Inverse final observation problems for Maxwell’s equations in the quasi-stationary magnetic approximation and stable sequential Lagrange principles for their solving
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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.
KeywordsMaxwell’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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- 2.J.-L. Coulomb and J.-K. Sabonnadiere, CAO en electrotechnique (Hermes, Paris, 1985; Mir, Moscow, 1988).Google Scholar
- 7.V. G. Romanov and S. I. Kabanikhin, Inverse Problems in Geoelectrics (Nauka, Moscow, 1991) [in Russian].Google Scholar
- 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
- 24.G. V. Alekseev, Optimization in Stationary Problems of Heat and Mass Transfer and Magnetic Hydrodynamics (Nauchnyi Mir, Moscow, 2010) [in Russian].Google Scholar
- 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
- 31.F. P. Vasil’ev, Optimization Methods (MTsNMO, Moscow, 2011) [in Russian].Google Scholar
- 34.M. I. Sumin, Ill-Posed Problems and Solution Methods (Nizhegorod. Gos. Univ., Nizhny Novgorod, 2009) [in Russian].Google Scholar
- 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
- 44.A. B. Bakushinskii and A. V. Goncharskii, Iterative Methods for Ill-Posed Problems (Nauka, Moscow, 1989) [in Russian].Google Scholar