یک تقریب موثر برای تجزیه و تحلیل حساسیت جهانی بر اساس واریانس

The paper presents a fairly efficient approximation for the computation of variance-based sensitivity measures associated with a general, n-dimensional function of random variables. The proposed approach is based on a multiplicative version of the dimensional reduction method (M-DRM), in which a given complex function is approximated by a product of low dimensional functions. Together with the Gaussian quadrature, the use of M-DRM significantly reduces the computation effort associated with global sensitivity analysis. An important and practical benefit of the M-DRM is the algebraic simplicity and closed-form nature of sensitivity coefficient formulas. Several examples are presented to show that the M-DRM method is as accurate as results obtained from simulations and other approximations reported in the literature.

In the context of a probabilistic analysis, the system response is typically represented by a function of random variables. The sensitivity of the response to input random variables can be quantified by the contribution of a random variable to the total variance of the response. This is the essence of the variance-based global sensitivity analysis in the literature [1]. The analytical basis for the global sensitivity analysis comes from ANOVA (Analysis of Variance) decomposition of the response variance [2]. Although ANOVA decomposition is conceptually simple, the computation of variance components of a general response function is rather a challenging task. The reason is that it involves a series of high-dimensional integrations for each global sensitivity coefficient. Therefore, the minimization of computational efforts is a primary area of research in the variance-based global sensitivity analysis, and several studies have already been presented in the literature. The Monte Carlo simulation is the most effective method for global sensitivity analysis of a general response function. Smart simulation algorithms have been developed to evaluate high-dimensional integrals [3] and [4]. In case of a complex model however, the simulation method can be so time consuming that it can deter applications of sensitivity analysis in day to day engineering practice. This has motivated the development of simple approximations for the sensitivity analysis. The most popular approach is based on the concept of high dimensional model representation (HDMR) [5], in which a complex function is decomposed into a hierarchy of low dimensional functions in an additive expansion. The HDMR basically creates a surrogate model, which simplifies the computation [6]. Tarantola et al. [7] proposed the random balance design (RBD) for sensitivity analysis of a nuclear waste disposal system. Sudret [8] reviewed polynomial chaos expansion on surrogate model construction, in which computation of global sensitivity coefficients is directly related to expansion coefficients of a PCE model [9]. Given the vast literature related to sensitivity analysis, the readers are referred to monographs for a detailed review of methods of sensitivity analysis [10] and [11].

The paper presents an effective approximate method for the computation of variance-based sensitivity coefficients. The proposed method is based on multiplicative dimensional reduction method (M-DRM), in which the response function is approximated as a product of univariate functions. The most notable aspect of the proposed M-DRM is that simple algebraic formulas can be derived for the primary, high-order (on joint effect) and the total sensitivity coefficients. Based on an N-point Gaussian quadrature, M-DRM requires nN function evaluations only in the sensitivity analysis of a function of n random variables, which implies that the proposed method significantly reduces the number of functional evaluations required for the sensitivity analysis. The performance of M-DRM is evaluated by analyzing six examples taken from the literature. In all the cases, sensitivity coefficients obtained from M-DRM are in excellent agreement with analytical or simulation-based reference solutions. In summary, the multiplicative dimensional reduction method provides a simple and efficient alternative for global sensitivity analysis in a practical setting.