Analysis and numerical methods for parameters identification in partial differential equations

Abstract

This thesis is devoted to design and analyze the numerical algorithm for parameter identification problem utlizing theoretical results. In recent years, numerous numerical schemes for parameter identification problems were developed, analyzed and tested. Most of existing work emphasizes well-posedness, convergence (with respect to the noise level), and convergence rates under various source conditions. In practice, the inversion formulations are further discretized, traditionally via the Galerkin finite element methods (FEMs) or, more recently, neural networks (NNs). However, discretization introduces additional errors that affect reconstruction quality, and rigorous error bounds for numerical inversion algorithms remain underexplored. After some background introduction and preliminaries in Chapters 1 and 2, we investigate the reconstruction of both the diffusion and reaction coefficients present in an elliptic/parabolic equation in Chapter 3. A decoupled algorithm is constructed to sequentially recover these two parameters. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in L²(Ω) for the recovered diffusion and reaction coefficients. Next, in Chapter 4, we focus on the numerical analysis of quantitative photoacoustic tomography (QPAT). The stability of the inverse problem significantly depends on a non-zero condition in the internal observations, a condition that can be met using randomly chosen boundary excitation data. Utilizing these randomly generated boundary data, we provide a rigorous error estimate in L²(Ω) norm for the numerical reconstruction. In Chapter 5, we propose a hybrid FEM-NN scheme, where the finite element method is employed to approximate the state and neural networks act as a smoothness prior to approximate the unknown parameter. We demonstrate that the hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias/approximation properties of NNs. In Chapter 6, we concern with numerically recovering multiple parameters simultaneously in the subdiffusion model from one single lateral measurement on a part of the boundary, while in an incompletely known medium. We prove a uniqueness result for special cases of diffusion coefficients and boundary excitations. The uniqueness analysis further inspires the development of a robust numerical algorithm for recovering the unknown parameters. Finally, in Chapter 7, we summarize our work and mention possible future research topics. Throughout, extensive numerical experiments are provided to illustrate the efficiency and reliability of the proposed algorithms.