تجزیه و تحلیل حساسیت جهانی برای خروجی های چند متغیره با استفاده از توسعه آشوب چندجمله ای

Many mathematical and computational models used in engineering produce multivariate output that shows some degree of correlation. However, conventional approaches to Global Sensitivity Analysis (GSA) assume that the output variable is scalar. These approaches are applied on each output variable leading to a large number of sensitivity indices that shows a high degree of redundancy making the interpretation of the results difficult. Two approaches have been proposed for GSA in the case of multivariate output: output decomposition approach [9] and covariance decomposition approach [14] but they are computationally intensive for most practical problems. In this paper, Polynomial Chaos Expansion (PCE) is used for an efficient GSA with multivariate output. The results indicate that PCE allows efficient estimation of the covariance matrix and GSA on the coefficients in the approach defined by Campbell et al. [9], and the development of analytical expressions for the multivariate sensitivity indices defined by Gamboa et al. [14].

Important models of interest in engineering produce vector or multivariate output, such as partial differential equations (PDE) that describe spatial and temporal changes of variables of interest. Conventional methodologies for Global Sensitivity Analysis (GSA), including Sobol decomposition [24] and moment-independent approach [8], were designed for scalar output. However, these methodologies are applied to each one of the variables comprising the multivariate output resulting in a large number of sensitivity measures. If the correlation in the output is strong, there is a high degree of redundancy in the estimated indices, situation in which it is difficult to interpret the results of the sensitivity analysis. Saltelli and Tarantola [23] warned about this problem “A possible cause of difficulty for SA is when the model consists of a large set (k) of input factors and, simultaneously, has many output variables (e.g., m). In such a case, a complete analysis would require the estimation of the sensitivity of each output to every input, thus returning m×k indices. We believe that the analysis can be made more effective by focusing not on the model output per se but on the problem that such output is supposed to solve. To this end, model use should be declared before uncertainty and sensitivity analyses are performed”. In other words, the suggestion proposed by Saltelli and Tarantola [23] is to simplify the original problem defining a scalar variable of interest to apply the GSA. Although this approach can be applied in many cases, there are some situations where this reduction is not possible due to the specific nature of the problem. In the case of PDE, the use of the sensitivity measures designed for scalar output ignores the important characteristics of spatial and/or temporal correlation that is generated from the physical processes encoded by mathematical model. There is a growing need for GSA methodologies specifically designed for multivariate output due to the increasing role of complex quantitative models used by engineers/scientists to support decision making. There are two approaches for the application of GSA in the case of multivariate output. In the first approach, Campbell et al. [9] proposed a methodology for GSA when the model output can be represented as functions that can be extended to multivariate output. In this approach the model output is decomposed in an orthogonal basis and then GSA is applied to the coefficients of this expansion. Lamboni et al. [18] applied this approach to mathematical models of crop growth, where the output displayed temporal variations. The orthonormal basis used in this case was the eigenvectors of the covariance matrix, and the sensitivities of the coefficients were estimated using conventional ANOVA decomposition. Following this work, Lamboni et al. [19] proposed a new set of sensitivity indices for multivariate output that can be applied in the approach defined by Campbell et al. [9]. In the second approach, Gamboa et al. [14] defined a new set of sensitivity measures based on decomposition of the covariance of the model output that are equivalent to the Sobol indices in the scalar case. This approach does not require the spectral decomposition of the covariance matrix as in the output decomposition approach [18] and [19] and therefore it is expected to be more efficient in computational terms. In the previous approaches, estimation of sensitivity measures is based on Monte Carlo simulation [24], [21] and [22], an approach that is simple to apply but in some cases requires a large number of model evaluations. Metamodels are good alternatives for estimation of sensitivity indices because they are simpler to evaluate than the original model and can be more accurate than the conventional Monte Carlo simulation in the case of small to moderate sample sizes [26]. There are different types of metamodels such as multiple linear and nonlinear regression, cubic splines, Artificial Neural Networks, Gaussian Processes and orthogonal polynomials. A specific type of orthogonal polynomial metamodel is the so-called Polynomial Chaos Expansion (PCE), that is, a series expansion of a random variable using orthogonal basis that depends on the predetermined Probability Density Functions (PDF) [17]. A typical example is the use of Hermite polynomials to represent normal random variables. The importance of PCE for GSA of scalar output is that analytical expressions for Sobol indices can be obtained from the coefficients of PCE, a result established by Sudret [27]. However, the use of PCE in the case of GSA for multivariate output has not been studied to date. Therefore in this paper, PCE is applied to two approaches of GSA for multivariate output: 1. In the decomposition of output approach, PCE is used to estimate efficiently the covariance matrices used to define the orthogonal decomposition of the output, and to obtain the Sobol decomposition of the coefficients in the resulting expansion. 2. In the covariance decomposition approach, PCE allows the development of analytical expressions for the multivariate sensitivity indices proposed by Gamboa et al. [14] in a similar way to the approach proposed by Sudret [27] for the scalar case. In addition, these two approaches are applied to a simple problem of reactive transport in porous media, allowing for a comparison with the goal of identifying advantages and disadvantages of these methodologies. This paper is organized as follows: Section 2 includes a description of the main concepts used in this paper (scalar and multivariate GSA in Section 2.1, a review of the PCE in Section 2.3, the application of PCE to scalar and multivariate GSA in Section 2.4), the proposed approach is tested on a simple problem of multicomponent transport in porous media in Section 3, and finally results and discussion are included in Section 4.

