Inverse final observation problems for Maxwell’s equations in the quasi-stationary magnetic approximation and stable sequential Lagrange principles for their solving

  • A. V. Kalinin
  • M. I. Sumin
  • A. A. Tyukhtina


An initial–boundary value problem for Maxwell’s equations in the quasi-stationary magnetic approximation is investigated. Special gauge conditions are presented that make it possible to state the problem of independently determining the vector magnetic potential. The well-posedness of the problem is proved under general conditions on the coefficients. For quasi-stationary Maxwell equations, final observation problems formulated in terms of the vector magnetic potential are considered. They are treated as convex programming problems in a Hilbert space with an operator equality constraint. Stable sequential Lagrange principles are stated in the form of theorems on the existence of a minimizing approximate solution of the optimization problems under consideration. The possibility of applying algorithms of dual regularization and iterative dual regularization with a stopping rule is justified in the case of a finite observation error.


Maxwell’s equations in quasi-stationary magnetic approximation vector potential gauge conditions inverse final observation problem retrospective inverse problem convex programming Lagrange principle dual regularization iterative dual regularization stopping rule 


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  1. 1.
    L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Fizmatlit, Moscow, 1982; Butterworth-Heinemann, Oxford, 1984).MATHGoogle Scholar
  2. 2.
    J.-L. Coulomb and J.-K. Sabonnadiere, CAO en electrotechnique (Hermes, Paris, 1985; Mir, Moscow, 1988).Google Scholar
  3. 3.
    A. Alonso Rodriguez and A. Valli, Eddy Current Approximation of Maxwell Equations: Theory, Algorithms, and Applications (Springer-Verlag, Milan, Italia, 2010).CrossRefMATHGoogle Scholar
  4. 4.
    M. P. Galanin and Yu. P. Popov, Quasi-stationary Electromagnetic Fields in Inhomogeneous Media (Fizmatlit, Moscow, 1995) [in Russian].MATHGoogle Scholar
  5. 5.
    V. I. Dmitriev, A. S. Il’inskii, and A. G. Sveshnikov, “The developments of mathematical methods for the study of direct and inverse problems in electrodynamics,” Russ. Math. Surv. 31 (6), 133–152 (1976).CrossRefMATHGoogle Scholar
  6. 6.
    V. G. Romanov, S. I. Kabanikhin, and T. P. Pukhnacheva, Inverse Problems in Electrodynamics (Vychisl. Tsentr Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 1984) [in Russian].MATHGoogle Scholar
  7. 7.
    V. G. Romanov and S. I. Kabanikhin, Inverse Problems in Geoelectrics (Nauka, Moscow, 1991) [in Russian].Google Scholar
  8. 8.
    P. Ola, L. Paivarinta, and E. Somersalo, “An inverse boundary value problem in electrodynamics,” Duke Math. J. 70, 617–653 (1993).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    S. Li and M. Yamamoto, “An inverse problem for Maxwell’s equations in anisotropic media,” Chin. Ann. Math. B 28 (1), 35–54 (2007).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    A. Alonso Rodriguez, J. Camano, and A. Valli, “Inverse source problems for eddy current equations,” Inverse Probl. 28 (1), 015006-1–015006-15 (2012).Google Scholar
  11. 11.
    Y. Shestopalov and Y. Smirnov, “Existence and uniqueness of a solution to the inverse problem of the complex permittivity reconstruction of a dielectric body in a waveguide,” Inverse Probl. 26 (10), 1–14 (2010).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    G. V. Alekseev and R. V. Brizitskii, “Theoretical analysis of boundary control extremal problems for Maxwell’s equations,” J. Appl. Ind. Math. 5, 478–490 (2011).MathSciNetCrossRefGoogle Scholar
  13. 13.
    P. Fernandes, “General approach to prove the existence and uniqueness of the solution in vector potential formulations of 3-D eddy current problems,” IEE Proc.-Sci. Meas. Technol. 142, 299–306 (1995).CrossRefGoogle Scholar
  14. 14.
    O. Biro and A. Valli, “The Coulomb gauged vector potential formulation for the eddy-current problem in general geometry: Well-posedness and numerical approximation,” Comput. Meth. Appl. Mech. Eng. 196, 890–904 (2007).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    P. Fernandes and A. Valli, “Lorenz-gauged vector potential formulations for the time-harmonic eddy-current problem with L8-regularity of material properties,” Math. Methods Appl. Sci. 31, 71–98 (2008).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    T. Kang and K. I. Kim, “Fully discrete potential-based finite element methods for a transient eddy current problem,” Computing 85, 339–362 (2009).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    J. Camano and R. Rodriguez, “Analysis of a FEM-BEM model posed on the conducting domain for the timedependent eddy current problem,” J. Comput. Appl. Math. 236, 3084–3100 (2012).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1979; Mir, Moscow, 1981).MATHGoogle Scholar
