{"title":"改进了原始功率基下数字网络增益系数的边界","authors":"Takashi Goda , Kosuke Suzuki","doi":"10.1016/j.jco.2022.101722","DOIUrl":null,"url":null,"abstract":"<div><p><span>We study randomized<span> quasi-Monte Carlo integration by scrambled nets. The scrambled net quadrature has long gained its popularity because it is an unbiased estimator of the true integral, allows for a practical error estimation, achieves a high order decay of the variance for smooth functions, and works even for </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-functions with any <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>. The variance of the scrambled net quadrature for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-functions can be evaluated through the set of the so-called <em>gain coefficients.</em></p><p>In this paper, based on the system of Walsh functions and the concept of dual nets, we provide improved upper bounds on the gain coefficients for digital nets in general prime power base. Our results explain the known bound by Owen (1997) for Faure sequences, the recently improved bound by Pan and Owen (2022) for digital nets in base 2 (including Sobol' sequences as a special case), and their finding that all the nonzero gain coefficients for digital nets in base 2 must be powers of two, all in a unified way.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"76 ","pages":"Article 101722"},"PeriodicalIF":1.8000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Improved bounds on the gain coefficients for digital nets in prime power base\",\"authors\":\"Takashi Goda , Kosuke Suzuki\",\"doi\":\"10.1016/j.jco.2022.101722\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span>We study randomized<span> quasi-Monte Carlo integration by scrambled nets. The scrambled net quadrature has long gained its popularity because it is an unbiased estimator of the true integral, allows for a practical error estimation, achieves a high order decay of the variance for smooth functions, and works even for </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-functions with any <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>. The variance of the scrambled net quadrature for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-functions can be evaluated through the set of the so-called <em>gain coefficients.</em></p><p>In this paper, based on the system of Walsh functions and the concept of dual nets, we provide improved upper bounds on the gain coefficients for digital nets in general prime power base. Our results explain the known bound by Owen (1997) for Faure sequences, the recently improved bound by Pan and Owen (2022) for digital nets in base 2 (including Sobol' sequences as a special case), and their finding that all the nonzero gain coefficients for digital nets in base 2 must be powers of two, all in a unified way.</p></div>\",\"PeriodicalId\":50227,\"journal\":{\"name\":\"Journal of Complexity\",\"volume\":\"76 \",\"pages\":\"Article 101722\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2023-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Complexity\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0885064X22000875\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Complexity","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0885064X22000875","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Improved bounds on the gain coefficients for digital nets in prime power base
We study randomized quasi-Monte Carlo integration by scrambled nets. The scrambled net quadrature has long gained its popularity because it is an unbiased estimator of the true integral, allows for a practical error estimation, achieves a high order decay of the variance for smooth functions, and works even for -functions with any . The variance of the scrambled net quadrature for -functions can be evaluated through the set of the so-called gain coefficients.
In this paper, based on the system of Walsh functions and the concept of dual nets, we provide improved upper bounds on the gain coefficients for digital nets in general prime power base. Our results explain the known bound by Owen (1997) for Faure sequences, the recently improved bound by Pan and Owen (2022) for digital nets in base 2 (including Sobol' sequences as a special case), and their finding that all the nonzero gain coefficients for digital nets in base 2 must be powers of two, all in a unified way.
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
• Biomedical computing
• Compressed computing and sensing
• Computational finance
• Computational number theory
• Computational stochastics
• Control theory
• Cryptography
• Design of experiments
• Differential equations
• Discrete problems
• Distributed and parallel computation
• High and infinite-dimensional problems
• Information-based complexity
• Inverse and ill-posed problems
• Machine learning
• Markov chain Monte Carlo
• Monte Carlo and quasi-Monte Carlo
• Multivariate integration and approximation
• Noisy data
• Nonlinear and algebraic equations
• Numerical analysis
• Operator equations
• Optimization
• Quantum computing
• Scientific computation
• Tractability of multivariate problems
• Vision and image understanding.