Walsh Figure of Merit for Digital Nets: An Easy Measure for Higher Order Convergent QMC
Walsh Figure of Merit for Digital Nets: An Easy Measure for Higher Order Convergent QMC
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数字网络的 Walsh 品质因数：高阶收敛 QMC 的简单测量
DOI：
10.1007/978-3-319-33507-0_5
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Ryuichi Ohori
中科院分区：
文献类型：
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作者：
Makoto Matsumoto;Ryuichi Ohori
修正一个整数。设为可积函数。设为有限点集。准蒙特卡罗积分是指近似于三维立方体上积分的积分的平均值。柯克斯玛-赫拉夫卡不等式告诉我们，通过明智地选择，人们可能会预计误差大致会减小。对于任一点集，J.C.Dick给出了一个点集的构造，使得对于光滑，收敛速度是保证的。作为M-Saito-Matoba理论的粗略版本，M-Saito-Matoba引入了沃尔什功绩(WAFOM)，给出了收敛速度。WAFOM是高效可计算的。通过对低WAFOM点集的暴力搜索，我们观察到了几个测试被积数对和8的一个阶的收敛速度。
Fix an integers. Letbe an integrable function. Letbe a finite point set. Quasi-Monte Carlo integration offbyis the average value offoverthat approximates the integration offover thes-dimensional cube. Koksma–Hlawka inequality tells that, by a smart choice of, one may expect that the error decreases roughly. For any, J. Dick gave a construction of point sets such that for-smoothf, convergence rateis assured. As a coarse version of his theory, M-Saito-Matoba introduced Walsh figure of Merit (WAFOM), which gives the convergence rate. WAFOM is efficiently computable. By a brute-force search of low WAFOM point sets, we observe a convergence rate of orderwith, for several test integrands forand 8.