Abstract
We state sufficient conditions for the uniqueness of minimizers of Tikhonov-type functionals. We further explore a connection between such results and the well-posedness of Morozov-like discrepancy principle. Moreover, we find appropriate conditions to apply such results to the local volatility surface calibration problem.
Highlights
In Tikhonov-type regularization, finding a balancing between data misfit and penalization is crucial, since, otherwise reconstructions may reproduce noise or incorporate too many bias
The aim of this article was to study the connection between the well-posedness of discrepancy principles and the uniqueness of reconstructions in Tikhonov-type regularization
We started by presenting variational techniques, and we found a positive lower bound for regularization parameters to uniqueness of Tikhonov minimizers to hold
Summary
In Tikhonov-type regularization, finding a balancing between data misfit and penalization is crucial, since, otherwise reconstructions may reproduce noise or incorporate too many bias. In discrepancy-based techniques, the regularization parameter is chosen whenever its corresponding data misfit has the same order of the noise level This is interesting since some convergence and convergence-rate results can be obtained without requirements on the behavior of the regularization parameter as a function of the noise level, in contrast to the standard Tikhonov-type regularization approach [20]. Vinicius Albani and Adriano De Cezaro the corresponding minimizers of the Tikhonov-type functional This phenomena, in particular, implies that the map that relates regularization parameter to data misfit can be multi-valued. The concept of fa∗ -minimizing solutions is introduced to restrict the set of all possible solutions, selecting the ones that present some desirable feature or satisfying some a priori information
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