Heuristic parameter choice in Tikhonov method from minimizers of the quasi-optimality function
Provides a theoretically guaranteed heuristic parameter choice for Tikhonov regularization, solving a known failure case for practitioners in inverse problems.
The paper addresses the failure of the quasi-optimality criterion for parameter selection in Tikhonov regularization when noise level is unknown, proving that a local minimizer of the quasi-optimality function always yields a good regularization parameter and proposing an algorithm to find it.
We consider choice of the regularization parameter in Tikhonov method in the case of the unknown noise level of the data. From known heuristic parameter choice rules often the best results were obtained in the quasi-optimality criterion where the parameter is chosen as the global minimizer of the quasi-optimality function. In some problems this rule fails, the error of the Tikhonov approximation is very large. We prove, that one of the local minimizers of the quasi-optimality function is always a good regularization parameter. We propose an algorithm for finding a proper local minimizer of the quasi-optimality function.