Reliable Adaptive Cubature Using Digital Sequences
Reliable Adaptive Cubature Using Digital Sequences
复制标题
使用数字序列的可靠自适应体积
DOI：
10.1007/978-3-319-33507-0_18
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Lluís Antoni Jiménez Rugama
中科院分区：
文献类型：
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作者：
F. J. Hickernell;Lluís Antoni Jiménez Rugama
准蒙特卡罗容积法通常使用Sobol'（或其他数字）序列对被积函数进行采样，以获得比IID采样更高的精度。一个重要的问题是如何保守地估计数字序列体积的误差，以便在达到所需的公差时可以终止采样。我们提出了一个基于被积函数离散沃尔什系数的误差界，并利用这个误差界构造了一个自适应数字序列体积算法。对于真沃尔什系数满足一定锥条件的被积函数，保证了误差界和相应算法的有效性。直觉上，这些锥条件意味着有序的沃尔什系数不会在很长一段时间内下降，然后又跳回来。我们的新算法的成本上界给出的沃尔什系数的\n {未知}衰减率。
Quasi-Monte Carlo cubature methods often sample the integrand using Sobol' (or other digital) sequences to obtain higher accuracy than IID sampling. An important question is how to conservatively estimate the error of a digital sequence cubature so that the sampling can be terminated when the desired tolerance is reached. We propose an error bound based on the discrete Walsh coefficients of the integrand and use this error bound to construct an adaptive digital sequence cubature algorithm. The error bound and the corresponding algorithm are guaranteed to work for integrands whose true Walsh coefficients satisfy certain cone conditions. Intuitively, these cone conditions imply that the ordered Walsh coefficients do not dip down for a long stretch and then jump back up. An upper bound on the cost of our new algorithm is given in terms of the \emph{unknown} decay rate of the Walsh coefficients.