Inverse final observation problems for Maxwell’s equations in the quasi-stationary magnetic approximation and stable sequential Lagrange principles for their solving. (English. Russian original) Zbl 1369.49038
Comput. Math. Math. Phys. 57, No. 2, 189-210 (2017); translation from Zh. Vychisl. Mat. Mat. Fiz. 57, No. 2, 187-209 (2017).
Summary: 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.
MSC:
| 49M25 | Discrete approximations in optimal control |
| 90C25 | Convex programming |
| 49N45 | Inverse problems in optimal control |
Keywords:
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 ruleReferences:
| [1] | L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Fizmatlit, Moscow, 1982; Butterworth-Heinemann, Oxford, 1984). · Zbl 0122.45002 |
| [2] | J.-L. Coulomb and J.-K. Sabonnadiere, CAO en electrotechnique (Hermes, Paris, 1985; Mir, Moscow, 1988). |
| [3] | A. Alonso Rodriguez and A. Valli, Eddy Current Approximation of Maxwell Equations: Theory, Algorithms, and Applications (Springer-Verlag, Milan, Italia, 2010). · Zbl 1204.78001 · doi:10.1007/978-88-470-1506-7 |
| [4] | M. P. Galanin and Yu. P. Popov, Quasi-stationary Electromagnetic Fields in Inhomogeneous Media (Fizmatlit, Moscow, 1995) [in Russian]. · Zbl 0872.00013 |
| [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). · Zbl 0366.35068 · doi:10.1070/RM1976v031n06ABEH001582 |
| [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]. · Zbl 0595.35103 |
| [7] | V. G. Romanov and S. I. Kabanikhin, Inverse Problems in Geoelectrics (Nauka, Moscow, 1991) [in Russian]. |
| [8] | P. Ola, L. Paivarinta, and E. Somersalo, “An inverse boundary value problem in electrodynamics,” Duke Math. J. 70, 617-653 (1993). · Zbl 0804.35152 · doi:10.1215/S0012-7094-93-07014-7 |
| [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). · Zbl 1121.35141 · doi:10.1007/s11401-005-0572-3 |
| [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). · Zbl 1247.65143 |
| [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). · Zbl 1426.78025 · doi:10.1088/0266-5611/26/10/105002 |
| [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). · doi:10.1134/S1990478911040028 |
| [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). · doi:10.1049/ip-smt:19951922 |
| [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). · Zbl 1173.78300 · doi:10.1016/j.cma.2006.10.008 |
| [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). · Zbl 1143.35377 · doi:10.1002/mma.900 |
| [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). · Zbl 1196.78025 · doi:10.1007/s00607-009-0049-4 |
| [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). · Zbl 1243.78043 · doi:10.1016/j.cam.2012.01.030 |
| [18] | R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1979; Mir, Moscow, 1981). · Zbl 0426.35003 |
| [19] | V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations (Springer-Verlag, New York, 1986). · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5 |
| [20] | A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics (Marcel Dekker, New York, 2000). · Zbl 0947.35173 |
| [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). · Zbl 1188.65074 · doi:10.1515/jiip.2007.045 |
| [22] | V. Isakov, Inverse Problems for Partial Differential Equations (Springer, New York, 2006). · Zbl 1092.35001 |
| [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). · Zbl 07811678 · doi:10.1134/S096554250707007X |
| [24] | G. V. Alekseev, Optimization in Stationary Problems of Heat and Mass Transfer and Magnetic Hydrodynamics (Nauchnyi Mir, Moscow, 2010) [in Russian]. |
| [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). · Zbl 1121.49038 |
| [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). · Zbl 1047.78004 |
| [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). · Zbl 1249.78005 · doi:10.1134/S0965542512030116 |
| [28] | M. I. Sumin, “Regularized parametric Kuhn-Tucker theorem in a Hilbert space,” Comput. Math. Math. Phys. 51 (9), 1489-1509 (2011). · Zbl 1274.90266 · doi:10.1134/S0965542511090156 |
| [29] | Sumin, M. I., On the stable sequential Kuhn-Tucker theorem and its applications, 1334-1350 (2012) |
| [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). · Zbl 1313.90252 · doi:10.1134/S0965542514010138 |
| [31] | F. P. Vasil’ev, Optimization Methods (MTsNMO, Moscow, 2011) [in Russian]. |
| [32] | A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977; Nauka, Moscow, 1986). · Zbl 0354.65028 |
| [33] | M. I. Sumin, “Duality-based regularization in a linear convex mathematical programming problem,” Comput. Math. Math. Phys. 47 (4), 579-600 (2007). · Zbl 1210.49022 · doi:10.1134/S0965542507040045 |
| [34] | M. I. Sumin, Ill-Posed Problems and Solution Methods (Nizhegorod. Gos. Univ., Nizhny Novgorod, 2009) [in Russian]. |
| [35] | K. Arrow, L. Hurwicz, and H. Uzawa, Studies in Nonlinear Programming (Stanford Univ. Press, Stanford, 1958; Inostrannaya Literatura, Moscow, 1962). · Zbl 0091.16002 |
| [36] | V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control (Nauka, Moscow, 1979) [in Russian]. · Zbl 0516.49002 |
| [37] | H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974; Mir, Moscow, 1978). · Zbl 0289.47029 |
| [38] | C. Weber, “A local compactness theorem for Maxwell’s equations,” Math. Methods Appl. Sci. 2, 12-25 (1980). · Zbl 0432.35032 · doi:10.1002/mma.1670020103 |
| [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). · Zbl 0943.49018 |
| [40] | J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972; Nauka, Moscow, 1977). · Zbl 0253.49001 |
| [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). |
| [42] | L. V. Kantorovich and G. P. Akilov, Functional Analysis (Pergamon, Oxford, 1982; Nevskii Dialekt, St. Petersburg, 2004). · Zbl 0484.46003 |
| [43] | P. D. Loewen, Optimal Control via Nonsmooth Analysis (Am. Math. Soc., Providence, R.I., 1993). · Zbl 0874.49002 |
| [44] | A. B. Bakushinskii and A. V. Goncharskii, Iterative Methods for Ill-Posed Problems (Nauka, Moscow, 1989) [in Russian]. · Zbl 0676.65050 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
