论文标题
Toeplitz Monte Carlo
Toeplitz Monte Carlo
论文作者
论文摘要
我们主要是由具有随机系数的部分微分方程应用的，我们引入了一类新的蒙特卡洛估计量，称为Toeplitz Monte Carlo（TMC）估计器，以近似于相同的单位单位概率指标的多元功能与直接乘积的多元功能的积分。 TMC估算器生成一个序列$ x_1，x_2，\ ldots $ i.i.d.。一个随机变量的示例，然后使用$（x_ {n+s-1}，x_ {n+s-2} \ ldots，x_n）$，用$ n = 1,2，\ ldots $作为正交点，其中$ s $表示尺寸。尽管连续点具有一定的依赖性，但所有正交淋巴结的串联均由toeplitz矩阵表示，该矩阵允许快速矩阵矢量乘法。在本文中，我们研究了TMC估计器的差异及其对尺寸$ s $的依赖性。数值实验证实了针对具有随机系数的部分微分方程的标准蒙特卡洛估计器的效率，尤其是当尺寸$ s $大时。
Motivated mainly by applications to partial differential equations with random coefficients, we introduce a new class of Monte Carlo estimators, called Toeplitz Monte Carlo (TMC) estimator for approximating the integral of a multivariate function with respect to the direct product of an identical univariate probability measure. The TMC estimator generates a sequence $x_1,x_2,\ldots$ of i.i.d. samples for one random variable, and then uses $(x_{n+s-1},x_{n+s-2}\ldots,x_n)$ with $n=1,2,\ldots$ as quadrature points, where $s$ denotes the dimension. Although consecutive points have some dependency, the concatenation of all quadrature nodes is represented by a Toeplitz matrix, which allows for a fast matrix-vector multiplication. In this paper we study the variance of the TMC estimator and its dependence on the dimension $s$. Numerical experiments confirm the considerable efficiency improvement over the standard Monte Carlo estimator for applications to partial differential equations with random coefficients, particularly when the dimension $s$ is large.
