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

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 with y - y . In this method, the iterative sequence {x k} is defined successively by x k+1 = x k - (kI + F (x k) F (x k))-1 F (x k) (F (x k) - y ) + k(x k - x0) , where x 0 := x0 is an initial guess of the exact solution x and {k} is a given decreasing sequence of positive numbers admitting suitable properties. When x 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 k determined by this rule it is proved that x k - x C inf xk - x + k : k = 0, 1, . . . with a constant C independent of , where xk denotes the iterative solution corresponding to the noise free case. As a consequence of this result, the convergence of x k is obtained, and moreover the rate of conver...

• Type: Journal
• Year: 2000
• Where: MOC
• Authors: Jin Qi-nian
• Keywords: Ill-posed Problems, MOC 2000, Solution, Stable Solutions