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Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

2014; Wiley; Volume: 288; Issue: 7 Linguagem: Inglês

10.1002/mana.201400048

1522-2616

Đinh Dũng, Tino Ullrich,

Advanced Numerical Analysis Techniques

Mathematische NachrichtenVolume 288, Issue 7 p. 743-762 Original Paper Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square Dinh Dũng, Dinh Dũng dinhzung@gmail.com Vietnam National University, Hanoi, Information Technology Institute, 144, Xuan Thuy, Hanoi, VietnamSearch for more papers by this authorTino Ullrich, Corresponding Author Tino Ullrich Hausdorff-Center for Mathematics and Institute for Numerical Simulation, 53115 Bonn, GermanyCorresponding author: e-mail: tino.ullrich@hcm.uni-bonn.deSearch for more papers by this author Dinh Dũng, Dinh Dũng dinhzung@gmail.com Vietnam National University, Hanoi, Information Technology Institute, 144, Xuan Thuy, Hanoi, VietnamSearch for more papers by this authorTino Ullrich, Corresponding Author Tino Ullrich Hausdorff-Center for Mathematics and Institute for Numerical Simulation, 53115 Bonn, GermanyCorresponding author: e-mail: tino.ullrich@hcm.uni-bonn.deSearch for more papers by this author First published: 31 October 2014 https://doi.org/10.1002/mana.201400048Citations: 23Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness in the case , and , where is either the d-dimensional torus or the d-dimensional unit cube . In addition, we prove upper bounds for QMC integration on the Fibonacci-lattice for bivariate periodic functions from in the case , and . A non-periodic modification of this classical formula yields upper bounds for if . In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from and indicate that a corresponding result is most likely also true in case . This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids can never achieve the optimal worst-case error. Citing Literature Volume288, Issue7May 2015Pages 743-762 RelatedInformation