Abstract

Our objective is to investigate the inverse problem of identifying variable parameters in certain variational and quasi-variational inequalities. To this end we extend a trilinear form based optimization framework that has been used quite effectively for parameter identification in variational equations emerging from partial differential equations. An abstract nonsmooth regularization approach is developed that encompasses the total variation regularization and permits the identification of discontinuous parameters. We investigate the inverse problem in an optimization setting using the output-least squares formulation. We give existence and convergence results for the optimization problem. We also penalize the variational inequality and arrive at an optimization problem for which the constraint variational inequality is replaced by the penalized equation. For this case, the smoothness of the parameter-to-solution map is studied and convergence analysis and optimality conditions are given. We also discretize the identification problem for quasi-variational inequalities and give the convergence analysis for the discrete problems. Examples are given to justify the theoretical framework.

Author information

Contact details are reproduced from the original publication and may be historical.

Joachim Gwinner

Institute of Mathematics, Department of Aerospace Engineering, Universität der Bundeswehr, Werner-Heisenberg-Weg 39, 85577 Neubiberg - Munich, Germany

joachim.gwinner@unibw.de

Baasansuren Jadamba

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

bxjsma@rit.edu

Akhtar A. Khan

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

aaksma@rit.edu

Miguel Sama

Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, 28040 Madrid, Spain

Keywords

  • Inverse problems
  • ill-posed problems
  • regularization
  • total variation
  • parameter identification
  • output least-squares
  • variational inequalities
  • quasi-variational inequalities
  • penalization
  • finite elements

