@inproceedings{40de394e64564d699105360f7623072e,
title = "Convex Optimization with Inexact Gradients in Hilbert Space and Applications to Elliptic Inverse Problems",
abstract = "In this paper, we propose the gradient descent type methods to solve convex optimization problems in Hilbert space. We apply it to solve the ill-posed Cauchy problem for the Poisson equation and make a comparative analysis with the Landweber iteration and steepest descent method. The theoretical novelty of the paper consists in the developing of a new stopping rule for accelerated gradient methods with inexact gradient (additive noise). Note that up to the moment of stopping the method “doesn{\textquoteright}t feel the noise”. But after this moment the noise starts to accumulate and the quality of the solution becomes worse for further iterations.",
keywords = "Convex optimization, Gradient method, Inexact oracle, Inverse and ill-posed problem",
author = "Vladislav Matyukhin and Sergey Kabanikhin and Maxim Shishlenin and Nikita Novikov and Artem Vasin and Alexander Gasnikov",
note = "Funding Information: The research of V.V. Matyukhin and A.V. Gasnikov in Sects. 1,2,3,4 was supported by Russian Science Foundation (project No. 21-71-30005). The research of S.I. Kabanikhin, M.A. Shishlenin and N.S. Novikov in the last section was supported by RFBR 19-01-00694 and by the comprehensive program of fundamental scientific researches of the SB RAS II.1, project No. 0314-2018-0009. The work of A. Vasin was supported by Andrei M. Raigorodskii Scholarship in Optimization. Publisher Copyright: {\textcopyright} 2021, Springer Nature Switzerland AG. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.; 20th International Conference on Mathematical Optimization Theory and Operations Research, MOTOR 2021 ; Conference date: 05-07-2021 Through 10-07-2021",
year = "2021",
doi = "10.1007/978-3-030-77876-7_11",
language = "English",
isbn = "9783030778750",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Science and Business Media Deutschland GmbH",
pages = "159--175",
editor = "Panos Pardalos and Michael Khachay and Alexander Kazakov",
booktitle = "Mathematical Optimization Theory and Operations Research - 20th International Conference, MOTOR 2021, Proceedings",
address = "Germany",
}