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

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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.

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Funding

This work was supported by the Russian Foundation for Basic Research (project nos. 17-01-00159-a and 19-51-53005-GFEN-a) and by the Program of Competitiveness Enhancement for the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) (contract no. 02.a03.21.0005 of August 27, 2013).

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    A. S. Leonov

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Translated by I. Ruzanova

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Leonov, A.S. Extra-Optimal Methods for Solving Ill-Posed Problems: Survey of Theory and Examples. Comput. Math. and Math. Phys. 60, 960–986 (2020). https://doi.org/10.1134/S0965542520060068

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