Journal of Mathematical Chemistry, Vol. 43, No. 3, March 2008 (© 2007) DOI: 10.1007/s10910-007-9250-x
Regularized random-sampling high dimensional model representation (RS-HDMR) Genyuan Li and Herschel Rabitz∗ Department of Chemistry, Princeton University, Princeton, NJ 08544, USA E-mail:
[email protected]
Jishan Hu Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Zheng Chen and Yiguang Ju Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Received 24 November 2006; Revised 19 January 2007
High Dimensional Model Representation (HDMR) is under active development as a set of quantitative model assessment and analysis tools for capturing high-dimensional input–output system behavior. HDMR is based on a hierarchy of component functions of increasing dimensions. The Random-Sampling High Dimensional Model Representation (RS-HDMR) is a practical approach to HDMR utilizing random sampling of the input variables. To reduce the sampling effort, the RS-HDMR component functions are approximated in terms of a suitable set of basis functions, for instance, orthonormal polynomials. Oscillation of the outcome from the resultant orthonormal polynomial expansion can occur producing interpolation error, especially on the input domain boundary, when the sample size is not large. To reduce this error, a regularization method is introduced. After regularization, the resultant RS-HDMR component functions are smoother and have better prediction accuracy, especially for small sample sizes (e.g., often few hundred). The ignition time of a homogeneous H2 /air combustion system within the range of initial temperature, 1000 < T0 < 1500 K, pressure, 0.1 < P < 100 atm and equivalence ratio of H2 /O2 , 0.2 < R < 10 is used for testing the regularized RS-HDMR. KEY WORDS: high dimensional model representation (HDMR), orthonormal polynomials, regularization, smoothing, combustion, ignition AMS(MOS) subject classifications: 33C50, 41A10, 57R12
∗ Corresponding author.
1207 0259-9791/08/0003-1207/0 © 2007 Springer Science+Business Media, LLC
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Introduction
High Dimensional Model Representation (HDMR) [1–22] is under development as a set of quantitative model assessment and analysis tools for capturing high-dimensional input–output system behavior. The HDMR techniques have been successfully applied in a variety of applications including semiconductor formulation [2], amino acid mutations of proteins [6], atmospheric chemistry [7], atmospheric solar radiation transport [8], molecular dynamics simulations [17], rate constant determination from concentration observations [18], and optimal control of molecular motion [19]. Recently, the capabilities of the HDMR technique were extended through the introduction of Random Sampling (RS)HDMR, which is a practical procedure based on RS of the input variables. RS-HDMR is very efficient for treating high dimensional input–output mapping problems and has been successfully utilized in several modeling applications, e.g., atmospheric chemistry [9], environmental metal bioremediation [11], integrated exposure and dose studies [12], bio-kinetics modeling [21]. As the impact of the multiple input variables on the output can be independent and cooperative, HDMR expresses the output f (x) as a finite hierarchical correlated function expansion in terms of the input variables x = (x1 , x2 , . . . , xn ): f (x) = f0 +
n
fi (xi ) +
i=1
+
fij (xi , xj ) + · · ·
1≤i
