{"title":"张量乘空间积分的难解性结果","authors":"Erich Novak , Friedrich Pillichshammer","doi":"10.1016/j.jco.2024.101901","DOIUrl":null,"url":null,"abstract":"<div><div>We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number <em>N</em> of function evaluations that is necessary such that the <em>N</em>-th minimal error is less than a factor <em>ε</em> times the initial error, i.e., the error for <span><math><mi>N</mi><mo>=</mo><mn>0</mn></math></span>, where <em>ε</em> belongs to <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We are interested to which extent the information complexity depends on the number <em>d</em> of variables of the integrands. If the information complexity grows exponentially fast in <em>d</em>, then the integration problem is said to suffer from the curse of dimensionality.</div><div>Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101901"},"PeriodicalIF":1.8000,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Intractability results for integration in tensor product spaces\",\"authors\":\"Erich Novak , Friedrich Pillichshammer\",\"doi\":\"10.1016/j.jco.2024.101901\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number <em>N</em> of function evaluations that is necessary such that the <em>N</em>-th minimal error is less than a factor <em>ε</em> times the initial error, i.e., the error for <span><math><mi>N</mi><mo>=</mo><mn>0</mn></math></span>, where <em>ε</em> belongs to <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We are interested to which extent the information complexity depends on the number <em>d</em> of variables of the integrands. If the information complexity grows exponentially fast in <em>d</em>, then the integration problem is said to suffer from the curse of dimensionality.</div><div>Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.</div></div>\",\"PeriodicalId\":50227,\"journal\":{\"name\":\"Journal of Complexity\",\"volume\":\"86 \",\"pages\":\"Article 101901\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-10-16\",\"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/S0885064X24000785\",\"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/S0885064X24000785","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
摘要
我们证明了张量乘空间中数值积分最坏情况误差的下限。信息复杂度是函数求值的最小次数 N,即 N 次最小误差小于初始误差的系数 ε 倍,即 N=0 时的误差,其中 ε 属于 (0,1)。我们感兴趣的是,信息复杂度在多大程度上取决于积分变量的数量 d。如果信息复杂度在 d 的范围内呈指数增长,那么积分问题就会受到维度诅咒的影响。在单变量问题存在最坏情况函数的假设下,我们提出了两种提供信息复杂度下限的方法。第一种方法基于对最坏情况函数的适当分解,可视为可分解再现核方法的一般化。第二种方法虽然只适用于正二次函数规则,但不需要对最坏情况函数进行适当分解。相反,它以最坏情况函数的样条近似为基础,可用于解析函数。本文介绍了这两种方法的几种应用。
Intractability results for integration in tensor product spaces
We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number N of function evaluations that is necessary such that the N-th minimal error is less than a factor ε times the initial error, i.e., the error for , where ε belongs to . We are interested to which extent the information complexity depends on the number d of variables of the integrands. If the information complexity grows exponentially fast in d, then the integration problem is said to suffer from the curse of dimensionality.
Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.
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.