This work is concerned with inverse problems under the assumption that the contrast on the value of the relevant physical coefficient between the defect and the matrix is not very big. This is the so-called small amplitude, small contrast or small aspect ratio assumption. Then, following the idea developed by Allaire and Gutiérrez (in Math. Modell. Num. Anal. 2007) for optimal design problems, we make an asymptotic expansion up to second order with respect to the aspect ratio, which allows us to greatly simplify the inverse problem by seeing it as an optimal design problem. We are then able to derive a steepest descent optimization method to minimize the discrepancy between the given boundary measurement and that produced by solving the boundary value problem for a certain spatial distribution of the inclusion. Numerical results show that in the context of heat or electric conduction, the method is very efficient in terms of detecting the location of the inclusion and estimating its volume if it is located not too far from the boundary.
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