This text offers a simultaneous presentation of the theory and numerical treatment of inverse problems for Maxwell's equations. Inverse problems are central to many areas of science and technology such as geophysical exploration, remote sensing, near-surface radar location, di-electric logging and medical imaging. The basic idea of inverse methods is to extract from the evaluation of measured electromagnetic fields the details of the medium considered. The inverse problems are investigated not only for Maxwell's equations, but also for their quasistationary approximation and, in the case of harmonic dependence, in time. Starting with the simplest one-dimensional inverse problems, the book leads its readers to more complicated multi-dimensional ones studied for media of various kinds. The unique solvability of a number of the considered problems is shown, as well as the stability of their solutions. The numberical analysis ranges from the finited-difference scheme inversion to the linearization methods, and to a dynamic variant of the Gel'fand-Levitan methods. The book should be of interest to researchers in the fields of applied mathematics and geophysics.
The applications in the book should be of use to experimentalists and engineers.
Part 1 Cauchy problem for Maxwell's equations: Maxwell's equations as a hyperbolic symmetric system; structure of the Cauchy problem solution in case of the current located on the media interface. Part 2 One-dimensional inverse problems: structure of the Fourier-image of the Cauchy problem solution for one-dimensional medium in case of the current located at a point; the problem of determining the medium permittivity; the problem of determining the conductivity co-efficient; the problem of determining all the co-efficients of Maxwell's equations. Part 3 Multi-dimensional inverse problems: linearization method applied to the inverse problems; investigation of the linearized problem of determining the permittivity co-efficient; unique solvability theorem for a two-dimensional problem of determining the conductivity co-efficient analytic in one variable; on the uniqueness of the solution of three-dimensional inverse problems. Part 4 Inverse problems in the case of source periodic in time: one-dimensional inverse problems; linear one-dimensional inverse problem; linearized three-dimensional inverse problems. Part 5 Inverse problems for quasi-stationary Maxwell's equations: on correspondence between the solutions of quasi-stationary and wave Maxwell's equations; a one-dimensional inverse problem of determining the conductivity and permeability co-efficients; the one-dimensional inverse problem for wave-quasi-stationary system of equations. Part 6 The inverse problems for the simplest anisotropic media: on the uniqueness of determination of permittivity and permeability in anisotropic media; on the problem of determining permittivity and conductivity tensors. Part 7 Numerical methods. Part 8 Convergence results. Part 9 Examples (Part contents)
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