基于正交多项式的解不适定算子方程的隐式迭代法

该文研究了基于Chebyshev和Jacobi多项式的解不适定算子方程的隐式迭代法.建立了隐式迭代法和由Hanke提出的显式迭代法之间的关系. 给出了与Chebyshev第一和第二多项式相关的迭代格式的残差有理式的一个重要引理. 对精确和扰动的数据, 研究了方程的收敛性和收敛速率. 利用Morozov残差原则, 给出了一个可执行的强健的正则化算法.最后还给出了一些数值例子, 数值结果与理论分析基本一致.

Implicit Iterative Methods Based on Some Orthogonal Polynomials for Ill Posed Operator Equations

In this paper, implicit iterative methods (IIMs) based on Chebyshev and Jacobi polynomials for ill-posed operator equations are investigated. The relation between IIMs and the explicit iterative methods (EIMs) developed by Hanke is established. An important lemma about residual rational formula of the iterative schemes related to the first and the second Chebyshev polynomials is presented. For nonperturbed and perturbed data, the convergence properties and convergence rate are studied. An implementable algorithm is given by using Morozov's discrepancy principle, which is a robust regularization algorithm. Finally, numerical examples are also given, which coincide well with theoretical results.