A Nonlinear Inverse Source Problem for Poisson's Equation with Point Source and Insulating Boundary Condition

Abstract

Inverse source problems for Poisson's equation lie at the heart of many industrial and medical imaging applications. This Thesis studies an ill-posed non-linear inverse source problem for Poisson's equation with point sources, in both 2 dimensional and 3 dimensional domains. We compare two different approaches to solving the inverse problem of finding the source location and strength, namely the Spectral method and the BLASSO method. Using the Spectral method, we establish a new analytical theory on the bound of the source number. Furthermore, we use numerical experiments to show the bound is sharp. Also, we compare the difference of the bound in 2D and 3D and provide analytical justification for such a difference. We also propose an original regularization scheme which improves resolution of the source separation problem when compared to other regularisation schemes, such as the Tikhonov regularization. Lastly we implement the BLASSO method in the final Chapter of this thesis. We show that the BLASSO method produces a superior solution to the inverse problem in terms of the resolution when compared to the Spectral method, but it is more sensitive to noise and harder to implement.

Date of Award	13 Dec 2021
Original language	English
Awarding Institution	University of Bath
Supervisor	Chris Budd (Supervisor) & Silvia Gazzola (Supervisor)