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On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems


Author: Jin Qi-nian
Journal: Math. Comp. 69 (2000), 1603-1623
MSC (1991): Primary 65J20, 45G10
DOI: https://doi.org/10.1090/S0025-5718-00-01199-6
Published electronically: February 18, 2000
MathSciNet review: 1680887
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Abstract:

The iteratively regularized Gauss-Newton method is applied to compute the stable solutions to nonlinear ill-posed problems $F(x)=y$ when the data $y$ is given approximately by $y^\delta$with $\Vert y^\delta-y\Vert\le\delta$. In this method, the iterative sequence $\{x_k^\delta\}$ is defined successively by

\begin{displaymath}x_{k+1}^\delta=x_k^\delta-(\alpha_k I+F'(x_k^\delta)^*F'(x_k^... ...lta)^*(F(x_k^\delta)-y^\delta) +\alpha_k(x_k^\delta-x_0)\Big), \end{displaymath}

where $x_0^\delta:=x_0$ is an initial guess of the exact solution $x^\dag $ and $\{\alpha_k\}$ is a given decreasing sequence of positive numbers admitting suitable properties. When $x_k^\delta$ is used to approximate $x^\dag $, the stopping index should be designated properly. In this paper, an a posteriori stopping rule is suggested to choose the stopping index of iteration, and with the integer $k_\delta$ determined by this rule it is proved that

\begin{displaymath}\Vert x_{k_\delta}^\delta-x^\dag\Vert\le C\inf\Big\{\Vert x_k-x^\dag\Vert +\frac{\delta}{\sqrt{\alpha_k}}:k=0,1,\ldots\Big\} \end{displaymath}

with a constant $C$ independent of $\delta$, where $x_k$ denotes the iterative solution corresponding to the noise free case. As a consequence of this result, the convergence of $x_{k_\delta}^\delta$is obtained, and moreover the rate of convergence is derived when $x_0-x^\dag $ satisfies a suitable ``source-wise representation". The results of this paper suggest that the iteratively regularized Gauss-Newton method, combined with our stopping rule, defines a regularization method of optimal order for each $0<\nu\le 1$. Numerical examples for parameter estimation of a differential equation are given to test the theoretical results.


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  • 1. A. B. Bakushinskii, The problems of the convergence of the iteratively regularized Gauss-Newton method, Comput. Math. Math. Phys., 32(1992), 1353-1359. MR 93k:65049
  • 2. H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Basel: Birkhäuser, 1989. MR 91b:93085
  • 3. B. Blaschke, A. Neubauer and O. Scherzer, On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numer. Anal., 17(1997), 421-436. MR 98f:65066
  • 4. F. Colonius and K. Kunisch, Stability for parameter estimation in two point boundary value problems, J. Reine Angew. Math., 370(1986), 1-29. MR 88j:93027
  • 5. H. W. Engl, Regularization methods for the stable solutions of inverse problems, Surv. Math. Ind., 3(1993), 71-143. MR 94g:65064
  • 6. H. W. Engl and H. Gfrerer, A posteriori parameter choice for general regularization methods for solving ill-posed problems, Appl. Numer. Math., 4(1988), 395-417. MR 89i:65060
  • 7. H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5(1989), 523-540. MR 91k:65102
  • 8. H. Gfrerer, An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, Math. Comp., 49(1987), 507-522. MR 88k:65049
  • 9. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equation of the First Kind, (Boston, MA: Pitman), 1984. MR 85k:45020
  • 10. C. W. Groetsch, Inverse Problems in Mathematical Sciences, Vieweg, Wiesbaden, 1993. MR 94m:00008
  • 11. M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72(1995), 21-37. MR 96i:65046
  • 12. B. Hofmann, Regularization for Applied Inverse and Ill-Posed Problems, Leipzig, Teubner, 1986. MR 88i:65001
  • 13. Q. N. Jin and Z. Y. Hou, On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems, Numer. Math., 83(1999), 139-159. CMP 99:16
  • 14. B. Kaltenbacher, Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 13(1997), 729-754. MR 98h:65025
  • 15. A. K. Louis, Inverse und Schlecht Gestellte Probleme, Teubner, Stutgart, 1989. MR 90g:65075
  • 16. A. Neubauer, On converse and saturation results for Tikhonov regularization of linear ill-posed problems, SIAM J. Numer. Anal., 34(1997), 517-527. MR 98d:65081
  • 17. R. Plato and H. Hämarik, On pseudo-optimal parameter choice and stopping rules for regularization methods in Banach spaces, Numer. Funct. Anal. Optimiz., 17(1996), 181-195. MR 97g:65124
  • 18. O. Scherzer, A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems, Numer. Funct. Anal. Optimiz., 17(1996), 197-214. MR 97g:65125
  • 19. O. Scherzer, H. W. Engl and K. Kunisch, Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J. Numer. Anal., 30(1993), 1796-1838. MR 95a:65104
  • 20. T. I. Seidman and C. R. Vogel, Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems, Inverse Problems, 5(1989), 227-238. MR 90d:65117
  • 21. A. N. Tikhonov and V. Y. Arsenin, Solutions of ill-posed problems. Winston, Washington, DC, 1977. MR 56:13604
  • 22. G. M. Vainikko and A. Y. Veretennikov, Iteration Procedures in Ill-Posed Problems (in Russian), Nauka, Moscow, 1986. MR 88c:47019
  • 23. V. V. Vasin and A. L. Ageev, Ill-Posed Problems with A Priori Information, Inverse and Ill-Posed Problems Series, VSP, Utrecht, The Netherlands, 1995. MR 97j:65100

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Additional Information

Jin Qi-nian
Affiliation: Institute of Mathematics, Nanjing University, Nanjing 210008, P. R. China
Email: galgebra@nju.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-00-01199-6
Keywords: Nonlinear ill-posed problems, the iteratively regularized Gauss-Newton method, stopping rule, convergence, rates of convergence.
Received by editor(s): March 17, 1998
Received by editor(s) in revised form: January 4, 1999
Published electronically: February 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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