{"title":"Numerical weighted integration of functions having mixed smoothness","authors":"Dinh Dũng","doi":"10.1016/j.jco.2023.101757","DOIUrl":null,"url":null,"abstract":"<div><p>We investigate the approximation of weighted integrals over <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span><span> for integrands<span><span> from weighted Sobolev spaces of mixed smoothness. We prove </span>upper and lower bounds of the convergence rate of optimal quadratures with respect to </span></span><em>n</em> integration nodes for functions from these spaces. In the one-dimensional case <span><math><mo>(</mo><mi>d</mi><mo>=</mo><mn>1</mn><mo>)</mo></math></span>, we obtain the right convergence rate of optimal quadratures. For <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"78 ","pages":"Article 101757"},"PeriodicalIF":1.8000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Complexity","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0885064X23000262","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the approximation of weighted integrals over for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to n integration nodes for functions from these spaces. In the one-dimensional case , we obtain the right convergence rate of optimal quadratures. For , the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain .
期刊介绍：
The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited.
Areas Include:
• Approximation theory
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• Multivariate integration and approximation
• Noisy data
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• Operator equations
• Optimization
• Quantum computing
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• Tractability of multivariate problems
• Vision and image understanding.
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