Efficient method for variance-based sensitivity analysis

Abstract

Presented is an efficient method for variance-based sensitivity analysis. It provides a general approach to transforming a sensitivity problem into one uncertainty propagation process, so that various existing approximation techniques (for uncertainty propagation) can be applied to speed up the computation. In this paper, formulations are deduced to implement the proposed approach with one specific technique named Univariate Reduced Quadrature (URQ). This implementation was evaluated with a number of numerical test-cases. Comparison with the traditional (benchmark) Monte Carlo approach demonstrated the accuracy and efficiency of the proposed method, which performs particularly well on the linear models, and reasonably well on most non-linear models. The current limitations with regard to non-linearity are mainly due to the limitations of the URQ method used.

Introduction

In the context of Uncertainty Quantification and Management (UQ&M), sensitivity analysis is used to identify the contribution of different uncertainty sources on the total variance of system/model outputs [1]. This is particularly useful for large scale simulation or design problems, where it is normally impractical to consider all the factors, especially at the outset. Various techniques have been developed for sensitivity analysis. Systematic reviews can be found in [1], [2], [3], [4] . Among these techniques, the variance-based method, also referred to as the Sobol’ Indices, is widely used. It has the benefits of being ‘global’ and ‘model-independent’ [1]; where ‘global’ refers to analysing all the factors simultaneously over the entire region of interest, while ‘model-independent’ means that the approach is sufficiently general to handle different problems, without the need of knowing the inner structure of the models (i.e. models are treated as ‘’black-boxes”).

The development of variance-based sensitivity analysis dates back to 1970s, when Cukier et al [5], [6], [7], Schaibly and Shuler [8], proposed the method of Fourier Amplitude Sensitivity Test (FAST), in which the Fourier Transformation and searching curves were used to decompose the output variances. Similar problems were also referred to as ‘Importance Measure’ by Hora and Iman [9], [10], Ishigami and Homma [11], and Saltelli et al [12], [13]; or ‘Top/Bottom Marginal Variance’ by Jansen [14]. In parallel, Sobol’ adopted the so-called ANOVA (Analysis of Variance)-representation to decompose a function, so that the portions of total variance caused by different factors can be formulated separately [15], [16], [17], [18], [19]. The numerical implementation is based on Monte Carlo Simulation along with multiple sampling sets (also referred as pick-freeze scheme [20]). It was later pointed out by Saltelli that all these methods calculate an equivalent statistical quantity [21], and that with this regard, the Sobol's approach is the most general one [22].

Further research has been focusing on the computational efficiency, which includes: improved sampling strategies (Sobol’ sequences [23], Latin Hyper Cube [24], and Random Balance Design (RBD) [25], [26], [27]); improved formulation of estimators (Jansen [28], Saltelli [29], Sobol’ et al [30]); approximation techniques (quadrature plus Latin Hyper Cube [31], grid quadrature [32]); Bayesian approach based on Gaussian processes (Oakley and O'hagan [33]); and Polynomial Chaos Expansion (PCE) [34], [35], [36], [37], [38], [39], [40] (where the polynomial coefficients are used to obtain the Sobol’ indices), etc.

In general, for most of the aforementioned techniques (except [26], [27]), the computational cost is related to the number of uncertainty sources, and becomes very expensive for high dimensional problems. Thus improving efficiency (i.e. the calculation speed), is still an area requiring further research, especially for early stage computational design, where the problem scale is large, and fast assessments are required.

In this research, a general approach is proposed to approximate the sensitivity indices based on the formulation from Saltelli [1], [2], [29]. In particular, we propose one implementation of the proposed approach, using the Univariate Reduced Quadrature (URQ) method [41], which was originally developed for uncertainty propagation.

The remaining part of the paper is structured as follows. Section 2 contains a background on variance-based sensitivity analysis and a brief description of the URQ method. In Section 3, the general approach for approximation is presented, followed by the detailed formulations incorporated with URQ, which include: the first order, second order, and total effect indices. The method is evaluated in Section 4, using a number of test-cases and is compared to the traditional (benchmark) MCS approach. Finally conclusions and future work are presented in Section 5.