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

We prove the logarithmic convergence rate of the families of usual and modified iterative Runge-Kutta methods for nonlinear ill-posed problems between Hilbert spaces under the logarithmic source condition, and numerically verify the obtained results. The iterative regularization is terminated by the a posteriori discrepancy principle.

[1]  Otmar Scherzer,et al.  A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions , 1998 .

[2]  Marlis Hochbruck,et al.  On the convergence of a regularizing Levenberg–Marquardt scheme for nonlinear ill-posed problems , 2010, Numerische Mathematik.

[3]  Thorsten Hohage,et al.  Logarithmic convergence rates of the iteratively regularized Gauss - Newton method for an inverse potential and an inverse scattering problem , 1997 .

[4]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[5]  U. Tautenhahn.  On the asymptotical regularization of nonlinear ill-posed problems , 1994 .

[6]  Thorsten Hohage,et al.  Regularization of exponentially ill-posed problems , 2000 .

[7]  C. Böckmann,et al.  Iterative Runge–Kutta-type methods for nonlinear ill-posed problems , 2008 .

[8]  C. Böckmann,et al.  Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems , 2016 .

[9]  Ye Zhang,et al.  On the second-order asymptotical regularization of linear ill-posed inverse problems , 2018, Applicable Analysis.

[10]  C. Böckmann,et al.  Convergence rate analysis of the first-stage Runge?Kutta-type regularizations , 2010 .

[11]  C. Böckmann,et al.  Convergence Rate of the Modified Landweber Method for Solving Inverse Potential Problems , 2020 .

[12]  Otmar Scherzer,et al.  A Modified Landweber Iteration for Solving Parameter Estimation Problems , 1998 .

[13]  Martin Hanke,et al.  The regularizing Levenberg-Marquardt scheme is of optimal order , 2010 .

[14]  M. Hanke,et al.  A convergence analysis of the Landweber iteration for nonlinear ill-posed problems , 1995 .

[15]  Christine Böckmann,et al.  Logarithmic convergence rate of Levenberg–Marquardt method with application to an inverse potential problem , 2011 .

[16]  M. Thamban Nair,et al.  A simplified generalized Gauss-Newton method for nonlinear ill-posed problems , 2009, Math. Comput..

[17]  R. Pinnau,et al.  Regularized fixed-point iterations for nonlinear inverse problems , 2006 .

[18]  Qinian Jin.  On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems , 2010, Numerische Mathematik.

[19]  C. Böckmann,et al.  A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems , 2019, Mathematics.