Bertrand Iooss

An efficient methodology for modeling complex computer codes with Gaussian processes

Complex computer codes are often too time expensive to be directly used to perform uncertainty propagation studies, global sensitivity analysis or to solve optimization problems. A well known and widely used method to circumvent this inconvenience consists in replacing the complex computer code by a reduced model, called a metamodel, or a response surface that represents the computer code and requires acceptable calculation time. One particular class of metamodels is studied: the Gaussian process model that is characterized by its mean and covariance functions. A specific estimation procedure is developed to adjust a Gaussian process model in complex cases (non-linear relations, highly dispersed or discontinuous output, high-dimensional input, inadequate sampling designs, etc.). The efficiency of this algorithm is compared to the efficiency of other existing algorithms on an analytical test case. The proposed methodology is also illustrated for the case of a complex hydrogeological computer code, simulating radionuclide transport in groundwater.

Calculations of Sobol indices for the Gaussian process metamodel

Global sensitivity analysis of complex numerical models can be performed by calculating variance-based importance measures of the input variables, such as the Sobol indices. However, these techniques, requiring a large number of model evaluations, are often unacceptable for time expensive computer codes. A well-known and widely used decision consists in replacing the computer code by a metamodel, predicting the model responses with a negligible computation time and rending straightforward the estimation of Sobol indices. In this paper, we discuss about the Gaussian process model which gives analytical expressions of Sobol indices. Two approaches are studied to compute the Sobol indices: the first based on the predictor of the Gaussian process model and the second based on the global stochastic process model. Comparisons between the two estimates, made on analytical examples, show the superiority of the second approach in terms of convergence and robustness. Moreover, the second approach allows to integrate the modeling error of the Gaussian process model by directly giving some confidence intervals on the Sobol indices. These techniques are finally applied to a real case of hydrogeological modeling.

Response surfaces and sensitivity analyses for an environmental model of dose calculations

Abstract A parametric sensitivity analysis is carried out on GASCON, a radiological impact software describing the radionuclides transfer to the man following a chronic gas release of a nuclear facility. An effective dose received by age group can thus be calculated according to a specific radionuclide and to the duration of the release. In this study, we are concerned by 18 output variables, each depending of approximately 50 uncertain input parameters. First, the generation of 1000 Monte-Carlo simulations allows us to calculate correlation coefficients between input parameters and output variables, which give a first overview of important factors. Response surfaces are then constructed in polynomial form, and used to predict system responses at reduced computation time cost; this response surface will be very useful for global sensitivity analysis where thousands of runs are required. Using the response surfaces, we calculate the total sensitivity indices of Sobol by the Monte-Carlo method. We demonstrate the application of this method to one site of study and to one reference group near the nuclear research Center of Cadarache (France), for two radionuclides: iodine 129 and uranium 238. It is thus shown that the most influential parameters are all related to the food chain of the goats milk, in decreasing order of importance: dose coefficient “effective ingestion”, goats milk ration of the individuals of the reference group, grass ration of the goat, dry deposition velocity and transfer factor to the goats milk.

Global sensitivity analysis of stochastic computer models with joint metamodels

The global sensitivity analysis method used to quantify the influence of uncertain input variables on the variability in numerical model responses has already been applied to deterministic computer codes; deterministic means here that the same set of input variables always gives the same output value. This paper proposes a global sensitivity analysis methodology for stochastic computer codes, for which the result of each code run is itself random. The framework of the joint modeling of the mean and dispersion of heteroscedastic data is used. To deal with the complexity of computer experiment outputs, nonparametric joint models are discussed and a new Gaussian process-based joint model is proposed. The relevance of these models is analyzed based upon two case studies. Results show that the joint modeling approach yields accurate sensitivity index estimators even when heteroscedasticity is strong.

Application of the control variate technique to estimation of total sensitivity indices

Abstract Global sensitivity analysis is widely used in many areas of science, biology, sociology and policy planning. The variance-based methods also known as Sobol׳ sensitivity indices has become the method of choice among practitioners due to its efficiency and ease of interpretation. For complex practical problems, estimation of Sobol׳ sensitivity indices generally requires a large number of function evaluations to achieve reasonable convergence. To improve the efficiency of the Monte Carlo estimates for the Sobol׳ total sensitivity indices we apply the control variate reduction technique and develop a new formula for evaluation of total sensitivity indices. Presented results using well known test functions show the efficiency of the developed technique.

Screening and metamodeling of computer experiments with functional outputs. Application to thermal-hydraulic computations

To perform uncertainty, sensitivity or optimization analysis on scalar variables calculated by a cpu time expensive computer code, a widely accepted methodology consists in first identifying the most influential uncertain inputs (by screening techniques), and then in replacing the cpu time expensive model by a cpu inexpensive mathematical function, called a metamodel. This paper extends this methodology to the functional output case, for instance when the model output variables are curves. The screening approach is based on the analysis of variance and principal component analysis of output curves. The functional metamodeling consists in a curve classification step, a dimension reduction step, then a classical metamodeling step. An industrial nuclear reactor application (dealing with uncertainties in the pressurized thermal shock analysis) illustrates all these steps.

Poincaré inequalities on intervals – application to sensitivity analysis

The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincare inequalities. Typically two kind of sensitivity indices are linked by a Poincare type inequality , which provide upper bounds of the most interpretable index by using the other one, cheaper to compute. This allows performing a low-cost screening of unessential variables. The efficiency of this screening then highly depends on the accuracy of the upper bounds in Poincare inequalities. The novelty in the questions concern the wide range of probability distributions involved, which are often truncated on intervals. After providing an overview of the existing knowledge and techniques, we add some theory about Poincare constants on intervals, with improvements for symmetric intervals. Then we exploit the spectral interpretation for computing exact value of Poincare constants of any admissible distribution on a given interval. We give semi-analytical results for some frequent distributions (truncated exponential, triangular, truncated normal), and present a numerical method in the general case. Finally, an application is made to a hydrological problem, showing the benefits of the new results in Poincare inequalities to sensitivity analysis.

Building Probability of Detection Curves via Metamodels

Probability of Detection (POD) curves is a standard tool in several industries to evaluate the performance of Non Destructive Testing (NDT) procedures. However, the classical methods for POD determination rely on strong statistical assumptions (lin earity, residuals normality and homoscedasticity). In the context of numerical POD estimation (with data coming from numerical simulations of the system), we study classic and novel model-based approaches. Applications are performed on Eddy Current Non Destructive Examination numerical data.

An efficient methodology for the analysis and modeling of computer experiments with large number of inputs

Complex computer codes are often too time expensive to be directly used to perform uncertainty, sensitivity, optimization and robustness analyses. A widely accepted method to circumvent this problem consists in replacing cpu-time expensive computer models by cpu inexpensive mathematical functions, called metamodels. For example, the Gaussian process (Gp) model has shown strong capabilities to solve practical problems , often involving several interlinked issues. However, in case of high dimensional experiments (with typically several tens of inputs), the Gp metamodel building process remains difficult, even unfeasible, and application of variable selection techniques cannot be avoided. In this paper, we present a general methodology allowing to build a Gp metamodel with large number of inputs in a very efficient manner. While our work focused on the Gp metamodel, its principles are fully generic and can be applied to any types of metamodel. The objective is twofold: estimating from a minimal number of computer experiments a highly predictive metamodel. This methodology is successfully applied on an industrial computer code.