Forward and Inverse Problems Under Uncertainty

par Wenlong Zhang

Projet de thèse en Mathématiques

Sous la direction de Habib Ammari et de Zhiming Chen.

Thèses en préparation à Paris Sciences et Lettres en cotutelle avec l'Academy of Mathematics and Systems Science, Chinese Academy of Sciences , dans le cadre de Sciences Mathématiques de Paris Centre , en partenariat avec DMA - Département de Mathématiques et Applications (laboratoire) et de Ecole normale supérieure (établissement de préparation de la thèse) depuis le 01-10-2014 .


  • Résumé

    Nous faisions des recherches pour comprendre les facteurs qui influent sur la qualité de l'imagerie de conductivité, y compris la limite du domaine, la structure des tissus, et l'anisotropie. Je termine déjà le projet sur l'imagerie multi-fréquence, anisotropie de tissu dans le temps et l'imagerie d'anisotropie avec DTI. La défense est prévue mai 2017.

  • Titre traduit

    Forward and Inverse Problems Under Uncertainty


  • Résumé

    This thesis contains two di erent subjects. In first part, two cases are considered. One is the the thin plate spline smoother model and the other one is the elliptic boundary equations with uncertain boundary data. In this part, stochastic convergences of the finite element methods are proved for each problem. A nonconforming Morley finite element method to approximate thin plate spline model is considered. We provide the optimal choice of smoothing parameter and propose a self-consistent iterative algorithm to determine the smoothing parameter based on our theoretical analysis. We propose a finite element method for solving elliptic equations with the observational Neumann boundary data which may subject to random noises. In second part, we provide a mathematical analysis of the linearized inverse problem in multifrequency electrical impedance tomography and multi-wavelength model for Di use Optical Spectroscopy Imaging. We present a mathematical and numerical framework for a procedure of imaging anisotropic electrical conductivity tensor using a novel technique called Di usion Tensor Magneto-acoustography and propose an optimal control approach for reconstructing the cross-property factor relating the di usion tensor to the anisotropic electrical conductivity tensor. We prove convergence and Lipschitz type stability of the algorithm and present numerical examples to illustrate its accuracy. The cell model for Electropermeabilization is demonstrated. We study e ective parameters in a homogenization model. We start from a physiological cell model for electropermeabilization and analyze its well-posedness. For a dynamical homogenization scheme, we prove convergence and then analyze the e ective parameters. We demonstrate numerically the sensitivity of these e ective parameters to critical microscopic parameters governing electropermeabilization.