Toeplitz Monte Carlo

Abstract

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.