Mathematics Subject Classification

49J40, 49N45, 90C26

References

  1. J. Abouchabaka, R. Aboulaïch, A. Nachaoui, A. Souissi: Quasi-variational inequality and shape optimization for solution of a free boundary problem, COMPEL 18(2) (1999) 143–164.https://doi.org/10.1108/03321649910264154Google Scholar
  2. S. Adly, M. Bergounioux, M. Ait Mansour: Optimal control of a quasi-variational obstacle problem, J. Global Optim. 47(3) (2010) 421–435.https://doi.org/10.1007/s10898-008-9366-yGoogle Scholar
  3. H. Attouch, G. Buttazzo, G. Michaille: Variational Analysis in Sobolev and BV Spaces, 2nd edition, MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia (2014).https://doi.org/10.1137/1.9781611973488Google Scholar
  4. A. Barbagallo: Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems, J. Global Optim. 40(1-3) (2008) 29–39.https://doi.org/10.1007/s10898-007-9194-5Google Scholar
  5. J. W. Barrett, L. Prigozhin: A quasi-variational inequality problem in superconductivity, Math. Models Methods Appl. Sci. 20(5) (2010) 679–706.https://doi.org/10.1142/S0218202510004404Google Scholar
  6. J. W. Barrett, L. Prigozhin: A quasi-variational inequality problem arising in the modeling of growing sandpiles, ESAIM Math. Model. Numer. Anal. 47(4) (2013) 1133–1165.https://doi.org/10.1051/m2an/2012062Google Scholar
  7. G. Bayada, M. El Alaoui Talibi: An application of the control by coefficients in a variational inequality for hydrodynamic lubrication, Nonlinear Anal. Real World Appl. 1(2) (2000) 315–328.https://doi.org/10.1016/S0362-546X(99)00108-XGoogle Scholar
  8. A. Borzì, C. Kanzow: Formulation and numerical solution of Nash equilibrium multiobjective elliptic control problems, SIAM J. Control Optim. 51(1) (2013) 718–744.https://doi.org/10.1137/120864921Google Scholar
  9. S. Boulaaras, M. A. Bencheikh le Hocine, M. Haiour: The finite element approximation in a system of parabolic quasi-variational inequalities related to management of energy production with mixed boundary condition, Comput. Math. Model. 25(4) (2014) 530–543.https://doi.org/10.1007/s10598-014-9247-9Google Scholar
  10. R. A. Capatina, M. Cocu: Internal approximation of quasi-variational inequalities, Numer. Math. 59(4) (1991) 385–398.https://doi.org/10.1007/BF01385787Google Scholar
  11. Z. Chen, J. Zou: An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems, SIAM J. Control Optim. 37(3) (1999) 892–910.https://doi.org/10.1137/S0363012997318602Google Scholar
  12. M. Cocu: Existence of solutions of Signorini problems with friction, Int. J. Eng. Sci. 22 (1984) 567–575.https://doi.org/10.1016/0020-7225(84)90058-2Google Scholar
  13. P. Daniele: A remark on a dynamic model of a quasi-variational inequality, Rend. Circ. Mat. Palermo (2) Suppl. 48 (1997) 91–100.Google Scholar
  14. A. Dreves, J. Gwinner: Jointly convex generalized Nash equilibria and elliptic multiobjective optimal control, J. Optim. Theory Appl. 168(3) 1065–1086.https://doi.org/10.1007/s10957-015-0788-7Google Scholar
  15. C. Eck, J. Jarusek: Solvability of static contact problems with Coulomb friction for orthotropic material, J. Elasticity 93(1) (2008) 93–104.https://doi.org/10.1007/s10659-008-9168-yGoogle Scholar
  16. T. Fukao, N. Kenmochi: Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint, Discrete Contin. Dyn. Syst. 35(6) (2015) 2523–2538.https://doi.org/10.3934/dcds.2015.35.2523Google Scholar
  17. F. Giannessi: Separation of sets and gap functions for quasi-variational inequalities, in: Variational inequalities and network equilibrium problems, Erice 1994, Plenum Publishers, New York (1995) 101–121.https://doi.org/10.1007/978-1-4899-1358-6_9Google Scholar
  18. F. Giannessi, A. A. Khan: Regularization of non-coercive quasi variational inequalities, Control Cybernet. 29(1) (2000) 91–110.Google Scholar
  19. M. S. Gockenbach, A. A. Khan: An abstract framework for elliptic inverse problems. I: An output least-squares approach, Math. Mech. Solids 12(3) (2007) 259–276.https://doi.org/10.1177/1081286505055758Google Scholar
  20. G. A. González: Theoretical framework of an identification problem for an elliptic variational inequality with bilateral restrictions, J. Comput. Appl. Math. 197(1) (2006) 245–252.https://doi.org/10.1016/j.cam.2005.11.002Google Scholar
  21. J. Gwinner: On the penalty method for constrained variational inequalities, in: Optimization: Theory and Algorithms, Confolant 1981, Lecture Notes Pure Appl. Math. 86 (1983) 197–211.Google Scholar
  22. J. Gwinner: A discretization theory for monotone semicoercive problems and finite element convergence for p-harmonic Signorini problems, Z. Angew. Math. Mech. 74(9) (1994) 417–427.https://doi.org/10.1002/zamm.19940740917Google Scholar
  23. J. Gwinner, N. Ovcharova: From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics, Optimization 64(8) (2015) 1683–1702.https://doi.org/10.1080/02331934.2014.1001758Google Scholar
  24. P. T. Harker: Generalized Nash games and quasi-variational inequalities, Eur. J. Oper. Res. 54(1) (1991) 81–94.https://doi.org/10.1016/0377-2217(91)90325-PGoogle Scholar
  25. M. Hintermüller: Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization, M2AN Math. Model. Numer. Anal. 35(1) (2001) 129–152.https://doi.org/10.1051/m2an:2001109Google Scholar
