On a class of frozen regularized Gauss-Newton methods for nonlinear inverse problems

Jin, Qinian

doi:10.1090/S0025-5718-10-02359-8

On a class of frozen regularized Gauss-Newton methods for nonlinear inverse problems
HTML articles powered by AMS MathViewer

by Qinian Jin PDF

Math. Comp. 79 (2010), 2191-2211 Request permission

Abstract:

In this paper we consider a class of regularized Gauss-Newton methods for solving nonlinear inverse problems for which an a posteriori stopping rule is proposed to terminate the iteration. Such methods have the frozen feature that they require only the computation of the Fréchet derivative at the initial approximation. Thus the computational work is considerably reduced. Under certain mild conditions, we give the convergence analysis and derive various estimates, including the order optimality, on these methods.

References

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
A. B. Bakushinskiĭ, Iterative methods without saturation for solving degenerate nonlinear operator equations, Dokl. Akad. Nauk 344 (1995), no. 1, 7–8 (Russian). MR 1361018
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, DOI 10.1093/imanum/17.3.421
David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR 1635980, DOI 10.1007/978-3-662-03537-5
Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of inverse problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1408680, DOI 10.1007/978-94-009-1740-8
F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal. 37 (2000), no. 2, 587–620. MR 1740766, DOI 10.1137/S0036142998341246
Thorsten Hohage, Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems 13 (1997), no. 5, 1279–1299. MR 1474369, DOI 10.1088/0266-5611/13/5/012
Victor Isakov, Inverse problems for partial differential equations, 2nd ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006. MR 2193218
Qi-Nian Jin, On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems, Math. Comp. 69 (2000), no. 232, 1603–1623. MR 1680887, DOI 10.1090/S0025-5718-00-01199-6
Qinian Jin, A convergence analysis of the iteratively regularized Gauss-Newton method under the Lipschitz condition, Inverse Problems 24 (2008), no. 4, 045002, 16. MR 2425869, DOI 10.1088/0266-5611/24/4/045002
Qinian Jin and Ulrich Tautenhahn, On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems, Numer. Math. 111 (2009), no. 4, 509–558. MR 2471609, DOI 10.1007/s00211-008-0198-y
Barbara Kaltenbacher, A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems, Numer. Math. 79 (1998), no. 4, 501–528. MR 1631677, DOI 10.1007/s002110050349
Barbara Kaltenbacher, Andreas Neubauer, and Otmar Scherzer, Iterative regularization methods for nonlinear ill-posed problems, Radon Series on Computational and Applied Mathematics, vol. 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. MR 2459012, DOI 10.1515/9783110208276
Pallavi Mahale and M. Thamban Nair, A simplified generalized Gauss-Newton method for nonlinear ill-posed problems, Math. Comp. 78 (2009), no. 265, 171–184. MR 2448702, DOI 10.1090/S0025-5718-08-02149-2
Plamen Stefanov and Gunther Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal. 256 (2009), no. 9, 2842–2866. MR 2502425, DOI 10.1016/j.jfa.2008.10.017
Ulrich Tautenhahn, On a general regularization scheme for nonlinear ill-posed problems, Inverse Problems 13 (1997), no. 5, 1427–1437. MR 1474375, DOI 10.1088/0266-5611/13/5/020
G. M. Vaĭnikko and A. Yu. Veretennikov, Iteratsionnye protsedury v nekorrektnykh zadachakh, “Nauka”, Moscow, 1986 (Russian). MR 859375