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http://cimat.repositorioinstitucional.mx/jspui/handle/1008/917| NUMERICAL SOLUTION OF THE INVERSE SCATTERING PROBLEM USING HIGH LEVEL REPRESENTATIONS | |
| MARIA LUISA DAZA TORRES | |
| Acceso Abierto | |
| Atribución-NoComercial | |
| PROBLEMAS INVERSOS INFERENCIA BAYESIANO | |
| In this thesis, we consider the inverse scattering problem of reconstructing a penetrable homogeneous obstacle from near-field measurements of a scattered wave in 2D. This classical inverse scattering problem is challenging because it is both, ill-posed and nonlinear. On the other hand, since data is polluted with noise, the inverse problem may be regarded as a statistical inference problem. Consequently, we have opted for a Bayesian inferential framework. The Bayesian approach has allowed us to introduce a priori knowledge about the nature of the scatterer, i.e., we assume a star-shaped obstacle with constant refractive index. We rely on Markov Chain Monte Carlo (MCMC) methods to explore the arising posterior distribution. Each sample from the posterior distribution requires the solution of the direct problem, which is equivalent to integrating the Lippmann-Schwinger equation on the support of an approximating scatterer. We approximate the support of the scattering obstacle using a point cloud, which is a high-level representation. Of note, high-level representations are a well established topic of computational geometry. Our motivation to use high-level representations is two-fold: in the one hand, the direct problem has a threshold property, i.e., the derivative of the direct problem can be arbitrarily approximated by a finite rank operator, giving rise to a low dimensional data-informed subspace. On the other hand, high level representations are intrinsically low dimensional, i.e. we design a suitable low dimensional parameter space to make inference easier. In this thesis, we have introduced an efficient method to solve the Lippmann-Schwinger equation and a probability transition kernel that commutes with affine transformations of space to sample the posterior distribution. We offer numerical evidence of the following facts: First, we can recover simultaneously both, the (non)convex support of a star-shaped scattering obstacle and its constant refractive index. Second, we show that the MCMC separates the multiple scales of the inverse problem. Indeed, the MCMC first locates the position of the scatterer, and adjusts the shape and refractive index afterwards. We believe that our methods can be extended to multifrequency data. | |
| 02-03-2018 | |
| Trabajo de grado, doctorado | |
| OTRAS | |
| Versión aceptada | |
| acceptedVersion - Versión aceptada | |
| Aparece en las colecciones: | Tesis del CIMAT |
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| Fichero | Descripción | Tamaño | Formato | |
|---|---|---|---|---|
| TE 775.pdf | 2.51 MB | Adobe PDF | Visualizar/Abrir |