Third-order Iterative Method For Choices Of Regularization Parameter In Linear Inverse Problem

Inverse problems are widely applied in many fields, but this kind of problems are often ill-posed in the sense of Hadamard. That brings much difficulty to solve this kind of problems. Small perturbations in the observation data may lead to large effects on the considered solutions. Thus to ensure a feasible and stable numerical solution, sonic kind of methods have been developed, such as pulse sp(?)ctrum tech-nology(PST), generalized pulse spectrum technology(GPST), Monte Carlo method, kind of optimization methods and regularization methods. The most important and effective methods is Tikhonov regularization method.We consider a linear ill-posed inverse problem of the formwhere K : X→Y is a bounded linear operator from space X to the observation space Y. Here we call the problem ill-posed in the sense that the solution does not depend continuously on the right-hand side observation data which arc often corrupted by error. Hence, in order to transform the problem into a well-posed problem and approximate the solution in a stable way, regularization methods should be applied. Among these methods, Tikhonov approach is the most well known one, which consists of replacing the least squares problem by the problem of a suitably chosen Tikhonov functional. Here the version we choose has the following formwhereα> 0 is the regularization parameter. In practice, the effectiveness of a regularization method depends strongly on the choice of a good regularization parameter. In regularization theory, an appropriate regularization parameters is the key of solving these problems because if the rcgularization parameter is too small the stability of the problem can not be guaranteed and if the regularization parameter is too larger the accuracy of the problem can not be guaranteed.The purpose of this paper is to describe some third-order iterative methods of selecting appropriate regularization parameters for regularization methods of linear ill-posed problems. The tool is Morozov principle and some third-order iterative methods, classical-Chebyshev, Halley and Super-Halley methods. We want to get a more effective method for finding reasonable regularization parameters.Chebyshev's method. Given an initial guessα₀ generate the Chcbyshev's sequence ₁,α₂…, bywhere L(α_n) = G'(α_n)^-1G"(α_n)G'(α_n)^-1G(α_n).Halley's method. Given an initial guessα₀, generate the Halley's sequenceα₁,α₂…, bywhere L(α_n) = G'(α_n)^-1G"(α_n)G'(α_n)^-1G(α_n).Super-Halley's method. Given an initial guessα₀, generate the Super-Halley's sequenceα₁,α₂…, bywhere L(α_n) = G'(α_n)^-1G"(α_n)G'(α_n)^-1G(α_n).We use the follow lemma to proof these methods are all of third-order convergence.Lemma 1 Ifφ^p(α)(p = 1,2,…) is continuous in a neighborhood of the solutionα^*, and then the iterative methodα_n+1 =φ(α_n) is of pth-order convergence.Theorem 1 The Chebyshev's method, the Hallcy's method and the Super-Halley's methods are of third-order convergence.In the end, we apply these iterative methods to deal with examples and compare the results with the Newton's method. We get a more effective method for finding reasonable regularization parameters in linear inverse problems.