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Inverse problems for quasi-variational inequalities

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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.

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References

  1. Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10, 1217–1229 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barbagallo, A., Mauro, P.: A general quasi-variational problem of Cournot–Nash type and its inverse formulation. J. Optim. Theory Appl. 170(2), 476–492 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barrett, J.W., Prigozhin, L.: A quasi-variational inequality problem arising in the modeling of growing sandpiles. ESAIM Math. Model. Numer. Anal. 47, 1133–1165 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bensoussan, A., Lions, J.-L.: Optimal impulse and continuous control: method of nonlinear quasivariational inequalities. Trudy Mat. Inst. Steklov. 134, 5–22 (1975)

    MathSciNet  MATH  Google Scholar 

  5. Boiger, R., Kaltenbacher, B.: An online parameter identification method for time dependent partial differential equations. Inverse Probl. 32(4), 045006 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ciarciá, C., Daniele, P.: New existence theorems for quasi-variational inequalities and applications to financial models. Eur. J. Oper. Res. 251, 288–299 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clason, C.: \(L^{\infty }\) fitting for inverse problems with uniform noise. Inverse Probl. 28(10), 104007 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Giannessi, F., Khan, A.A.: Regularization of non-coercive quasi variational inequalities. Control Cybern. 29, 91–110 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Gockenbach, M.S., Khan, A.A.: An abstract framework for elliptic inverse problems. I. An output least-squares approach. Math. Mech. Solids 12, 259–276 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hintermüller, M.: Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization. M2AN Math. Model. Numer. Anal. 35, 129–152 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jadamba, B., Khan, A.A., Rus, G., Sama, M., Winkler, B.: A new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location. SIAM J. Appl. Math. 74, 1486–1510 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kano, R., Kenmochi, N., Murase, Y.: Existence theorems for elliptic quasi-variational inequalities in Banach spaces. In: Chipot, M., Lin, C.-S., Tsai, D.-H. (eds.) Recent Advances in Nonlinear Analysis, pp. 149–169. World Scientific Publishing, Hackensack (2008)

    Chapter  Google Scholar 

  13. Khan, A.A., Motreanu, D.: Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities. J. Optim. Theory Appl. 167(3), 1136–1161 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khan, A.A., Tammer, C., Zalinescu, C.: Regularization of quasi-variational inequalities. Optimization 64, 1703–1724 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kluge, R. (ed.): On some parameter determination problems and quasi-variational inequalities. In: Theory of Nonlinear Operators, vol. 6, pp. 129–139. Akademie-Verlag, Berlin (1978).

  16. Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: A class of quasi-variational inequalities for adaptive image denoising and decomposition. Comput. Optim. Appl. 54(2), 371–398 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, F., Nashed, M.Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set Valued Anal. 6, 313–344 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mosco, U.: Implicit variational problems and quasi variational inequalities. In: Nonlinear Operators and the Calculus of Variations, Lecture Notes in Mathematics, pp. 83–156. Springer, Berlin (1976)

  19. Motreanu, D., Sofonea, M.: Quasivariational inequalities and applications in frictional contact problems with normal compliance. Adv. Math. Sci. Appl. 10, 103–118 (2000)

    MathSciNet  MATH  Google Scholar 

  20. Nashed, M.Z., Scherzer, O.: Least squares and bounded variation regularization with nondifferentiable functionals. Numer. Funct. Anal. Optim. 19, 873–901 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21, 1303–1314 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Scrimali, L.: Evolutionary quasi-variational inequalities and the dynamic multiclass network equilibrium problem. Numer. Funct. Anal. Optim. 35(7–9), 1225–1244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the reviewers for the valuable remarks. Akhtar Khan is supported by National Science Foundation Grant 005613-002 and RIT’s COS FEAD Grant for 2016–2017.

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Authors and Affiliations

  1. Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY, 14623, USA

    Akhtar A. Khan

  2. Département de Mathématiques, Université de Perpignan, 66860, Perpignan, France

    Dumitru Motreanu

Authors
  1. Akhtar A. Khan
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  2. Dumitru Motreanu
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Correspondence to Akhtar A. Khan.

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Khan, A.A., Motreanu, D. Inverse problems for quasi-variational inequalities. J Glob Optim 70, 401–411 (2018). https://doi.org/10.1007/s10898-017-0597-7

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  • DOI: https://doi.org/10.1007/s10898-017-0597-7

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