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Ratio control variate method for efficiently determining high‐dimensional model representations
Author(s) -
Li Genyuan,
Rabitz Herschel
Publication year - 2006
Publication title -
journal of computational chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.907
H-Index - 188
eISSN - 1096-987X
pISSN - 0192-8651
DOI - 10.1002/jcc.20435
Subject(s) - control variates , random variate , monte carlo method , interpolation (computer graphics) , mathematics , estimator , function (biology) , component (thermodynamics) , monte carlo integration , hybrid monte carlo , statistics , computer science , random variable , markov chain monte carlo , physics , animation , computer graphics (images) , evolutionary biology , biology , thermodynamics
The High‐Dimensional Model Representation (HDMR) technique is a family of approaches to efficiently interpolate high‐dimensional functions. RS(Random Sampling)‐HDMR is a practical form of HDMR based on randomly sampling the overall function, and utilizing orthonormal polynomial expansions to approximate the RS‐HDMR component functions. The determination of the expansion coefficients for the component functions employs Monte Carlo integration, which controls the accuracy of the RS‐HDMR interpolation. The control variate method is an established approach to improve the accuracy of Monte Carlo integration. However, this method is often not practical for an arbitrary function f ( x ) because there is no general way to find the analytical control variate function h ( x ), which needs to be very similar to f ( x ). In this article, we show that truncated RS‐HDMR expansions can be used as h ( x ) for arbitrary f ( x ), and a more stable iterative ratio control variate method was developed for the determination of the expansion coefficients for the RS‐HDMR component functions. As the asymptotic error (standard deviation) of the estimator given by the ratio control variate method is proportional to 1/ N (sample size), it is more efficient than the direct Monte Carlo integration, whose error is proportional to $1/\sqrt{N}$ . In an illustration of a four‐dimensional atmospheric model a few hundred random samples are sufficient to construct an RS‐HDMR expansion by the ratio control variate method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples. © 2006 Wiley Periodicals, Inc. J Comput Chem 27: 1112–1118, 2006