A noniterative reconstruction method based on higher-order topological derivatives for solving a class of inverse problems - Fernandez, Lucas dos Santos (2019)
This thesis concerns a class of inverse potential problems which, in turn, can be seen as reconstruction problems. In particular, the inverse problems, studied here, are characterized by the fact that the desired unknown is a geometrical subset contained in the interior of a reference domain and its reconstruction is obtained from partial measurements of the associated potential. Such type of inverse problem has many applications in physics and engineering. In fact, they are related to the location of pollution sources in a given environment; the identification of monopoles and dipoles in electroencephalography and magnetoencephalography; the detection of the epicenter of a given earthquake, knowing its effect on the earth surface as well as the detection of unknown anomalies present inside the core of the earth from measurements of the gravity field on its surface, for example. In the particular case of this thesis, we consider five inverse reconstruction problems. Two of them are governed by a modified Helmholtz equation. Another two are governed by a Helmholtz equation. The last problem is modeled by a convection-diffusion governing equation. Two problems who have the same governing equation differ from one another by the boundary condition and/or the region of the domain where the measurements of the scalar field of interest are collected. Since these problems are written in the form of ill-posed boundary value problems, we rewrite them as topology optimization problems. In particular, a shape functional measuring the misfit between the solution obtained from the model and the data taken from the boundary measurements is minimized with respect to a set of ball-shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting truncated expansion is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to the noisy data and independent of any initial guess. Furthermore, since this algorithm can accurately approximate the set of unknown geometrical subsets by several balls, it can be used for supplying a good initial guess for more complex iterative approaches such as the ones based on level-sets methods, for instance. We further emphasize that, as part of the topological derivatives-based approach, estimates of the remaining terms arising from the asymptotic expansion of the topologically perturbed functional are given in detail providing a complete asymptotic analysis for each of the problems addressed here. Finally, in order to show the effectiveness of the devised reconstruction algorithm, some numerical experiments in two spatial dimensions are presented, taking into account the reconstruction of multiple anomalies in different scenarios.