Skip to main content
SpringerLink
Log in

Stable sequential Lagrange principles in the inverse final observation problem for the system of Maxwell equations in the quasistationary magnetic approximation

  • Partial Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We justify the possibility of using stable, with respect to errors in the input data, algorithms of dual regularization and iterative dual regularization for solving the inverse final observation problem for the system of Maxwell equations in the quasistationary magnetic approximation under general conditions on the coefficients, which is treated as an optimal control problem for the differential equation describing the magnetic field intensity with an operator equality constraint. We state a classical parametric Lagrange principle and stable Lagrange principles in sequential form for the posed problem. We present a stopping rule for the iterative process for the stable sequential Lagrange principle in iterative form in the case of finite fixed error in the input data.

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. Alonso Rodriguez, A. and Valli, A., Eddy Current Approximation of Maxwell Equations. Theory, Algorithms and Applications, Milan, 2010.

    Book  MATH  Google Scholar 

  2. Coulomb, J.-L. and Sabonnadiere, J.-K., CAO en electrotechnique, Paris: Hermes Pub., 1985. Translated under the title SAPR v elektrotekhnike, Moscow: Mir, 1988.

    Google Scholar 

  3. Galanin, M.P. and Popov, Yu.P., Kvazistatsionarnye elektromagnitnye polya v neodnorodnykh sredakh (Quasistationary Electromagnetic Fields in Inhomogeneous Media), Moscow: Nauka, 1995.

    Google Scholar 

  4. Prilepko, A., Piskarev, S., and Shaw, S.-Y., On Approximation of Inverse Problem for Abstract Parabolic Differential Equations in Banach Spaces, J. Inverse Ill-Posed Probl., 2007, vol. 15, no. 8, pp. 831–851.

    Article  MathSciNet  MATH  Google Scholar 

  5. Temam, R., Navier–Stokes Equations. Theory and Numerical Analysis, Amsterdam: North-Holland, 1977. Translated under the title Uravneniya Nav’e–Stoksa. Teoriya i chislennyi analiz, Moscow: Mir, 1981.

    MATH  Google Scholar 

  6. Girault, V. and Raviart, P., Finite Element Methods for Navier–Stokes Equations, Berlin, 1986.

    Book  MATH  Google Scholar 

  7. Sumin, M.I., A Regularized Gradient Dual Method for Solving Inverse Problem of Final Observation for a Parabolic Equation, Comput. Math. Math. Phys., 2004, vol. 44, no. 11, pp. 1903–1921.

    MathSciNet  MATH  Google Scholar 

  8. Isakov, V., Inverse Problems for Partial Differential Equations, New York, 2006.

    MATH  Google Scholar 

  9. Agoshkov, V.I. and Botvinovskii, E.A., Numerical Solution of the Nonstationary Stokes System by the Methods of the Theory of Conjugate Equations and Optimal Control Theory, Zh. Vychisl. Mat. Mat. Fiz., 2007, vol. 47, no. 7, pp. 1192–1207.

    MathSciNet  Google Scholar 

  10. Dmitriev, V.I., Il’inskii, A.S., and Sveshnikov, A.G., Development of Mathematical Methods for the Study of Direct and Inverse Problems of Electrodynamics, Uspekhi Mat. Nauk, 1976, vol. 31, no. 6 (192), pp. 123–141.

    MathSciNet  Google Scholar 

  11. Romanov, V.G., Kabanikhin, S.I., and Pukhnacheva, T.P., Obratnye zadachi elektrodinamiki (Inverse Problems of Electrodynamics), Novosibirsk, 1984.

    MATH  Google Scholar 

  12. Li, S. and Yamamoto, M., An Inverse Problem for Maxwell’s Equations in Anisotropic Media, Chinese Ann. Math., 2007, vol. 28B, no. 1, pp. 35–54.

    Article  MathSciNet  MATH  Google Scholar 

  13. Alekseev, G.V., Optimizatsiya v statsionarnykh zadachakh teplomassoperenosa i magnitnoi gidrodinamiki (Optimization in Stationary Problems of Heat and Mass Transport and Magnetic Hydrodynamics), Moscow, 2010.

    Google Scholar 

  14. Rodriguez, A.A, Camano, J., and Valli, A., Inverse Source Problems for Eddy Current Equations, Inverse Problems, 2012, vol. 28, no. 1, pp. 015006–1–015006-15.

    Article  MathSciNet  MATH  Google Scholar 

  15. Sumin, M.I., Regularized Parametric Kuhn–Tucker Theorem in a Hilbert Space, Comput. Math. Math. Phys., 2011, vol. 51, no. 9, pp. 1489–1509.

    Article  MathSciNet  MATH  Google Scholar 

  16. Sumin, M.I., On the Stable Sequential Kuhn–Tucker Theorem and Its Applications, Appl. Math., 2012, vol. 3, no. 10A, pp. 1334–1350.

