Méthodes de Monte-Carlo pour les diffusions discontinues : application à la tomographie par impédance électrique

Résumé : This thesis deals with the development of Monte-Carlo methods to compute Feynman-Kac representations involving divergence form operators with a piecewise constant diffusion coefficient. The proposed methods are variations around the walk on spheres method inside the regions with a constant diffusion coefficient and stochastic finite differences techniques to treat the interface conditions as well as the different kinds of boundary conditions. By combining these two techniques, we build random walks which score computed along the walk gives us a biased estimator of the solution of the partial differential equation we consider. We prove that the global bias is in general of order two with respect to the finite difference step. These methods are then applied for tumour detection to the forward problem in electrical impedance tomography. A variance reduction technique is also proposed in this case. Finally, we treat the inverse problem of tumours detection from surface measurements using two stochastics algorithms based on a spherical parametric representation of the tumours. Many numerical tests are proposed and show convincing results in the localization of the tumours.
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Thèse
Mathématiques [math]. Aix-Marseille Université, 2015. Français
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https://hal.inria.fr/tel-01387004
Contributeur : Thi Quynh Giang Nguyen <>
Soumis le : mardi 25 octobre 2016 - 06:34:21
Dernière modification le : jeudi 18 janvier 2018 - 02:23:27

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  • HAL Id : tel-01387004, version 1

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Thi Quynh Giang Nguyen. Méthodes de Monte-Carlo pour les diffusions discontinues : application à la tomographie par impédance électrique. Mathématiques [math]. Aix-Marseille Université, 2015. Français. 〈tel-01387004〉

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