Inverse problems for multi-valued quasi variational inequalities and noncoercive variational inequalities with noisy data

ABSTRACT

We study the inverse problem of identifying a variable parameter in variational and quasi-variational inequalities. We consider a quasi-variational inequality involving a multi-valued monotone map and give a new existence result. We then formulate the inverse problem as an optimization problem and prove its solvability. We also conduct a thorough study of the inverse problem of parameter identification in noncoercive variational inequalities which appear commonly in applied models. We study the inverse problem by posing optimization problems using the output least-squares and the modified output least-squares. Using regularization, penalization, and smoothing, we obtain a single-valued parameter-to-selection map and study its differentiability. We consider optimization problems using the output least-squares and the modified output least-squares for the regularized, penalized and smoothened variational inequality. We give existence results, convergence analysis, and optimality conditions. We provide applications and numerical examples to justify the proposed framework.

KEYWORDS:

• Inverse problems
• variational inequalities
• quasi-variational inequalities
• regularization

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35R30
• 49N45
• 65J20
• 65J22
• 65M30

Acknowledgements

We are grateful to the reviewers for the careful reading that brought substantial improvements to our manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Akhtar Khan is supported by the National Science Foundation under Award No. 1720067. The research of Stanislaw Migorski is supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 CONMECH, and National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262. The research of Miguel Sama is partially supported by Ministerio de Economia y Competitividad (Spain), project MTM2015-68103-P and [grant number 2019-MAT12] (ETSI Industriales, UNED).