    Article  Google Scholar 

  17. Sumin, M.I., On the Stable Sequential Lagrange Principle in Convex Programming and Its Application in the Solution of Unstable Problems, Tr. Inst. Mat. Mekh., 2013, vol. 19, no. 4, pp. 231–240.

    MathSciNet  Google Scholar 

  18. Sumin, M.I., Stable Sequential Convex Programming in a Hilbert Space and Its Application for Solving Unstable Problems, Comput. Math. Math. Phys., 2014, vol. 54, no. 1, pp. 22–44.

    Article  MathSciNet  MATH  Google Scholar 

  19. Sumin, M.I., Duality-Based Regularization in a Linear Convex Mathematical Programming Problem, Comput. Math. Math. Phys., 2007, vol. 47, no. 4, pp. 579–600.

    Article  MathSciNet  MATH  Google Scholar 

  20. Arrow, K.J., Hurwicz, L., and Uzawa, H., Studies in Linear and Non-Linear Programming, Stanford: Stanford Univ. Press, 1958. Translated under the title Issledovaniya po lineinomu i nelineinomu programmirovaniyu, Moscow: Inostrannaya Literatura, 1962.

    MATH  Google Scholar 

  21. Landau, L.D. and Lifshits, E.M., Teoreticheskaya fizika. T. 8. Elektrodinamika sploshnykh sred (Theoretical Physics. Vol. 8. Electrodynamics of Continuum), Moscow, 1982.

    Google Scholar 

  22. Cessenat, M., Mathematical Methods in Electromagnetism: Linear Theory and Applications, Singapore, 1996.

    Book  MATH  Google Scholar 

  23. Weber, C., A Local Compactness Theorem for Maxwell’s Equations, Math. Methods Appl. Sci., 1980, vol. 2, pp. 12–25.

    Article  MathSciNet  MATH  Google Scholar 

  24. Kalinin, A.V. and Kalinkina, A.A., L p-Estimates for Vector Fields, Izv. Vyssh. Uchebn. Zaved. Mat., 2004, no. 3, pp. 26–35.

    MathSciNet  MATH  Google Scholar 

  25. Kalinin, A.V. and Kalinkina, A.A., Quasistationary Initial–Boundary Value Problems for the System of Maxwell Equations, Vestnik Nizhegorod. Gos. Univ. Mat. Model. Optim. Upravl., 2003, no. 1 (26), pp. 21–38.

    MATH  Google Scholar 

  26. Sumin, M.I., Nekorrektnye zadachi i metody ikh resheniya. Materialy k lektsiyam dlya studentov starshikh kursov (Ill-Posed Problems and Solution Methods), Nizhny Novgorod State University, 2009.

    Google Scholar 

  27. Gajewskii, H., Gr¨oger, K., and Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Berlin: Akademie-Verlag, 1974. Translated under the title Nelineinye operatornye uravneniya i operatornye differentsial’nye uravneniya, Moscow: Mir, 1978.

    Google Scholar 

  28. Loewen, P.D., Optimal Control via Nonsmooth Analysis, CRM Proceedings and Lecture Notes, vol. 2., Amer. Math. Soc., Providence, RI, 1993.

    Google Scholar 

  29. Alekseev, V.M., Tikhomirov, V.M., and Fomin, S.V., Optimal’noe upravlenie (Optimal Control), Moscow: Nauka, 1979.

    MATH  Google Scholar 

  30. Vasil’ev, F.P., Metody optimizatsii (Optimization Methods), Moscow, 2001.

    Google Scholar 

  31. Bakushinskii, A.B. and Goncharskii, A.V., Nekorrektnye zadachi. Chislennye metody i prilozheniya (Ill-Posed Problems. Numerical Methods and Applications), Moscow, 1989.

    Google Scholar 

  32. Bakushinskii, A.B. and Goncharskii, A.V., Iterativnye metody resheniya nekorrektnykh zadach (Iterative Methods for Solving Ill-Posed Problems), Moscow: Nauka, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. N.I. Lobachevsky Nizhny Novgorod State University, Nizhny Novgorod, Russia

    A. V. Kalinin, M. I. Sumin & A. A. Tyukhtina

Authors
  1. A. V. Kalinin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. I. Sumin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Tyukhtina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. V. Kalinin or M. I. Sumin.

Additional information

Original Russian Text © A.V. Kalinin, M.I. Sumin, A.A. Tyukhtina, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 5, pp. 608–624.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kalinin, A.V., Sumin, M.I. & Tyukhtina, A.A. Stable sequential Lagrange principles in the inverse final observation problem for the system of Maxwell equations in the quasistationary magnetic approximation. Diff Equat 52, 587–603 (2016). https://doi.org/10.1134/S0012266116050062

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266116050062

Search

Navigation