Ratio control variate method for efficiently determining high-dimensional model representations

Genyuan Li, Herschel Rabitz

Research output: Contribution to journalArticlepeer-review

29 Scopus citations

Abstract

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/√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.

Original languageEnglish (US)
Pages (from-to)1112-1118
Number of pages7
JournalJournal of Computational Chemistry
Volume27
Issue number10
DOIs
StatePublished - Jul 30 2006

All Science Journal Classification (ASJC) codes

  • Chemistry(all)
  • Computational Mathematics

Keywords

  • Atmospheric chemistry
  • Control variate
  • HDMR
  • High-dimensional systems
  • Monte Carlo integration
  • Random sampling

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