Complex computer codes, for instance simulating physical phenomena, 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 consis
ts in replacing cpu time expensive computer models by cpu inexpensive mathematical functions, called metamodels. In this paper, we focus on the Gaussian process metamodel and two essential steps of its definition phase. First, the initial design of the computer code input variables (which allows to fit the metamodel) has to honor adequate space filling properties. We adopt a numerical approach to compare the performance of different types of space filling designs, in the class of the optimal Latin hypercube samples, in terms of the predictivity of the subsequent fitted metamodel. We conclude that such samples with minimal wrap-around discrepancy are particularly well-suited for the Gaussian process metamodel fitting. Second, the metamodel validation process consists in evaluating the metamodel predictivity with respect to the initial computer code. We propose and test an algorithm which optimizes the distance between the validation points and the metamodel learning points in order to estimate the true metamodel predictivity with a minimum number of validation points. Comparisons with classical validation algorithms and application to a nuclear safety computer code show the relevance of this new sequential validation design.
The global sensitivity analysis of a complex numerical model often calls for the estimation of variance-based importance measures, named Sobol indices. Metamodel-based techniques have been developed in order to replace the cpu time-expensive computer
code with an inexpensive mathematical function, which predicts the computer code output. The common metamodel-based sensitivity analysis methods are well-suited for computer codes with scalar outputs. However, in the environmental domain, as in many areas of application, the numerical model outputs are often spatial maps, which may also vary with time. In this paper, we introduce an innovative method to obtain a spatial map of Sobol indices with a minimal number of numerical model computations. It is based upon the functional decomposition of the spatial output onto a wavelet basis and the metamodeling of the wavelet coefficients by the Gaussian process. An analytical example is presented to clarify the various steps of our methodology. This technique is then applied to a real hydrogeological case: for each model input variable, a spatial map of Sobol indices is thus obtained.
