Abstract
We consider extremal problems introduced and investigated earlier by the author for calculating global and local a posteriori error estimates of approximate solutions to ill-posed inverse problems. For linear inverse problems in Hilbert spaces, they consist in maximization of quadratic functionals with two quadratic constraints. The article shows how under certain conditions these problems can be reduced to a problem of maximization of a special (written analytically) differentiable functional with one constraint. New algorithms for calculating global and local a posteriori error estimates based on the solution of these problems are proposed. Their effectiveness is illustrated by numerical experiments on a posteriori error estimation of solutions to the model two-dimensional inverse problem of potential continuation. Experiments show that the proposed algorithms give a posteriori error estimates close to the true error values. Proposed algorithms for global a posteriori error estimation turn out to be more rapid (3 to 5 times) than the previously known algorithms.
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Funding
The work is financially supported by Russian Foundation for Basic Research (projects nos. 17-01-00159-a and 19-51-53005-GFEN-a), and by the Program of competitive growth of National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), project no. 02.a03.21.0005 from 27.08.2013.
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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 2, pp. 29–38.
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Leonov, A.S. Effective Algorithms for Computing Global and Local Posterior Error Estimates of Solutions to Linear Ill-Posed Problems. Russ Math. 64, 26–34 (2020). https://doi.org/10.3103/S1066369X20020048
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DOI: https://doi.org/10.3103/S1066369X20020048