Coupling Topological Gradient and Gauss-Newton Method

摘要

Topological asymptotic analysis is an emerging method that has been applied with success to shape optimization and shape inverse problems. However, it is not suitable for solving severely ill-posed inverse problems. It is a short term approach and it fails when applied to small inclusions detection in elastographic imaging. We show in this paper that it is possible to solve this problem by coupling Gauss-Newton and topological gradient methods in a natural way. The data is the displacement under a small compression, only one component of the displacement is given. The inversion problem is solved in two steps. In a first step, we obtain a weight vector for the observations; this is performed by a dual Gauss-Newton method. The second step consists of computing the topological sensitivity relative to the insertion of an inclusion in the medium (a stiff disk), the cost function being weighted with the result of the first step. This method is applied to numerical experiments.