论文标题
沃尔什功能，炒$（0，m，s）$ - 网和负协方差：将符号计算应用于准蒙特卡洛集成
Walsh functions, scrambled $(0,m,s)$-nets, and negative covariance: applying symbolic computation to quasi-Monte Carlo integration
论文作者
论文摘要
我们调查了基本$ b $ walsh函数，基于$（0，m，s）$ - 基本$ b $中的net缩放估算器的差异小于或等于基于相同数量的蒙特 - 卡洛估计器的净值。首先，我们计算两个不同的点从炒$（t，m，s）$ - 基本$ b $中随机选择的两个不同点的联合概率密度函数的WALSH分解。使用此功能，我们根据函数的沃尔什系数获得了积分估计器协方差的表达式。最后，我们证明，当功能的沃尔什系数满足特定衰减条件时，积分估计器的协方差为负。为此，我们使用符号计算中的创意望远镜求解算法来为协方差术语找到一个等效封闭形式的表达式。
We investigate base $b$ Walsh functions for which the variance of the integral estimator based on a scrambled $(0,m,s)$-net in base $b$ is less than or equal to that of the Monte-Carlo estimator based on the same number of points. First we compute the Walsh decomposition for the joint probability density function of two distinct points randomly chosen from a scrambled $(t,m,s)$-net in base $b$ in terms of certain counting numbers and simplify it in the special case $t$ is zero. Using this, we obtain an expression for the covariance of the integral estimator in terms of the Walsh coefficients of the function. Finally, we prove that the covariance of the integral estimator is negative when the Walsh coefficients of the function satisfy a certain decay condition. To do this, we use creative telescoping and recurrence solving algorithms from symbolic computation to find a sign equivalent closed form expression for the covariance term.
