We consider in this paper the problem of building a fast-running approximation-also called surrogate model-of a complex computer code. The co-kriging based surrogate model is a promising tool to build such an approximation when the complex computer code can be run at different levels of accuracy. We present here an original approach to perform a multi-fidelity co-kriging model which is based on a recursive formulation. We prove that the predictive mean and the variance of the presented approach are identical to the ones of the original co-kriging model. However, our new approach allows to obtain original results. First, closed-form formulas for the universal co-kriging predictive mean and variance are given. Second, a fast cross-validation procedure for the multi-fidelity co-kriging model is introduced. Finally, the proposed approach has a reduced computational complexity compared to the previous one. The multi-fidelity model is successfully applied to emulate a hydrodynamic simulator.

Global sensitivity analysis is now established as a powerful approach for determining the key random input parameters that drive the uncertainty of model output predictions. Yet the classical computation of the so-called Sobol' indices is based on Monte Carlo simulation, which is not affordable when computationally expensive models are used, as it is the case in most applications in engineering and applied sciences. In this respect metamodels such as polynomial chaos expansions (PCE) and Gaussian processes (GP) have received tremendous attention in the last few years, as they allow one to replace the original, taxing model by a surrogate which is built from an experimental design of limited size. Then the surrogate can be used to compute the sensitivity indices in negligible time. In this chapter an introduction to each technique is given, with an emphasis on their strengths and limitations in the context of global sensitivity analysis. In particular, Sobol' (resp. total Sobol') indices can be computed analytically from the PCE coefficients. In contrast, confidence intervals on sensitivity indices can be derived straightforwardly from the properties of GPs. The performance of the two techniques is finally compared on three well-known analytical benchmarks (Ishigami, G-Sobol and Morris functions) as well as on a realistic engineering application (deflection of a truss structure).

Complex computer codes are widely used in science and engineering to model physical phenomena. Furthermore, it is common that they have a large number of input parameters. Global sensitivity analysis aims to identify those which have the most important impact on the output. Sobol indices are a popular tool to perform such analysis. However, their estimations require an important number of simulations and often cannot be processed under reasonable time constraint. To handle this problem, a Gaussian process regression model is built to surrogate the computer code and the Sobol indices are estimated through it. The aim of this paper is to provide a methodology to estimate the Sobol indices through a surrogate model taking into account both the estimation errors and the surrogate model errors. In particular, it allows us to derive non-asymptotic confidence intervals for the Sobol index estimations. Furthermore, we extend the suggested strategy to the case of multi-fidelity computer codes which can be run at different levels of accuracy. For such simulators, we use an extension of Gaussian process regression models for multivariate outputs.

This paper deals with the Gaussian process-based approximation of a code which can be run at different levels of accuracy. This method, which is a particular case of cokriging, allows us to improve a surrogate model of a complex computer code using fast approximations of it. In particular, we focus on the case of a large number of code levels on the one hand and on a Bayesian approach when we have two levels on the other hand. The main results of this paper are a new approach to estimating the model parameters which provides a closed form expression for an important parameter of the model (the scale factor), a reduction of the numerical complexity by simplifying the covariance matrix inversion, and a new Bayesian modeling that gives an explicit representation of the joint distribution of the parameters and that is not computationally expensive. A thermodynamic example is used to illustrate the comparison between 2-level and 3-level cokriging. Introduction.Computer codes are widely used in science and engineering to study physical systems since real experiments are often costly and sometimes impossible. Nevertheless, simulations can sometimes be costly and time-consuming as well. In this case, conception based on an exhaustive exploration of the input space of the code is generally impossible under reasonable time constraints. Therefore, a mathematical approximation of the output of the code-also called surrogate or metamodel-is often built with a few simulations to represent the real system.Gaussian process regression is a particular class of surrogate which makes the assumption that prior beliefs about the code can be modeled by a Gaussian process. It is very popular in the field of computer experiments since it provides a basis for statistical inference. We focus here on this metamodel and on its extension to multiple response models. The reader is referred to [16] and [14] for further detail about Gaussian process models.Actually, a computer code can often be run at different levels of complexity, and a hierarchy of levels of code can be available. The aim of this paper is to study the use of several levels of a code to predict the output of a costly computer code.A first metamodel for multilevel computer codes was built by Kennedy and O'Hagan [8] using a spatially stationary correlation structure. This multistage model is a particular case of cokriging which is a well-known geostatistical method. Then Forrester, Sobester, and Keane [4] went into more detail about the estimation of the model parameters. Furthermore,

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