Extra-Optimal Methods for Solving Ill-Posed Problems: Survey of Theory and Examples

Autor:	Alexander S. Leonov
Rok vydání:	2020
Předmět:	Well-posed problem 010102 general mathematics Order of accuracy Distinctive feature 01 natural sciences 010101 applied mathematics Computational Mathematics Quality (physics) Entropy (information theory) Applied mathematics A priori and a posteriori 0101 mathematics Optimal methods Mathematics
Zdroj:	Computational Mathematics and Mathematical Physics. 60:960-986
ISSN:	1555-6662 0965-5425
DOI:	10.1134/s0965542520060068
Popis:	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.
Databáze:	OpenAIRE
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