  19. 19.
    V. Girault and P. Raviart, Finite Element Methods for Navier–Stokes Equations (Springer-Verlag, New York, 1986).CrossRefMATHGoogle Scholar
  20. 20.
    A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics (Marcel Dekker, New York, 2000).MATHGoogle Scholar
  21. 21.
    A. Prilepko, S. Piskarev, and S.-Y. Shaw, “On approximation of inverse problem for abstract parabolic differential equations in Banach spaces,” J. Inverse Ill-Posed Probl. 15 (8), 831–851 (2007).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    V. Isakov, Inverse Problems for Partial Differential Equations (Springer, New York, 2006).MATHGoogle Scholar
  23. 23.
    V. I. Agoshkov and E. A. Botvinovskii, “Numerical solution of the nonstationary Stokes system by methods of adjoint-equation theory and optimal control theory,” Comput. Math. Math. Phys. 47 (7), 1142–1157 (2007).MathSciNetCrossRefGoogle Scholar
  24. 24.
    G. V. Alekseev, Optimization in Stationary Problems of Heat and Mass Transfer and Magnetic Hydrodynamics (Nauchnyi Mir, Moscow, 2010) [in Russian].Google Scholar
  25. 25.
    M. I. Sumin, “A regularized gradient dual method for solving inverse problem of final observation for a parabolic equation,” Comput. Math. Math. Phys. 44 (11), 1903–1921 (2004).MathSciNetMATHGoogle Scholar
  26. 26.
    A. V. Kalinin and A. A. Kalinkina, “Quasi-stationary initial–boundary value problems for Maxwell’s equations,” Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr. 26 (1), 21–38 (2003).MATHGoogle Scholar
  27. 27.
    M. I. Ivanov, I. A. Kremer, and M. V. Urev, “Regularization method for solving the quasi-stationary Maxwell equations in an inhomogeneous conducting medium,” Comput. Math. Math. Phys. 52 (3), 476–488 (2012).MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    M. I. Sumin, “Regularized parametric Kuhn–Tucker theorem in a Hilbert space,” Comput. Math. Math. Phys. 51 (9), 1489–1509 (2011).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    M. I. Sumin, “On the stable sequential Kuhn–Tucker theorem and its applications,” Appl. Math. A 3 (10), special issue “Optimization,” 1334–1350 (2012).Google Scholar
  30. 30.
    M. I. Sumin, “Stable sequential convex programming in a Hilbert space and its application for solving unstable problems,” Comput. Math. Math. Phys. 54 (1), 22–44 (2014).MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    F. P. Vasil’ev, Optimization Methods (MTsNMO, Moscow, 2011) [in Russian].Google Scholar
  32. 32.
    A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977; Nauka, Moscow, 1986).MATHGoogle Scholar
  33. 33.
    M. I. Sumin, “Duality-based regularization in a linear convex mathematical programming problem,” Comput. Math. Math. Phys. 47 (4), 579–600 (2007).MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    M. I. Sumin, Ill-Posed Problems and Solution Methods (Nizhegorod. Gos. Univ., Nizhny Novgorod, 2009) [in Russian].Google Scholar
  35. 35.
    K. Arrow, L. Hurwicz, and H. Uzawa, Studies in Nonlinear Programming (Stanford Univ. Press, Stanford, 1958; Inostrannaya Literatura, Moscow, 1962).MATHGoogle Scholar
  36. 36.
    V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control (Nauka, Moscow, 1979) [in Russian].MATHGoogle Scholar
  37. 37.
    H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974; Mir, Moscow, 1978).MATHGoogle Scholar
  38. 38.
    C. Weber, “A local compactness theorem for Maxwell’s equations,” Math. Methods Appl. Sci. 2, 12–25 (1980).MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    M. I. Sumin, “Suboptimal control of distributed-parameter systems: Minimizing sequences and the value function,” Comput. Math. Math. Phys. 37 (1), 21–39 (1997).MathSciNetMATHGoogle Scholar
  40. 40.
    J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972; Nauka, Moscow, 1977).MATHGoogle Scholar
  41. 41.
    A. V. Kalinin and A. A. Tyukhtina, “On the uniqueness of the solution to a retrospective inverse problem for Maxwell’s equations in the quasi-stationary magnetic approximation,” Vestn. Nizhegorod. Gos. Univ., No. 4, 263–270 (2014).Google Scholar
  42. 42.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis (Pergamon, Oxford, 1982; Nevskii Dialekt, St. Petersburg, 2004).MATHGoogle Scholar
  43. 43.
    P. D. Loewen, Optimal Control via Nonsmooth Analysis (Am. Math. Soc., Providence, R.I., 1993).MATHGoogle Scholar
  44. 44.
    A. B. Bakushinskii and A. V. Goncharskii, Iterative Methods for Ill-Posed Problems (Nauka, Moscow, 1989) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • A. V. Kalinin
    • 1
  • M. I. Sumin
    • 1
  • A. A. Tyukhtina
    • 1
  1. 1.Nizhny Novgorod State UniversityNizhny NovgorodRussia

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