Augmented Tikhonov regularization

This paper proposes a regularizing functional of Tikhonov type that determines the regularization parameter and the noise level along with the solutions for linear inverse problems in the Bayesian framework. The existence of minimizers to the functional is shown, and properties of the minimizers are studied. An alternating iterative algorithm is suggested for efficiently solving the resulting nonlinear optimization problem, and its convergence is established. Numerical results for both mildly and severely ill-posed benchmark examples are presented to illustrate relevant features of the functional.