تجزیه و تحلیل حساسیت جهانی از مدل های کامپیوتری با ورودی های کاربردی

Global sensitivity analysis is used to quantify the influence of uncertain model inputs on the response variability of a numerical model. The common quantitative methods are appropriate with computer codes having scalar model inputs. This paper aims at illustrating different variance-based sensitivity analysis techniques, based on the so-called Sobol's indices, when some model inputs are functional, such as stochastic processes or random spatial fields. In this work, we focus on large cpu time computer codes which need a preliminary metamodeling step before performing the sensitivity analysis. We propose the use of the joint modeling approach, i.e., modeling simultaneously the mean and the dispersion of the code outputs using two interlinked generalized linear models (GLMs) or generalized additive models (GAMs). The “mean model” allows to estimate the sensitivity indices of each scalar model inputs, while the “dispersion model” allows to derive the total sensitivity index of the functional model inputs. The proposed approach is compared to some classical sensitivity analysis methodologies on an analytical function. Lastly, the new methodology is applied to an industrial computer code that simulates the nuclear fuel irradiation.

Modern computer codes that simulate physical phenomenas often take as inputs a high number of numerical parameters and physical variables, and return several outputs–scalars or functions. For the development and the use of such computer models, sensitivity analysis (SA) is an invaluable tool. The original technique, based on the derivative computations of the model outputs with respect to the model inputs, suffers from strong limitations for computer models simulating non-linear phenomena. More recent global SA techniques take into account the entire range of variation of the model inputs and aim to apportion the whole output uncertainty to the model input uncertainties [1]. The global SA methods can also be used for model calibration, model validation, decision making process, i.e., any process where it is useful to know which are the variables that mostly contribute to the output variability. The common quantitative methods are applicable to computer codes with scalar model inputs. For example, in the nuclear engineering domain, global SA tools have been applied to numerous models where all the uncertain model inputs are modeled by random variables, possibly correlated—such as thermal-hydraulic system codes [2], waste storage safety studies [3], environmental model of dose calculations [4], reactor dosimetry processes [5]. Recent research papers have tried to consider more complex model inputs in the global SA process, especially in petroleum and environmental studies: • Tarantola et al. [6] work on an environmental assessment on soil models that use spatially distributed maps affected by random errors. This kind of uncertainty is modeled by a spatial random field (following a specified probability distribution), simulated at each code run. For the SA, the authors propose to replace the spatial model input by a “trigger” random parameter ξξ that governs the random field simulation. For some values of ξξ, the random field is simulated and for the other values, the random field values are put to zero. Therefore, the sensitivity index of ξξ is used to quantify the influence of the spatial model input. • Ruffo et al. [7] evaluate an oil reservoir production using a model that depends on different heterogeneous geological media scenarios. These scenarios, which are of limited number, are then identified by a discrete trigger spanning randomly the set of scenarios. Therefore, the sensitivity index of the trigger is used to identify the influence of the alternative scenarios. • Iooss et al. [8] study a groundwater radionuclide migration model which depend on several random scalar parameters and on a spatial random field (a geostatistical simulation of the hydrogeological layer heterogeneity). The authors propose to consider the spatial model input as an “uncontrollable” parameter. Therefore, they fit on a few simulation results of the computer model a double model, called a joint model: the first component models the effects of the scalar parameters while the second models the effects of the “uncontrollable” parameter. In this paper, we tackle the problem of the global SA for numerical models and when some model inputs ɛɛ are functional. ɛ(u)ɛ(u) is a one or multi-dimensional stochastic function where uu can be spatial coordinates, time scale or any other physical parameters. Our work focuses on models that depend on scalar parameter vector XX and involve some stochastic process simulations or random fields ɛ(u)ɛ(u) as inputs. The computer code output Y depends on the realizations of these random functions. These models are typically non-linear with strong interactions between model inputs. Therefore, we concentrate our methodology on the variance based sensitivity indices estimation; that is, the so-called Sobol's indices [1] and [9]. To deal with this situation, a first natural approach consists in using either all the discretized values of the functional model input ɛ(u)ɛ(u) or its decomposition into an appropriate basis of functions. Then, for all the new scalar variables related to ɛ(u)ɛ(u), sensitivity indices are computed. However, in the case of complex functional model inputs, this approach seems to be rapidly intractable as these variables cannot be represented by a small number of scalar variables [6]. Moreover, when dealing with non-physical variables (for example, coefficients of orthogonal functions used in the decomposition), sensitivity indices interpretation may be laborious. Indeed, most often, physicists would prefer to obtain one global sensitivity index related to ɛ(u)ɛ(u). Finally, a major drawback for the decomposition approach is related to the uncertainty modeling stage. More precisely, this approach needs to specify the probability density functions for the coefficients of the decomposition. The following section presents three different strategies to compute Sobol's indices with functional model inputs: (a) the macroparameter method, (b) the “trigger” parameter method, and (c) the proposed joint modeling approach. Section 3 compares the relevance of these three strategies on an analytical example: the WN-Ishigami function. Lastly, the proposed approach is illustrated on an industrial computer code simulating fuel irradiation in a nuclear reactor.

This paper has proposed a solution to perform global sensitivity analysis for time consuming computer models which depend on functional model inputs, such as a stochastic process or a random field. Our purpose concerned the computation of variance-based importance measures of the model output according to the uncertain model inputs. We have discussed a first natural solution which consists in integrating the functional input inside a macroparameter, and using standard Monte Carlo algorithms to compute sensitivity indices. This solution is not applicable to time consuming computer code. We have discussed another solution, used in previous studies, based on the replacement of the functional input by a “trigger” parameter that governs the integration or not of the functional input uncertainties. However, the estimated sensitivity indices are not the expected ones due to changes in the model structure carrying out by the method itself. Finally, we have proposed an innovative strategy, the joint modeling method, based on a preliminary step of double (and joint) metamodel fitting, which resolves the large cpu time problem of Monte Carlo methods. It consists in rejecting the functional inputs in noisy inputs. Then, two metamodels depending only on the scalar random inputs are simultaneously fitted: one for the mean function and one for the dispersion (variance) function. Tests on an analytical function have shown the relevance of the joint modeling method, which provides all the sensitivity indices of the scalar inputs and the total sensitivity index of the functional input. In addition, it reveals in a qualitative way the influential interactions between the functional input and the scalar inputs. It would be interesting in the future to be able to distinguish the contributions of several functional inputs that are currently totally mixed in one sensitivity index. This is the main drawback of the proposed method in its present form. In an industrial application, the usefulness and feasibility of our methodology have been established. Indeed, other methods are not applicable on this application because of large cpu time of the computer code. To a better understanding of the model behavior, the information brought by the global sensitivity analysis can be very useful to the physicist or the modeling engineer. Finally, the joint model can also be useful to propagate uncertainties in complex models, containing input random functions, to obtain some mean predictions with their confidence intervals.