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 , 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. . Up to now, the logarithmic convergence rate under logarithmic source condition has only been investigated for particular examples, namely, the Levenberg–Marquardt method  and the modified Landweber method . Here, we extended the results to the whole family of Runge-Kutta-type methods with and without modification.
 Hohage, T., Regularization of exponentially ill-posed problems. Numer. Funct. Anal. Optimiz. 2000, 21, 439–464.
 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.
 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.
 Pornsawad, P., Sungcharoen, P., Böckmann, C., Convergence rate of the modified Landweber method for solving inverse potential problems. Mathematics 2020, 8, 608.