Abstract
This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).
Highlights
This paper tries to complement the theory and practice of Tikhonov regularization with oversmoothing penalties for the stable approximate solution of nonlinear ill-posed problems in a Hilbert scale setting
For overcoming the remaining weaknesses of the parameter choice expressed in Equation (5), sequential versions of the discrepancy principle can be applied that approximate αdiscr, and we refer to [4,5,6] for more details
We present the sketch of an alternative proof for the order optimal Hölder convergence rates under the Hölder-type source condition x † ∈ X p for 0 < p < 1
Summary
This paper tries to complement the theory and practice of Tikhonov regularization with oversmoothing penalties for the stable approximate solution of nonlinear ill-posed problems in a Hilbert scale setting. For overcoming the remaining weaknesses of the parameter choice expressed in Equation (5), sequential versions of the discrepancy principle can be applied that approximate αdiscr , and we refer to [4,5,6] for more details Error estimates in the X-norm and convergence rates were proven under two-sided inequalities that characterize the degree of ill-posedness a > 0 of the problem We follow this approach and adapt it to the case of nonlinear problems throughout the subsequent sections and assume the inequality chain c a k x − x † k−a ≤ k F ( x ) − F ( x † )kY ≤ Ca k x − x † k− a for all x ∈ D (6). In that section (Section 6), the obtained numerical results are presented and interpreted based on a series of tables and figures
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
