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

Full-text PDF

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.

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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