Regularizing algorithms with optimal and extra-optimal quality
A.S. Leonov
National Research Nuclear University, 115409, Moscow, Kashirskoe shosse, 31
Keywords: некорректные задачи, регуляризующие алгоритмы, качество приближенного решения, апостериорная оценка качества, РА с экстраоптимальным качеством, ill-posed problems, regularizing algorithms, quality of approximate solution, a posteriori quality estimates, RA with extra-optimal quality
Abstract
The notion of a special quality for approximate solutions to ill-posed inverse problems is introduced. A posteriori estimates of the quality are studied for different regularizing algorithms (RA). Examples of typical quality functionals are provided, which arise in solving linear and nonlinear inverse problems. The techniques and the numerical algorithm for calculating a posteriori quality estimates for approximate solutions of general nonlinear inverse problems are developed. The new notions of optimal and extra-optimal quality of a regularizing algorithm are introduced. The theory of regularizing algorithms with optimal and extra-optimal quality is presented, which includes an investigation of optimal properties for estimation functions of the quality. Examples of regularizing algorithms with extra-optimal quality of solutions are given, as well as examples of regularizing algorithms without such property. The results of numerical experiments illustrate a posteriori quality estimation.
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