  26. M. Hintermüller, T. Surowiec: A PDE-constrained generalized Nash equilibrium problem with pointwise control and state constraints, Pac. J. Optim. 9(2) (2013) 251–273.Google Scholar
  27. K. H. Hoffmann, J. Sprekels: On the identification of parameters in general variational inequalities by asymptotic regularization, SIAM J. Math. Anal. 17(5) (1986) 1198–1217.https://doi.org/10.1137/0517083Google Scholar
  28. B. Hu: A quasi-variational inequality arising in elastohydrodynamics, SIAM J. Math. Anal. 21(1) (1990) 18–36.https://doi.org/10.1137/0521002Google Scholar
  29. K. Ito, K. Kunisch: The augmented Lagrangian method for parameter estimation in elliptic systems, SIAM J. Control Optim. 28(1) (1990) 113–136.https://doi.org/10.1137/0328006Google Scholar
  30. B. Jadamba, A. A. Khan, G. Rus, M. Sama, B. Winkler: 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(5) (2014) 1486–1510.https://doi.org/10.1137/130928261Google Scholar
  31. B. Jadamba, A. A. Khan, M. Sama: Inverse problems of parameter identification in partial differential equations, in: Mathematics in Science and Technology, World Sci. Publ., Hackensack (2011) 228–258.https://doi.org/10.1142/9789814338820_0009Google Scholar
  32. A. Kadoya, N. Kenmochi, M. Niezgódka: Quasi-variational inequalities in economic growth models with technological development, Adv. Math. Sci. Appl. 24(1) (2014) 185–214.Google Scholar
  33. T. Kaerkkaeinen: An equation error method to recover diffusion from the distributed observation, Inverse Problems 13(4) (1997) 1033–1051.https://doi.org/10.1088/0266-5611/13/4/009Google Scholar
  34. A. A. Khan, D. Motreanu: Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities, J. Optim. Theory Appl. 167(3) (2015) 1136–1161.https://doi.org/10.1007/s10957-015-0825-6Google Scholar
  35. A. A. Khan, C. Tammer, C. Zalinescu: Regularization of quasi-variational inequalities, Optimization 64(8) (2015) 1703–1724.https://doi.org/10.1080/02331934.2015.1028935Google Scholar
  36. I. Knowles: Parameter identification for elliptic problems, J. Comput. Appl. Math. 131(1-2) (2001) 175–194.https://doi.org/10.1016/S0377-0427(00)00275-2Google Scholar
  37. F. Lefèvre, N. Nassif: A quasi-variational inequality for the simulation of a MODFET transistor, COMPEL 18(2) (1999) 175–186.https://doi.org/10.1108/03321649910264181Google Scholar
  38. F. Lenzen, F. Becker, J. Lellmann, S. Petra, C. Schnörr: A class of quasi-variational inequalities for adaptive image denoising and decomposition, Comput. Optim. Appl. 54(2) (2013) 371–398.https://doi.org/10.1007/s10589-012-9456-0Google Scholar
  39. A. Maugeri: Variational and quasi-variational inequalities in network flow models. Recent developments in theory and algorithms, in: Variational Inequalities and Network Equilibrium Problems, Erice 1994, Plenum Publishers, New York (1995) 195–211.https://doi.org/10.1007/978-1-4899-1358-6_15Google Scholar
  40. A. Maugeri, F. Raciti: On existence theorems for monotone and nonmonotone variational inequalities, J. Convex Analysis 16(3-4) (2009) 899–911.https://doi.org/10.68381/jca16055Google Scholar
  41. D. Motreanu, M. Sofonea: Quasivariational inequalities and applications in frictional contact problems with normal compliance, Adv. Math. Sci. Appl. 10(1) (2000) 103–118.Google Scholar
  42. K. Naito: A system of quasivariational inequalities and its application to reaction diffusion equations, Publ. Res. Inst. Math. Sci. 21(4) (1985) 677–698.https://doi.org/10.2977/prims/1195178924Google Scholar
  43. N. R. Nassif: On the existence and uniqueness of solutions for a class of quasivariational inequalities to solve reverse-biased semiconductor devices, Ann. Mat. Pura Appl. (4) 144 (1986) 95–112.https://doi.org/10.1007/BF01760813Google Scholar
  44. N. Ovcharova, J. Gwinner: A study of regularization techniques of nondifferentiable optimization in view of application to hemivariational inequalities, J. Optim. Theory Appl. 162(3) (2014) 754–778.https://doi.org/10.1007/s10957-014-0521-yGoogle Scholar
  45. L. Prigozhin: Variational inequalities in critical-state problems, Physica D 197(3-4) (2004) 197–210.https://doi.org/10.1016/j.physd.2004.07.001Google Scholar
  46. J.-F. Rodrigues: On a quasi-variational inequality arising in semiconductor theory, Rev. Mat. Univ. Complut. Madrid 5(1) (1992) 137–151.https://doi.org/10.5209/rev_REMA.1992.v5.n1.17946Google Scholar
  47. M. Sofonea, A. Matei: History-dependent quasi-variational inequalities arising in contact mechanics, European J. Appl. Math. 22(5) (2011) 471–491.https://doi.org/10.1017/S0956792511000192Google Scholar
  48. B. A. Ton: Time-dependent quasi-variational inequalities and the Nash equilibrium, Nonlinear Anal. 41(7-8, Ser. A: Theory Methods) (2000) 1057–1081.https://doi.org/10.1016/S0362-546X(98)00328-9Google Scholar
  49. W. P. Ziemer: Weakly Differentiable Functions, Graduate Texts in Mathematics 120, Springer-Verlag, New York (1989).https://doi.org/10.1007/978-1-4612-1015-3Google Scholar

Suggested citation

J. Gwinner, B. Jadamba, A. A. Khan, M. Sama. “Identification in Variational and Quasi-Variational Inequalities.” Journal of Convex Analysis 25 (2018), No. 2, 545–569. https://doi.org/10.68381/jca25035

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.