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Nonlinear integral equations and the iterative solution for an inverse boundary value problem. (English) Zbl 1086.35139
Summary: Determining the shape of a perfectly conducting inclusion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modelled as an inverse boundary value problem for harmonic functions. We present a novel solution method for such inverse boundary value problems via a pair of nonlinear and ill-posed integral equations for the unknown boundary that can be solved by linearization, i.e., by regularized Newton iterations. We present a mathematical foundation of the method and illustrate its feasibility by numerical examples.

MSC:
35R30Inverse problems for PDE
35J25Second order elliptic equations, boundary value problems
45Q05Inverse problems (integral equations)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
65R20Integral equations (numerical methods)
65R32Inverse problems (integral equations, numerical methods)