Antithetic variates in higher dimensions

2009/02/24 by Sebastián del Baño Rollin, Rollin, Sebastian del Baño, Joan-Andreu Lázaro-Camí +1
Computer Science · Mathematics · Physics and Astronomy · #65C05 #68W20 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Probability (math.PR) #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.0902.4211

openalex publication_date 2009/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the concept of multidimensional antithetic as the absolute minimum of the covariance defined on the orthogonal group by A↦ Cov(f(ξ),f(Aξ)) where ξ is a standard N-dimensional normal random variable and f:ℝ^N→ℝ is an almost everywhere differentiable function. The antithetic matrix is designed to optimise the calculation of E[f(ξ)] in a Monte Carlo simulation. We present an iterative annealing algorithm that dynamically incorporates the estimation of the antithetic matrix within the Monte Carlo calculation.

Citations

• A Stochastic Approximation Method
• Asymptotics of the spectral gap with applications to the theory of simulated annealing

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