ON THE ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD FOR SOLVING NONLINEAR ILL-POSED PROBLEMS

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 yδ with �yδ − y � ≤δ. In this method, the iterative sequence {xδ k} is defined successively by x δ k+1 = xδk − (αkI + F ′ (x δ k)∗F ′ (x δ k))−1�F ′ (x δ k) ∗ (F (x δ k) − yδ)+αk(x δ � k − x0), where xδ 0: = x0 is an initial guess of the exact solution x † and {αk} is a given decreasing sequence of positive numbers admitting suitable properties. When xδ k is used to approximate x † , 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 kδ determined by this rule it is proved that �x δ k − x δ † � �≤C inf �xk − x † � + δ √ : k =0, 1,... αk with a constant C independent of δ, wherexkdenotes the iterative solution corresponding to the noise free case. As a consequence of this result, the convergence of xδ k is obtained, and moreover the rate of convergence is derived δ when x0 − x † 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 <ν ≤ 1. Numerical examples for parameter estimation of a differential equation are given to test the theoretical results. 1