22-24 September 2021
Alpen-Adria-Universität Klagenfurt
Europe/Vienna timezone

Convergence Rate of Runge-Kutta-Type Regularization for Nonlinear Ill-Posed Problems under Logarithmic Source Condition

23 Sep 2021, 10:20
30m
HS 1 (Alpen-Adria-Universität Klagenfurt)

HS 1

Alpen-Adria-Universität Klagenfurt

Speaker

Christine Böckmann (University of Potsdam, Institute of Mathematics, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Gemany)

Description

We present two families of regularization method for solving nonlinear ill-posed problems between Hilbert spaces by applying the family of Runge–Kutta methods to an initial value problem, in particular, to the asymptotical regularization method.
In Hohage [1], a systematic study of convergence rates for regularization methods under logarithmic source condition including the case of operator approximations for a priori and a posteriori stopping rules is provided.
We prove the logarithmic convergence rate of the families of usual and modified iterative Runge-Kutta methods under the logarithmic source condition, and numerically verify the obtained results. The iterative regularization is terminated by the a posteriori discrepancy principle, Pornsawad, et al. [2]. Up to now, the logarithmic convergence rate under logarithmic source condition has only been investigated for particular examples, namely, the Levenberg–Marquardt method [3] and the modified Landweber method [4]. Here, we extended the results to the whole family of Runge-Kutta-type methods with and without modification.

[1] Hohage, T., Regularization of exponentially ill-posed problems. Numer. Funct. Anal. Optimiz. 2000, 21, 439–464.
[2] Pornsawad, P., Resmerita, E., Böckmann, C., Convergence Rate of Runge-Kutta-Type Regularization for Nonlinear Ill-Posed Problems under Logarithmic Source Condition, Mathematics 2021, 9, 1042.
[3] Böckmann, C., Kammanee, A., Braunß, A., Logarithmic convergence rate of Levenberg–Marquardt method with application to an inverse potential problem. J. Inv. Ill-Posed Probl. 2011, 19, 345–367.
[4] Pornsawad, P., Sungcharoen, P., Böckmann, C., Convergence rate of the modified Landweber method for solving inverse potential problems. Mathematics 2020, 8, 608.

Primary author

Christine Böckmann (University of Potsdam, Institute of Mathematics, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Gemany)

Co-authors

Pornsarp Pornsawad (Department of Mathematics, Faculty of Science, Silpakorn University) Elena Resmerita (MATH)

Presentation Materials

There are no materials yet.