شناسایی ابعاد موثر مدل با استفاده از تجزیه و تحلیل حساسیت جهانی

It is shown that the effective dimensions can be estimated at reasonable computational costs using variance based global sensitivity analysis. Namely, the effective dimension in the truncation sense can be found by using the Sobol' sensitivity indices for subsets of variables. The effective dimension in the superposition sense can be estimated by using the first order effects and the total Sobol' sensitivity indices. The classification of some important classes of integrable functions based on their effective dimension is proposed. It is shown that it can be used for the prediction of the QMC efficiency. Results of numerical tests verify the prediction of the developed techniques.

Modern mathematical models of real systems in physics, chemistry, biology, economics and other areas often have high complexity with hundreds or even thousands of variables. Straightforward modelling using such models can be computationally costly or even impossible. There is a demand for complexity reduction techniques which are not only general and applicable to any complex non-linear model but also rigorous in that their application provides estimates of the approximation errors. Variance based global sensitivity analysis allows to develop such complexity reduction techniques. Recently a new class of measures was introduced by Borgonovo [1] and [2]. These measures are known as moment-independent. They are based on the entire distribution of the output without a specific reference to its moments. Potentially, moment-independent measures can also be used for complexity reduction. For modelling and complexity reduction purposes it is important to distinguish between the model nominal dimension and its effective dimension. The notions of the “effective dimension” in the truncation and superposition sense was introduced by Caflisch et al. in [3]. Quite often complex mathematical models have effective dimensions much lower than their nominal dimensions. The knowledge of model effective dimensions is very important as it allows to apply various complexity reduction techniques. The effective dimension in the truncation sense d T loosely speaking is equal to the number of important factors in the model. Identification of important and not important variables allows to fix not important variables at their nominal values. The resultant model would have lower complexity with dimensionality reduced from n to d T. A condition dT⪡ndT⪡n often occurs in practical problems. Another type of complexity reduction is associated with the effective dimension in the superposition sense dS: the function has the effective dimension in the superposition sense dS if it is almost a sum of s-dimensional function components in the ANOVA decomposition. For some problems such as path-dependent option pricing in mathematical finance changing the order in which input variables are sampled can dramatically decrease dT. Such techniques are known as dimension reduction. Most results on dimension reduction are empirical and qualitative (see for example [3]). A straightforward evaluation of the effective dimensions from their definitions is not practical in the general. Owen introduced the dimension distribution for a square integrable function [4]. The effective dimension can be defined through a quantile of the dimension distribution. He showed that for some classes of functions quantiles of the dimension distribution can be explicitly calculated but they are difficult to estimate in a general case. In this paper we show that global sensitivity analysis based on the Sobol' sensitivity indices (SI) allows to estimate the effective dimensions at reasonable computational costs. Evaluation of the Sobol' SI necessitates the computation of high-dimensional integrals. The classical grid methods become computationally impractical when the number of dimensions n increases because of “the curse of dimensionality”. The convergence rate of Monte Carlo (MC) integration methods does not depend on the number of dimensions n . However, the rate of convergence O(N−1/2)O(N−1/2), where N is the number of sampled points, attained by MC is rather slow. A higher rate of convergence can be obtained by using quasi-Monte Carlo (QMC) methods based on uniformly distributed sequences instead of pseudo-random numbers. Asymptotically, QMC can provide the rate of convergence O(N−1). For sufficiently large N , QMC should always outperform MC. However, in practice such sample sizes quite often are infeasible, especially when high-dimensional problems are concerned. Many numerical experiments demonstrated that the advantages of QMC can disappear for high-dimensional problems. There were claims that the degradation in performance of QMC occurs at n≥12n≥12 [5]. In contrast, other papers reported the superiority of QMC over MC for some integrands with n=360n=360 [6]. Some explanations for such inconsistent results were given using the notion of the effective dimension [3]. In [7] it was shown how the ANOVA components are linked to the effectiveness of QMC integration methods. Sloan and Wozniakowski [8] studied the efficiency of the quasi-Monte Carlo algorithms for high-dimensional integrals. They identified classes of functions for which the effect of the dimension is negligible. These are the so-called weighted classes in which the behavior in the successive dimensions is moderated by a sequence of weights. There is no computationally feasible technique that would predict the efficiency of QMC in high dimensions. In this paper we use Sobol' SI as a quantitative measure of the QMC efficiency. This paper is organized as follows. Section 2 briefly describes MC and QMC integration algorithms and issues concerning the possible degradation of QMC efficiency in higher dimensions. Section 3 gives a description of the Sobol' SI. Section 4 presents improved formulas for evaluation of the Sobol' SI. The notion of the effective dimension is introduced in Section 5. The classification of functions based on Sobol' SI is suggested in Section 6. It is shown how this classification can be used for the prediction of the QMC efficiency. Test examples and numerical results are considered in Section 7. Finally, conclusions are given in Section 8.

It has been shown that global sensitivity analysis allows the estimation of the effective dimensions at reasonable computational costs. Namely, dT can be found by calculation of the Sobol' sensitivity indices for subsets of variables. dS can be estimated by either using calculating the first order effects and the total Sobol' SI or by using the RS/QMC-HDMR method. Global sensitivity analysis can also be used to predict the efficiency of the QMC method. Functions with respect to their dependence on the input variables can be loosely divided into three categories: functions with not equally important variables (type A) for which dT⪡ndT⪡n; functions with equally important variables and with dominant low-order terms (type B) for which dS⪡ndS⪡n, and functions with equally important variables and with dominant interaction terms (type C) for which dS=dT=ndS=dT=n. For functions of type A and B, QMC is even in the high-dimensional case superior to MC while for functions of type C, QMC loses its advantage over MC because of the importance of higher order terms in the corresponding ANOVA decomposition. The results of numerical tests verify the prediction of the suggested classification.