A repulsion-based method for the definition and the enrichment of optimized space filling designs in constrained input spaces

Perrin, Guillaume; Cannamela, Claire

Journal de la société française de statistique, Tome 158 (2017) no. 1, pp. 37-67.

Profitant de l’essor considérable des puissances de calcul disponibles et de progrès importants en modélisation des phénomènes physiques, le rôle de la simulation n’est plus seulement descriptif, mais prédictif. Pour garantir cette capacité prédictive, il est alors nécessaire de développer des méthodes permettant d’associer à tout résultat numérique une précision, qui intègre les différentes sources d’incertitudes. Un des enjeux de ces méthodes de quantification des incertitudes concerne l’optimisation de l’exploration du domaine de variation des entrées de modélisation. Cette tâche peut s’avèrer difficile, en particulier lorsque le coût numérique associé à une simulation est élevé, ou lorsque le domaine d’entrée présente un certain nombre de contraintes, si bien qu’il ne peut plus être transformé en un hypercube via une bijection. Dans ce contexte, ce travail présente une méthode basée sur des répulsions permettant la définition de plans d’expériences optimisés dans des domaines contraints, dont les projections sur chaque paramètre d’entrée présentent de bonnes propriétés statistiques. Enfin, on montre que cette méthode permet également l’enrichissement de plans d’expériences déjà définis, tout en préservant un bon remplissage global du domaine de définition des entrées.

Due to increasing available computational resources and to a series of breakthroughs in the solving of nonlinear equations and in the modeling of complex mechanical systems, simulation nowadays becomes more and more predictive. Methods that could quantify the uncertainties associated with the results of the simulation are therefore needed to complete these predictions and widen the possibilities of simulation. One key step of these methods is the exploration of the whole space of the input variables, especially when the computational cost associated with one run of the simulation is high, and when there exists constraints on the inputs, such that the input space cannot be transformed into a hypercube through a bijection. In this context, the present work proposes an adaptive method to generate initial designs of experiments in any bounded convex input space, which are distributed as uniformly as possible on their definition space, while preserving good projection properties for each scalar input. Finally, it will be shown how this method can be used to add new elements to an initial design of experiments while preserving very interesting space filling properties.

Keywords: optimal design, design of experiments, computer experiment, Latin Hypercube Sampling, surrogate model
Mot clés : plans d’expériences optimisés, simulation numérique, plans hypercubes latins, metamodèle

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