Toeplitz Monte Carlo
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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,... of i.i.d. samples for one random variable, and then uses (x_n+s-1,x_n+s-2...,x_n) with n=1,2,... 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.
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