通过近似逆累积分布函数生成近似随机变量
Approximating inverse cumulative distribution functions to produce approximate random variables
AI总结：
本研究提出通过近似逆累积分布函数生成低成本近似随机变量的方法，给出两种高斯分布近似方案，验证误差收敛并测得计算加速，融入多层蒙特卡洛框架可实现5-7倍加速，扩展到相关场景加速可达250倍以上。
摘要：
针对通过逆变换法生成的随机变量，本文提出了近似随机变量的概念，这类随机变量通过对分布的逆累积分布函数进行近似来生成。这些近似方法的设计目标是计算成本低，远低于精确库函数，因此非常适用于蒙特卡洛模拟。本文针对高斯分布提出了两种近似方法：一种是等间隔区间上的分段常数近似，另一种是采用几何递减区间的分段线性近似。文中对近似的误差进行了有界分析并证明了收敛性，同时在C和C++实现中测量了计算节省量。研究考察了针对Intel和Arm硬件定制的实现，以及使用OpenMP构建的与硬件无关的实现。将这些节省量融入采用Euler-Maruyama格式的嵌套多层蒙特卡洛框架中，可在不损失精度的前提下利用速度提升，实现5到7倍的加速。这些思路还被经验性地扩展到Milstein格式，以及Cox-Ingersoll-Ross过程的非中心卡方分布场景中，可实现250倍及以上的加速。
英文摘要：
For random variables produced through the inverse transform method, approximate random variables are introduced, which are produced by approximations to a distribution's inverse cumulative distribution function. These approximations are designed to be computationally inexpensive, and much cheaper than exact library functions, and thus highly suitable for use in Monte Carlo simulations. Two approximations are presented for the Gaussian distribution: a piecewise constant on equally spaced intervals, and a piecewise linear using geometrically decaying intervals. The error of the approximations are bounded and the convergence demonstrated, and the computational savings measured for C and C++ implementations. Implementations tailored for Intel and Arm hardwares are inspected, alongside hardware agnostic implementations built using OpenMP. The savings are incorporated into a nested multilevel Monte Carlo framework with the Euler-Maruyama scheme to exploit the speed ups without losing accuracy, offering speed ups by a factor of 5--7. These ideas are empirically extended to the Milstein scheme, and the Cox-Ingersoll-Ross process' non central chi-squared distribution, which offer speed ups by a factor of 250 or more.
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