The Iterated Tikhonov Regularization Methods For Nonlinear Ill-Posed Problems With Monotone Operators

In many engineering fields, like telemetering and exploration, lots of inverse pro-blems arise. Usually, these inverse problems are ill-posed. In the paper, we introduce the general concept of ill-posed problems and the regularization strategy. And then, we mainly discuss methods for solving nonlinear ill-posed problems F(x)=y (1) with monotone operators in real Hilbert space H. Here, F is continuous and mono-tonous, i.e. F satisfies (F(x)-F(y),x-y)≥0, (?)x,y∈H. (2) Under some proper assumptions, we apply the iterated Tikhonov regularization with parameterαand the iterated Tikhonov regularization with parameter m to solve equation(1), we could get conclusions as follows:Theorem 1. Let r=(?)δ2+h2. For any fixed iterations m≥1, if regularization para-meterα:=α(δ, h) satisfies limα=0,limδ/α=0,limh/α=0, then lim‖xα,δ,hm-x*‖=0.Theorem 2.Let r=(?)δ2+h+.For any fixed iterations m≥1,Let constants C1>0, C2>0,η∈(0,1],ζ∈(0,1]satisfy C1δη+C2hζ>δ+h‖x*‖. Assuming‖Fh(0)-yδ‖>C1δη+C2hζ,there is aα*:=α*(δ,h)>0 which satisfies‖Fh(xα*,δ,hm-x*‖=0 If 0<η<1,0<ζ<1,then lim‖xα*,δ,hm-x*‖=0.Theorem 3.Let r=(?)δ2+h2.If regularization parameter m:=m(δ,h)satisfies lim m=∞, lim mδ=0, lim mh=0, then lim‖xmδ,h-x‖=0The methods,most frequently used in nonlinear ill-posed problems,require the knowledge of the Frechet derivative if F.Therefore,they are not applicable if F is not Frechet differentiable.In this paper,when F is not Frechet differentiable,we get a stable solution through the iterated Tikhonov regularization(3)and(4).Compared with the Tikhonov regularization,the iterated Tikhonov regularization(3)get a better convergence.At the same time,the iterated Tikhonov regularization(4)could also obtain a good convergence in few iterations.We need not to increase the iterations.This paper is organized as below:we introduce the ill-posed problems and the iterated Tikhonov regularization method in chapter 1; in chapter 2,we give the con-vergent analysis of the iterited Tikhonov regularization with parameterα; in chapter 3,we give the convergent analysis of the iterited Tikhonov regularization with para- meter m; in chapter 4, a numerical example is included to verify the availability of the two methods.