Fast CBC construction of randomly shifted lattice rules achieving O(n-1+δ) convergence for unbounded integrands over Rs in weighted spaces with POD weights
Fast CBC construction of randomly shifted lattice rules achieving O(n-1+δ) convergence for unbounded integrands over Rs in weighted spaces with POD weights
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DOI：
10.1016/j.jco.2014.02.004
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发表时间：
2014-08-01
影响因子：
1.7
通讯作者：
Kuo, Frances Y.
中科院分区：
文献类型：
--
作者：
Nichols, James A.;Kuo, Frances Y.
本文为组件（CBC）构建随机转移的晶格规则的理论基础是根据实用应用而产生的R-S量身定制的随机晶格规则。对于形式F（r）的积分，f（y）pi（j = 1）phi（y（j））dy具有单变量概率密度，我们的一般策略是首先将积分映射到该积分中使用PHI的累积分布函数的倒数，然后应用Quasi-Monte Carlo（QMC）方法。但是，单位立方体中的转换的集成数很少属于具有混合第一个衍生物的功能的Sobolev sobolev空间的标准QMC设置。因此，分析需要用于Kuo，Sloan，Wasilkowski和Waterhouse（2010）的R-S上方的非标准功能空间设置。本文由三个应用程序的需求激励，将上述论文的理论扩展到几个非平凡的方向，包括针对CBC构建具有一般非产品权重的晶格规则的新错误分析，引入了一个不常规的变体为了设置，在规范中使用坐标依赖性权重功能，以及使用POD（“产品和订单相关”）重量的快速CBC构建策略。 （c）2014 Elsevier Inc.保留所有权利。
This paper provides the theoretical foundation for the component-by-component (CBC) construction of randomly shifted lattice rules that are tailored to integrals over R-s arising from practical applications. For an integral of the form f(R), f(y) Pi(s)(j=1) phi(y(j)) dy with a univariate probability density phi, our general strategy is to first map the integral into the unit cube [0, 1](s) using the inverse of the cumulative distribution function of phi, and then apply quasi-Monte Carlo (QMC) methods. However, the transformed integrand in the unit cube rarely falls within the standard QMC setting of Sobolev spaces of functions with mixed first derivatives. Therefore, a nonstandard function space setting for integrands over R-s, previously considered by Kuo, Sloan, Wasilkowski and Waterhouse (2010), is required for the analysis. Motivated by the needs of three applications, the present paper extends the theory of the aforementioned paper in several non-trivial directions, including a new error analysis for the CBC construction of lattice rules with general non-product weights, the introduction of an unanchored variant for the setting, the use of coordinate-dependent weight functions in the norm, and the strategy for fast CBC construction with POD ("product and order dependent") weights. (C) 2014 Elsevier Inc. All rights reserved.