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.

主要是由于应用到具有随机系数的偏微分方程的推动下，我们引入了一种新的蒙特卡洛估计器，称为Toeplitz蒙特卡洛（TMC）估计器，用于相对于相同单变量概率的直接乘积来逼近多元函数的积分。测量。TMC估计器为一个随机变量生成iid样本的序列\（x_1，x_2，\ ldots \），然后使用\（（x_ {n + s-1}，x_ {n + s-2} \ ldots，x_n ）\），其中\（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，但是所有正交节点的级联由Toeplitz矩阵表示，这允许快速进行矩阵-矢量乘法。在本文中，我们研究了TMC估计量的方差及其对维数s的依赖性。数值实验证实，与标准的蒙特卡洛估计器相比，对于具有随机系数的偏微分方程的应用，效率有显着提高，尤其是当尺寸s大时。