Augmented Tikhonov Regularization With Nonnegative Constraints

This thesis focuses on linear inverse problems with nonnegative constraints. Based on the Bayesian framework, a Tikhonov-type regularization functional with nonnega-tive constraints is proposed, leading to a minimization problem, the solution of which determines the required regularization parameter, the noise level, and the barrier pa-rameter as well as the solution for the linear inverse problems with nonnegative con-straints. The existence of minimizers to the functional is demonstrated, an alternating iterative algorithm is devised to find the minimizer of the nonlinear functional, and the convergence of the algorithm is established. In the paper, it is also shown that the nonlinear system arising as the sub-problem of the previous algorithm has a unique nonnegative solution. To solve the sub-problem, a nonlinear Gauss-Seidel iterative method is suggested, which can ensure the positivity of the solution during the it-eration process. Moreover, it is proved that the nonlinear Gauss-Seidel method is converged to the unique exact solution for any starting point. A series of numerical results are also provided to illustrate the effectiveness of the regularization method, the alternating iterative algorithm and the nonlinear Gauss-Seidel method for regularized solution to linear inverse problems with nonnegative constraints.