Logarithmic convergence rate of Levenberg–Marquardt method with application to an inverse potential problem

Abstract We prove logarithmic convergence rate of the Levenberg–Marquardt method in a Hilbert space if a logarithmic source condition is satisfied. This method is applied to an inverse potential problem. Numerical implementations demonstrate the convergence rate.

[1]  Christine Böckmann,et al.  Padé iteration method for regularization , 2006, Appl. Math. Comput..

[2]  M. Thamban Nair,et al.  Tikhonov regularization of nonlinear ill-posed equations under general source condition , 2007 .

[3]  O. Scherzer,et al.  Numerical comparison of iterative regularization methods for a parameter estimation problem in a hyperbolic PDE , 2001 .

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

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

[6]  Bernd Hofmann,et al.  Analysis of Profile Functions for General Linear Regularization Methods , 2007, SIAM J. Numer. Anal..

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

[8]  Bernd Hofmann,et al.  Convergence rates for Tikhonov regularization based on range inclusions , 2005 .

[9]  C. Böckmann.  Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions. , 2001, Applied optics.

[10]  S. Kabanikhin.  Definitions and examples of inverse and ill-posed problems , 2008 .

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

[12]  A. Bakushinsky,et al.  Iterative Methods for Approximate Solution of Inverse Problems , 2005 .

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

[14]  Mark S. Gockenbach,et al.  Partial Differential Equations - Analytical and Numerical Methods (2. ed.) , 2011 .

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

[16]  M. Hanke.  A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems , 1997 .

[17]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[18]  Aslak Tveito,et al.  Introduction to Partial Differential Equations , 2004 .

[19]  Vitalii P. Tanana,et al.  Theory of Linear Ill-Posed Problems and its Applications , 2002 .

[20]  Barbara Kaltenbacher,et al.  Iterative Regularization Methods for Nonlinear Ill-Posed Problems , 2008, Radon Series on Computational and Applied Mathematics.

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

[22]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[23]  Victor Isakov,et al.  Inverse Source Problems , 1990 .

[24]  P. Mathé,et al.  Geometry of linear ill-posed problems in variable Hilbert scales Inverse Problems 19 789-803 , 2003 .

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

[26]  W. Rundell,et al.  Iterative methods for the reconstruction of an inverse potential problem , 1996 .

[27]  Bernd Hofmann,et al.  How general are general source conditions? , 2008 .