Date of Award
Master of Science (MS)
Brannan , James R
Liu , Shitao
Inverse Problems is a field of great interest for many applications, such as parameter identification and image reconstruction. The underlying models of inverse problems in many applications often involve Partial Differential Equations (PDEs). A Reduced Basis (RB) method for solving PDE based inverse problems is introduced in this thesis. The RB has been rigorously established as an efficient approach for solving PDEs in recent years. In this work, we investigate whether the RB method can be used as a regularization for solving ill-posed and nonlinear inverse problems using iterative methods. We rigorously analyze the RB method and prove convergence of the RB approximation to the exact solution. Furthermore, an iterative algorithm is proposed based on gradient method with RB regularization. We also implement the proposed method numerically and apply the algorithm to the inverse problem of Electrical Impedance Tomography (EIT) which is known to be a notoriously ill-posed and nonlinear. For the EIT example, we provide all necessary details and carefully explain each step of the RB method. We also investigate the limitations of the RB method for solving nonlinear inverse problems in general. We conclude that the RB method can be used to solve nonlinear inverse problems with appropriate assumptions however the assumptions are somewhat restrictive and may not be applicable for a wide range of problems.
Gralla, Phil, "Inverse Problems Using Reduced Basis Method" (2014). All Theses. 2007.