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In this paper, we consider an inverse problem of coefficient identification for the Schrodinger equation from the observation data on the exterior boundary. Our aim is to detect the number, the location, the size and the shape of the coefficient with piecewise constant within a body. This problem is nonlinear and ill-posed, thus we should apply stable and elegant reconstruction algorithms in order to improve the corresponding approximation. We give several examples to show the viability of our proposed methods.