Research Seminar in Mathematics - Regularization methods for inverse problems arising in bioluminescence tomography

Speaker

Prof. Rongfang Gong, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China.

Abstract

In the bioluminescence tomography (BLT) problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem is ill-posed and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter to balance the accuracy and stability of approximate solutions. In this talk, several parameter-dependent frameworks are proposed to reformulate the original problem. Then Tikhonov regularization is applied to the reduced inverse source problem governed by diffusion approximation (DA) equation or the radiative transfer equation (RTE). By properly adjusting the parameters introduced in Robin boundary conditions, we achieve one important property: the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy. The finite element method is used to discretize the DA equation-based BLT problem while the discrete-ordinate finite-element method is used to discretize the RTE-based BLT problem. Numerical results are provided to illustrate the performance of the proposed methods. This is a joint work with Xiaoliang Cheng of Zhejiang University, China; Joseph Eichholz of Rose-Hulman Institute of Technology, USA; Weimin Han of University of Iowa, USA.