimization from [10], regularize the nonsmooth part in the variational inequality and arrive at an optimization problem for which the constraint variational inequality is replaced by the regularized variational equation. For this case, the smoothness of the parameter-to-solution map is studied and convergence analysis and optimality conditions are given.

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“…We note here that-to the best of our knowledge-the study of the stability of minimal and maximal solutions of QVIs and the optimal control thereof, with both being focus topics of this work, have not yet been treated in the literature. We further note that the stability of the solution set is also of relevance in identification problems involving QVIs; see [27].…”

Section: Introductionmentioning

confidence: 97%

Stability of the solution set of quasi-variational inequalities and optimal control

Alphonse¹,

Hintermüller²,

Rautenberg³

2019

Preprint

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For a class of quasivariational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements to the solution set of the QVI with respect to monotonic perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.

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Variational and Quasi-Variational Inequalities

Cited by 9 publications

References 0 publications

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

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

An Optimization Approach to Parameter Identification in Variational Inequalities of Second Kind -- II

Stability of the solution set of quasi-variational inequalities and optimal control

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