Skip to main content

Advertisement

SpringerLink
Log in

An optimization approach to parameter identification in variational inequalities of second kind

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This paper 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. A new abstract framework covers frictional contact as well as other non-smooth problems from continuum mechanics. We investigate the dependance of the solution of the forward problem on these parameters and prove Lipschitz continuity results. We formulate an optimization approach to the parameter identification problem and provide a solvability result. Moreover we establish a convergence result for finite dimensional approximation in the optimization approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ben Abda, A., Chaabane, S., El Dabaghi, F., Jaoua, M.: On a non-linear geometrical inverse problem of Signorini type: identifiability and stability. Math. Math. Appl. Sci. 21, 1379–1398 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. Elsevier, Amsterdam (2003)

  3. Adly, S., Bergounioux, M., Ait Mansour, M.: Optimal control of a quasi-variational obstacle problem. J. Glob. Optim. 47(3), 421–435 (2010)

  4. Aßmann, U., Rösch, A.: Identification of an unknown parameter function in the main part of an elliptic partial differential equation. Z. Anal. Anwend. 32(2), 163–178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ayadi, M., Baffico, L., Khaled Gdoura, M., Sassi, T.: Error estimates for Stokes problem with Tresca friction conditions. ESAIM Math. Model. Numer. Anal. 48, 1413–1429 (2014)

  6. Baffico, L., Sassi, T.: Existence result for a fluid structure interaction problem with friction type slip boundary condition. ZAMM Z. Angew. Math. Mech. 95, 831–844 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Birkhäuser Boston Inc, Boston (1989)

    Book  MATH  Google Scholar 

  8. Baumeister, J.: Stable Solution of Inverse Problems. Vieweg, Braunschweig (1987)

    Book  MATH  Google Scholar 

  9. Daniele, P.: Dynamic Networks and Evolutionary Variational Inequalities. Edward Elgar, Cheltenham (2006)

    MATH  Google Scholar 

  10. Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku 888, 199–216 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Glowinski, R.: Numerical methods for nonlinear variational problems. Springer, Berlin (2008). (Reprint of the 1984 original)

  12. Glowinski, R., Wachs, A.: On the Numerical Simulation of Viscoplastic Fluid Flow. Handbook of Numerical Analysis. Vol XVI. Special Volume: Numerical Methods for non-Newtonian Fluids. Elsevier, Amsterdam (2017)

  13. Gockenbach, M.S., Khan, A.A.: An abstract framework for elliptic inverse problems. I. An output least-squares approach. Math. Mech. Solids 12(3), 259–276 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. González, G.A.: Theoretical framework of an identification problem for an elliptic variational inequality with bilateral restrictions. J. Comput. Appl. Math. 197, 245–252 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gwinner, J.: Finite-element convergence for contact problems in plane linear elastostatics. Q. Appl. Math. 50, 11–25 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gwinner, J.: Time dependent variational inequalities—some recent trends. In: Equilibrium Problems and Variational Models (Erice, 2000), Nonconvex Optim. Appl., vol. 68, pp. 225–264. Kluwer Academy Publisher, Norwell, MA (2003)

  17. Gwinner, J.: \(hp\)-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics. J. Comput. Appl. Math. 254, 175–184 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gwinner, J., Ovcharova, N.: From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics. Optimization 64, 1683–1702 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hasanov, A.: Inverse coefficient problems for elliptic variational inequalities with a nonlinear monotone operator. Inverse Problems 14, 1151–1169 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

  21. Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Translated from the Slovak by J. Jarník, Springer, New York (1988)

    Book  MATH  Google Scholar 

  22. Hoffmann, K.H., Sprekels, J.: On the identification of parameters in general variational inequalities by asymptotic regularization. SIAM J. Math. Anal. 17, 1198–1217 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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(5), 1486–1510 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kabanikhin, S.: Inverse and Ill-Posed Problems. de Gruyter, Berlin (2012)

    MATH  Google Scholar 

  25. Kinderlehrer, D. and Stampacchia, G.: An Introduction to Variational Inequalities and their Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Reprint of the 1980 original (2000)

  26. Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems, 2nd edn. Springer, New York (2011)

    Book  MATH  Google Scholar 

  27. Kluge, R.: On some parameter determination problems in variational inequalities and quasi-variational inequalities. Math. Nachr. 89, 127–148 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kress, R.: Linear Integral Equations. Applied Mathematical Sciences, vol. 82, 3rd edn. Springer, New York (2014)

  29. Le, Vy-Khoi., Ang, D.D.: Contact of a viscoelastic body with a rough rigid surface and identification of friction coefficients. J. Inv. Ill-Posed Problems 9, 75–94 (2001)

  30. Maugeri, A., Scrimali, L.: Global Lipschitz continuity of solutions to parameterized variational inequalities. Boll. Unione Mat. Ital. 2(9), 45–69 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Richter, M.: Inverse Probleme. Theorie und Anwendungsbeispiele. Springer, Heidelberg, Springer Spektrum, Grundlagen (2015)

    Book  MATH  Google Scholar 

  32. Rieder, A.: Keine Probleme mit Inversen Problemen. Vieweg, Braunschweig (2003)

    Book  MATH  Google Scholar 

  33. Yên, N.D.: Hölder continuity of solutions to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universitaet der Bundeswehr Muenchen, Neubiberg, Germany

    Joachim Gwinner

Authors
  1. Joachim Gwinner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joachim Gwinner.

Additional information

Dedicated to Professor K.-H. Hoffmann and to the memory of Professor W. Krabs

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gwinner, J. An optimization approach to parameter identification in variational inequalities of second kind. Optim Lett 12, 1141–1154 (2018). https://doi.org/10.1007/s11590-017-1173-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-017-1173-2

Keywords

Search

Navigation