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On the identification of star-shape sources from boundary measurements using a reciprocity functional. (English) Zbl 1166.65392
Summary: We consider the numerical problem of shape reconstruction of an unknown characteristic source inside a domain. We consider a steady-state conductivity problem modelled by the Poisson equation, where the heat source is the non-homogeneous characteristic function. A well-known result, back to 1938, by P. Novikov [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 18, 165–168 (1938; Zbl 0018.30901)] says that star-shaped sources can be reconstructed uniquely from the Cauchy boundary data (see also the work of V. Isakov, e.g., [Inverse problems for partial differential equations. Applied Mathematical Sciences. 127. New York, NY: Springer (1998; Zbl 0908.35134)]). Here we consider the reciprocity functional that maps harmonic functions to their integral in the unknown characteristic support. We connect the uniqueness result with the recovery of a function from a certain knowledge of the Fourier coefficients, by taking harmonic monomials as test functions. We also establish a numerical method that consists in an algebraic non-linear system of equations, leading to an approximation of the radial function that defines the boundary of the unknown source. Simulations showing the performance of the numerical method are presented.

MSC:
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
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