We present an analysis of convergence of a quasi-regression Monte Carlo method proposed by Glasserman and Yu (2004). We show that the method surely converges to the true price of an American option even under multiple underlyings via polynomial chaos expansion and weaker conditions than those used in Glasserman and Yu (2004). Further, we show the number of simulation paths grows exponentially in the number of basis functions to obtain convergence in implementing the method. Finally, we propose a rate of convergence considering regularity of value functions.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

In this paper, we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) framework and the polynomial chaos. The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third. When its steady symmetric solitary-wave-like solutions are randomly perturbed, the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations. By representing the perturbation with a random variable, the governing FKdV equation is interpreted as a stochastic equation. The polynomial chaos expansion of the random solution has been used for the study of stability in two ways. First it allows us to identify the stable solution of the stochastic governing equation. Secondly it is used to construct upper and lower bounding surfaces for unstable solutions, which encompass the fluctuations of waves.

A number of approaches for discretizing partial differential equations with random dataare based on generalized polynomial chaos expansions of random variables. These constitutegeneralizations of the polynomial chaos expansions introduced by Norbert Wiener toexpansions in polynomials orthogonal with respect to non-Gaussian probability measures. Wepresent conditions on such measures which imply mean-square convergence of generalizedpolynomial chaos expansions to the correct limit and complement these with illustrativeexamples.

L'identification des paramètres influants pour le comportementdes liaisons par contact (coefficients de frottement, jeux, ...) n'est pas chose aisée. La mesure de tels paramètres est biensouvent fortement entachée d'erreurs très importantes.Les phénomènes mis en jeux (contact et frottement)sont fortement non-linéaires. La construction par simulationsnumériques de réponses moyennes de l'assemblage ou bien desurfaces de réponse nécessite un grand nombre de calcul trèscoûteux. Des techniques efficaces pour l'obtention rapide de telles informations sont proposées dans cet article.Pour décrire les assemblages nous utilisons une décomposition detype sous-structures et interfaces. L'algorithme de résolution utilisé est basé sur la méthode LATIN.Pour construire à moindre frais la surface de réponse,nous utilisons un résultat de calcul obtenu préalablementpour un jeu de paramètres comme initialisation pour un calculavec un autre jeu de paramètres.Dans un second temps, nous proposons une méthode de constructiondes caractéristiques stochastiques de la réponse de l'assemblage,par projection de la solution dans une base de type chaospolynomial.