{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T08:39:59Z","timestamp":1771663199551,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T00:00:00Z","timestamp":1662681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education and Science of the Russian Federation","award":["N 075-15-2019-1621"],"award-info":[{"award-number":["N 075-15-2019-1621"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper discusses a method for taking into account rounding errors when constructing a stopping criterion for the iterative process in gradient minimization methods. The main aim of this work was to develop methods for improving the quality of the solutions for real applied minimization problems, which require significant amounts of calculations and, as a result, can be sensitive to the accumulation of rounding errors. However, this paper demonstrates that the developed approach can also be useful in solving computationally small problems. The main ideas of this work are demonstrated using one of the possible implementations of the conjugate gradient method for solving an overdetermined system of linear algebraic equations with a dense matrix.<\/jats:p>","DOI":"10.3390\/a15090324","type":"journal-article","created":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T20:52:25Z","timestamp":1663015945000},"page":"324","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Accounting for Round-Off Errors When Using Gradient Minimization Methods"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5140-3617","authenticated-orcid":false,"given":"Dmitry","family":"Lukyanenko","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia"},{"name":"Moscow Center for Fundamental and Applied Mathematics, 119234 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8365-7224","authenticated-orcid":false,"given":"Valentin","family":"Shinkarev","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6942-2138","authenticated-orcid":false,"given":"Anatoly","family":"Yagola","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"409","DOI":"10.6028\/jres.049.044","article-title":"Methods of conjugate gradients for solving linear systems","volume":"49","author":"Hestenes","year":"1952","journal-title":"J. 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Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"140206","DOI":"10.1098\/rsos.140206","article-title":"New stopping criteria for iterative root finding","volume":"1","author":"Nikolajsen","year":"2014","journal-title":"R. Soc. Open Sci."},{"key":"ref_6","unstructured":"Polyak, B., Kuruzov, I., and Stonyakin, F. (2022). Stopping rules for gradient methods for non-convex problems with additive noise in gradient. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kabanikhin, S. (2011). Inverse and Ill-Posed Problems: Theory and Applications, Walter de Gruyter.","DOI":"10.1515\/9783110224016"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Vasin, A., Gasnikov, A., Dvurechensky, P., and Spokoiny, V. (2022). Accelerated gradient methods with absolute and relative noise in the gradient. arXiv.","DOI":"10.1080\/10556788.2023.2212503"},{"key":"ref_9","unstructured":"Cohen, M., Diakonikolas, J., and Orecchia, L. (2018, January 9). On acceleration with noise-corrupted gradients. Proceedings of the 35th International Conference on Machine Learning, 2018, Stockholmsm\u00e4ssan, Sweden."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s10957-016-0999-6","article-title":"Stochastic intermediate gradient method for convex problems with stochastic inexact oracle","volume":"171","author":"Dvurechensky","year":"2016","journal-title":"J. Optim. Theory Appl."},{"key":"ref_11","unstructured":"Gasnikov, A., Kabanikhin, S., Mohammed, A., and Shishlenin, M. (2017). Convex optimization in Hilbert space with applications to inverse problems. arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.jcp.2017.09.033","article-title":"A stopping criterion for the iterative solution of partial differential equations","volume":"352","author":"Rao","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1137\/0613012","article-title":"Stopping Criteria for Iterative Solvers","volume":"13","author":"Arioli","year":"1992","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s100920170006","article-title":"Stopping criteria for iterative methods: Applications to PDE\u2019s","volume":"38","author":"Arioli","year":"2001","journal-title":"Calcolo"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00211-003-0500-y","article-title":"A stopping criterion for the conjugate gradient algorithm in a finite element method framework","volume":"97","author":"Arioli","year":"2004","journal-title":"Numer. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1007\/s00211-004-0568-z","article-title":"Stopping criteria for iterations in finite element methods","volume":"99","author":"Arioli","year":"2005","journal-title":"Numer. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1137\/080724071","article-title":"Stopping Criteria for the Iterative Solution of Linear Least Squares Problems","volume":"31","author":"Chang","year":"2009","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1002\/nla.244","article-title":"Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations","volume":"8","author":"Axelsson","year":"2001","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1007\/BF01934094","article-title":"A practical termination criterion for the conjugate gradient method","volume":"28","author":"Kaasschieter","year":"1988","journal-title":"BIT Numer. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1567","DOI":"10.1137\/08073706X","article-title":"A posteriori error estimates including algebraic error and stopping criteria for iterative solvers","volume":"32","year":"2010","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.apnum.2016.03.006","article-title":"A stopping criterion for iterative regularization methods","volume":"106","author":"Landi","year":"2016","journal-title":"Appl. Numer. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1134\/S1064562413040133","article-title":"Improved forms of iterative methods for systems of linear algebraic equations","volume":"88","author":"Kalitkin","year":"2013","journal-title":"Dokl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Tikhonov, A., Goncharsky, A., Stepanov, V., and Yagola, A. (1995). Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-8480-7"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/9\/324\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:28:49Z","timestamp":1760142529000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/9\/324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,9]]},"references-count":23,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["a15090324"],"URL":"https:\/\/doi.org\/10.3390\/a15090324","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,9]]}}}