Studies On Inverse Problem Theory And Algorithm With Sparsity Constraints

In many areas of life and production, people need to extract useful infor-mation by inverting the observed data. These issues belong to the category of inverse problems which are usually ill-posed. Classic Tikhonov regularization that established in the Hilbert space with complete theory and algorithm is a powerful tool to deal with these inverse problems. However, in order to preserve some of the main features of the inversion variables, regularization in Banach space occurred in the past ten years in the fields of signal process-ing, image processing, medical imaging, non-destructive detection, parameter identification in PDE, machine learning, statistical analysis, finance, bioinfor-matics and other areas. This thesis studies several modeling and computing problems raised in compressed sensing, variable selection, Robin parameter identification of elliptic partial differential equations with the main line of sparse regularization under the framework of Banach space regularization.This thesis is structured as follows:Chapter1, We summarize the relative works in the literature and give simple introduction of our research motivation.Chapter2, We proposed primal dual active set algorithm with continu-ation (PDASC) to the e1and non-convex regularized model in compressive sensing and variable selection. For the e1regularized model, we prove that PDAS enjoys the local one step convergence property while PDASC con-verges globally. For the e0regularized model we prove the uniqueness of the global minimizer and the global convergence of PDASC under mild assump-tions. And we discuss a posterior regularization parameter selection rules based on discrepancy principle, modified discrepancy principle and Bayesian information criterion.Chapter3, We consider inversion of Robin parameters in elliptic partial differential equations. By taking full account of the physical background and practical significance, we propose a new variational regularization model with sparsity constraint. We established the existence, stability and prior regu-larization parameter rule and the convergence of finite element discretization even if the model we considered is non-convex and non-smooth. We propose a simple but effective lagged Newton algorithm to solve the model and use the discrepancy principle to select the regularization parameter.