Abstract

A new direction in methods for solving ill-posed problems, namely, the theory of regularizing algorithms with approximate solutions of extra-optimal quality is surveyed. A distinctive feature of these methods is that they are optimal not only in the order of accuracy of resulting approximate solutions, but also with respect to a user-specified quality functional. Such functionals can be specified, for example, as an a posteriori estimate of the quality (accuracy) of approximate solutions, a posteriori estimates of various linear functionals of these solutions, and estimates of their mathematical entropy and multidimensional variations of chosen types. The relationship between regularizing algorithms that are extra-optimal and optimal in the order of quality is studied. Issues concerning the practical derivation of a posteriori estimates for the quality of approximate solutions are addressed, and numerical algorithms for finding such estimates are described. The exposition is illustrated by results of numerical experiments.

中文翻译：

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解决病态问题的最佳方法：理论和实例综述

摘要

提出了解决不适定问题的方法的新方向，即用最优质量的近似解对算法进行正则化的理论。这些方法的一个显着特征是，它们不仅在所得近似解的准确性顺序上是最优的，而且在用户指定的质量函数方面也是最优的。例如，可以将此类函数指定为近似解的质量（准确性）的后验估计，这些解决方案的各种线性函数的后验估计以及其数学熵和所选类型的多维变化的估计。研究了按质量顺序优化和优化的正则化算法之间的关系。解决了有关对近似解的质量进行后验估计的实际推导的问题，并描述了用于找到此类估计的数值算法。数值实验的结果说明了这一论述。