{"title":"Multidimensional Cubature Formulas with Superpower Convergence","authors":"A. A. Belov, M. A. Tintul","doi":"10.1134/S1064562423701478","DOIUrl":null,"url":null,"abstract":"<p>In many applications, multidimensional integrals over the unit hypercube arise, which are calculated using Monte Carlo methods. The convergence of the best of them turns out to be quite slow. In this paper, fundamentally new cubature formulas with superpower convergence based on improved Korobov grids and a special variable substitution are proposed. A posteriori error estimates are constructed, which are nearly indistinguishable from the actual accuracy. Examples of calculations illustrating the advantages of the proposed methods are given.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"108 3","pages":"514 - 518"},"PeriodicalIF":0.5000,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Doklady Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S1064562423701478","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In many applications, multidimensional integrals over the unit hypercube arise, which are calculated using Monte Carlo methods. The convergence of the best of them turns out to be quite slow. In this paper, fundamentally new cubature formulas with superpower convergence based on improved Korobov grids and a special variable substitution are proposed. A posteriori error estimates are constructed, which are nearly indistinguishable from the actual accuracy. Examples of calculations illustrating the advantages of the proposed methods are given.
期刊介绍：
Doklady Mathematics is a journal of the Presidium of the Russian Academy of Sciences. It contains English translations of papers published in Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences), which was founded in 1933 and is published 36 times a year. Doklady Mathematics includes the materials from the following areas: mathematics, mathematical physics, computer science, control theory, and computers. It publishes brief scientific reports on previously unpublished significant new research in mathematics and its applications. The main contributors to the journal are Members of the RAS, Corresponding Members of the RAS, and scientists from the former Soviet Union and other foreign countries. Among the contributors are the outstanding Russian mathematicians.
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