Stratified Monte Carlo quadrature for continuous random fields

Details Stratified Monte Carlo quadrature for continuous random fields Stratified Monte Carlo quadrature for continuous random fields

2011-04-26

arxiv.org/abs/1104.4920v2

Mathematics

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17 pages, 6 figures, typos and references corrected, results unchanged

Scientific paper

We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition (or strata) generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behavior is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the H\"{o}lder class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.

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