Two approaches for the estimation of the multivariate sensitivity indices from the PCE are presented. In the first approach, the output is decomposed using an orthogonal basis defined from the spectral decomposition of the covariance matrix, and then the GSA is applied on the coefficients of this expansion. The PCE is used in these two steps, first to estimate the covariance matrix that is used to define the orthogonal basis, and second to obtain the Sobol decomposition of the coefficients in this expansion. The computational cost of this approach lies in the estimation of the coefficients in the PCE and the spectral decomposition of the covariance matrix estimated from them. This last step can be computationally intensive if the number of output variables is large and in some cases can seriously reduce the applicability of this approach in realistic problems. In the second approach, PCE is used to obtain analytical expressions of the sensitivity indices for multivariate output that are defined from the covariance decomposition of the output. The main computational cost in this approach is the estimation of the coefficients in the PCE making it more efficient compared with the output decomposition approach. The fact that these multivariate sensitivity indices can be obtained directly from the coefficients in the PCE implies that a simple post-processing of these coefficients allows the application of the scalar and multivariate GSA from the same set of model evaluations, giving complete information about the sensitivities of the considered system. The two approaches of multivariate GSA previously described (output and covariance decomposition) produce multivariate sensitivity measures that are equivalent, as shown in Section 2.2. The selection of the methodology depends if the problem at hand requires only the multivariate sensitivity indices or a detailed view of the controlling process. In the output decomposition approach, the eigenvectors of the covariance matrix and the sensitivity measures associated with the coefficients in the resulting expansion give insight on the physical processes that control the variation in the multivariate output. A set of multivariate sensitivity indices can be obtained from the eigenvalues of the covariance matrix and the sensitivity indices from the coefficients (see Eqs. (35)). These additional products of the sensitivity analysis can be critical in cases where the main goal is the understanding of the controlling processes. In contrast, these multivariate sensitivity indices can be obtained in a direct way with the covariance decomposition approach. These two approaches have been applied and compared in a simple test problem of multicomponent reactive transport in porous media. The results indicate that the sensitivity indices for variation of the concentrations of chemical species in the whole domain estimated using conventional and sparse PCE are accurate when compared to the reference values with a reduced computational effort. From these results, the sparse PCE is the best alternative in terms of accuracy and the required number of model evaluations for the estimation of these multivariate sensitivity indices.