On the Identification of Star-Shape Sources From Boundary Measurements Using a Reciprocity Functional

Inverse Problems in Science and Engineering, 17(2) (2009), PII 905612898
http://dx.doi.org/10.1080/17415970802082799

In this article 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 Novikov (Novikov, Sur le probleme inverse du potentiel, Dokl. Akad. Nauk 18 (1938), pp. 165-168) says that star-shaped sources can be reconstructed uniquely from the Cauchy boundary data (see also the work of Isakov e.g. (Isakov, Inverse problems for partial diferential equations, Applied Mathematical Sciences, Vol. 127, Springer-Verlag, New York, 1998). 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.