Abstract

Results obtained in recent years concerning necessary and sufficient conditions for the convergence (at a given rate) of approximation methods for solutions of irregular operator equations are overviewed. The exposition is given in the context of classical direct and converse theorems of approximation theory. Due to the proximity of the resulting necessary and sufficient conditions to each other, the solutions on which a certain convergence rate of the methods is reached can be characterized nearly completely. The problems under consideration include irregular linear and nonlinear operator equations and ill-posed Cauchy problems for first- and second-order differential operator equations. Procedures for stable approximation of solutions of general irregular linear equations, classes of finite-difference regularization methods and the quasi-reversibility method for ill-posed Cauchy problems, and the class of iteratively regularized Gauss–Newton type methods for irregular nonlinear operator equations are examined.

中文翻译：

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求解不规则算子方程的迭代方法的正逆定理和不适定柯西问题的有限差分方法

摘要

概述了近年来获得的有关不规则算子方程解的逼近方法（以给定速率）收敛的必要和充分条件的结果。在近似理论的经典直接和逆定理的上下文中给出了说明。由于所产生的必要条件和充分条件彼此接近，因此可以接近完全地表征达到该方法的一定收敛速度的解决方案。正在考虑的问题包括不规则的线性和非线性算子方程，以及一阶和二阶微分算子方程的不适定柯西问题。一般不规则线性方程组解的稳定逼近程序，