Abstract
We present a new iterative method which does not involve inversion of the operators for obtaining an approximate solution for the nonlinear ill-posed operator equationF(x)=y. The proposed method is a modified form of Tikhonov gradient (TIGRA) method considered by Ramlau (2003). The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005). The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper.
Highlights
This paper is devoted to the study of nonlinear ill-posed problem F (x) = y, (1)where F : D(F) ⊆ X → Y is a nonlinear operator between the Hilbert spaces X and Y
We prove that xnδ,α converges to the unique solution xαδ of the equation
Throughout this paper we assume that the operator F satisfies the following assumptions
Summary
Where F : D(F) ⊆ X → Y is a nonlinear operator between the Hilbert spaces X and Y. In Tikhonov regularization, a solution of the problem (1) is approximated by a solution of the minimization problem. It is known [3] that the minimizer xαδ of the functional Jα(x) satisfies the Euler equation. It is known that [1] for properly chosen regularization parameter α, the minimizer xαδ of the functional Jα(x) is a International Journal of Mathematics and Mathematical Sciences good approximation to a solution xwith minimal distance from x0. Where x0δ = x0, βk is a scaling parameter and αk is a regularization parameter, which will change during the iteration and obtained a convergence rate estimate for the TIGRA algorithm under the following assumptions:. In the proposed method one needs to compute the Frechet derivative only at one point x0.
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