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

Published electronically:
February 18, 2000

MathSciNet review:
1680887

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

The iteratively regularized Gauss-Newton method is applied to compute the stable solutions to nonlinear ill-posed problems when the data is given approximately by with . In this method, the iterative sequence is defined successively by

where is an initial guess of the exact solution and is a given decreasing sequence of positive numbers admitting suitable properties. When is used to approximate , 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 determined by this rule it is proved that

with a constant independent of , where denotes the iterative solution corresponding to the noise free case. As a consequence of this result, the convergence of is obtained, and moreover the rate of convergence is derived when 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 . Numerical examples for parameter estimation of a differential equation are given to test the theoretical results.

**1.**A. B. Bakushinskiĭ,*On a convergence problem of the iterative-regularized Gauss-Newton method*, Zh. Vychisl. Mat. i Mat. Fiz.**32**(1992), no. 9, 1503–1509 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys.**32**(1992), no. 9, 1353–1359 (1993). MR**1185952****2.**H. T. Banks and K. Kunisch,*Estimation techniques for distributed parameter systems*, Systems & Control: Foundations & Applications, vol. 1, Birkhäuser Boston, Inc., Boston, MA, 1989. MR**1045629****3.**Barbara Blaschke, Andreas Neubauer, and Otmar Scherzer,*On convergence rates for the iteratively regularized Gauss-Newton method*, IMA J. Numer. Anal.**17**(1997), no. 3, 421–436. MR**1459331**, 10.1093/imanum/17.3.421**4.**Keith Glover and Jonathan R. Partington,*Bounds on the achievable accuracy in model reduction*, Modelling, robustness and sensitivity reduction in control systems (Groningen, 1986) NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., vol. 34, Springer, Berlin, 1987, pp. 95–118. MR**912132****5.**Heinz W. Engl,*Regularization methods for the stable solution of inverse problems*, Surveys Math. Indust.**3**(1993), no. 2, 71–143. MR**1225782****6.**Heinz W. Engl and Helmut Gfrerer,*A posteriori parameter choice for general regularization methods for solving linear ill-posed problems*, Appl. Numer. Math.**4**(1988), no. 5, 395–417. MR**948506**, 10.1016/0168-9274(88)90017-7**7.**Heinz W. Engl, Karl Kunisch, and Andreas Neubauer,*Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems*, Inverse Problems**5**(1989), no. 4, 523–540. MR**1009037****8.**Helmut 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), no. 180, 507–522, S5–S12. MR**906185**, 10.1090/S0025-5718-1987-0906185-4**9.**C. W. Groetsch,*The theory of Tikhonov regularization for Fredholm equations of the first kind*, Research Notes in Mathematics, vol. 105, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**742928****10.**Charles W. Groetsch,*Inverse problems in the mathematical sciences*, Vieweg Mathematics for Scientists and Engineers, Friedr. Vieweg & Sohn, Braunschweig, 1993. MR**1247696****11.**Martin Hanke, Andreas Neubauer, and Otmar Scherzer,*A convergence analysis of the Landweber iteration for nonlinear ill-posed problems*, Numer. Math.**72**(1995), no. 1, 21–37. MR**1359706**, 10.1007/s002110050158**12.**Bernd Hofmann,*Regularization for applied inverse and ill-posed problems*, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 85, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1986. A numerical approach; With German, French and Russian summaries. MR**906063****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.**Barbara Kaltenbacher,*Some Newton-type methods for the regularization of nonlinear ill-posed problems*, Inverse Problems**13**(1997), no. 3, 729–753. MR**1451018**, 10.1088/0266-5611/13/3/012**15.**Alfred Karl Louis,*Inverse und schlecht gestellte Probleme*, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], B. G. Teubner, Stuttgart, 1989 (German). MR**1002946****16.**Andreas Neubauer,*On converse and saturation results for Tikhonov regularization of linear ill-posed problems*, SIAM J. Numer. Anal.**34**(1997), no. 2, 517–527. MR**1442926**, 10.1137/S0036142993253928**17.**R. Plato and U. Hämarik,*On pseudo-optimal parameter choices and stopping rules for regularization methods in Banach spaces*, Numer. Funct. Anal. Optim.**17**(1996), no. 1-2, 181–195. MR**1391881**, 10.1080/01630569608816690**18.**Otmar Scherzer,*A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems*, Numer. Funct. Anal. Optim.**17**(1996), no. 1-2, 197–214. MR**1391882**, 10.1080/01630569608816691**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), no. 6, 1796–1838. MR**1249043**, 10.1137/0730091**20.**Thomas I. Seidman and Curtis R. Vogel,*Well-posedness and convergence of some regularisation methods for nonlinear ill posed problems*, Inverse Problems**5**(1989), no. 2, 227–238. MR**991919****21.**Andrey N. Tikhonov and Vasiliy Y. Arsenin,*Solutions of ill-posed problems*, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John; Scripta Series in Mathematics. MR**0455365****22.**G. M. Vaĭnikko and A. Yu. Veretennikov,*Iteratsionnye protsedury v nekorrektnykh zadachakh*, “Nauka”, Moscow, 1986 (Russian). MR**859375****23.**V. V. Vasin and A. L. Ageev,*Ill-posed problems with a priori information*, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1995. MR**1374573**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65J20,
45G10

Retrieve articles in all journals with MSC (1991): 65J20, 45G10

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