In this article, we provide a method to compute analytic expressions of the resolvent kernel of differential operators of the diffusion type with discontinuous coefficients in one dimension. Then we apply it when the coefficients are piecewise constant. We also perform the Laplace inversion of the resolvent kernel to obtain expressions of the transition density functions or fundamental solutions. We show how these explicit formula are useful to simulate advection-diffusion problems using particle tracking techniques.We are interested in deriving analytic expressions of the resolvent kernels (or of its Laplace inversion, the so-called transition density functions) of such operators and to use these analytic expressions in the context of particle tracking methods [13,14,43]. These analytic expressions may also provide other very interestings results as asymptotics in long time [49] and short time [10], various quantities such as option prices in finance [46], solution of inverse problems [28] or understanding of the underlying equations and their spectral properties [25,26].Due to the discontinuities of and/or , the derivation of analytic expressions of the resolvent kernel of ( ,( , , ) , Dom( ,( , , ) )) requires special methods. Commonly, these analytic expressions are derived using probabilistic methods or arguments [2,6,17,23,58]. However, the derivation is quite heavy and sometimes cumbersome. Other approaches, described in [25,26], in the same spirit of the method proposed in this paper, consider only the case = 0.Our first contribution consists in proposing a general computation method which provides explicit, closed form formula of the resolvent kernel when both and are discontinuous; extending the previous results we cited right above. It also generalizes the so-called method of images which was only usable when the symmetries in the coefficients allow it [60]. The method we propose is more universal and systematic than the existing ones with probabilistic arguments which tends to treat the problems case by case. In the situation where and are piecewise constant, we show how to end up, through a simple change of variable, to the study of a Skew Brownian motion [39] with drift (called the drifted Skew Brownian motion).Then we are interested in using the analytic expression of the resolvent kernel of , ( , , ) in the context of solving advection-diffusion problems with a particle tracking technique. We consider the classical advection-diffusion operator:with domain Dom(ℒ ). On purpose, we take the same coefficient and as the ones in the expression of , ( , , ) .The coefficient is classically called the diffusivity coefficient and the coefficient the drift. The operator (ℒ , Dom(ℒ )) is encountered in many applications: in (with only a few references) geophysics for soil [12,29,32,52,57], air [56] and ocean [54]; astrophysics [61]; molecular dynamics [8]; population ecology [9,36]; finance [11,24,59]. The discontinuities come generally from the medium, which contains permeable or semi-permeable barriers. F...
International audienceIn this paper, we present some investigations on the parallelization of stochastic Lagrangian simulations. The challenge is the proper management of the random numbers. We review two different object-oriented strategies: to draw the random numbers on the fly within each MPI's process or to use a different random number generator for each simulated path. We show the benefits of the second technique which is implemented in the PALMTREE software developed by the Project-team Sage of Inria Rennes. The efficiency of PALMTREE is demonstrated on two classical examples
We present a new Monte Carlo algorithm to simulate diffusion processes in presence of discontinuous convective and diffusive terms. The algorithm is based on the knowledge of close form analytic expressions of the resolvents of the diffusion processes which are usually easier to obtain than close form analytic expressions of the density. In the particular case of diffusion processes with piecewise constant coefficients, known as Skew Diffusions, such close form expressions for the resolvent are available. Then we apply our algorithm to this particular case and we show that the approximate densities of the particles given by the algorithm replicate well the particularities of the true densities (discontinuities, bimodality, ...) Besides, numerical experiments show a quick convergence.
Abstract. Shielding studies in neutron transport, with Monte Carlo codes, yield challenging problems of small-probability estimation. The particularity of these studies is that the small probability to estimate is formulated in terms of the distribution of a Markov chain, instead of that of a random vector in more classical cases. Thus, it is not straightforward to adapt classical statistical methods, for estimating small probabilities involving random vectors, to these neutron-transport problems. A recent interacting-particle method for small-probability estimation, relying on the Hastings-Metropolis algorithm, is presented. It is shown how to adapt the Hastings-Metropolis algorithm when dealing with Markov chains. A convergence result is also shown. Then, the practical implementation of the resulting method for small-probability estimation is treated in details, for a Monte Carlo shielding study. Finally, it is shown, for this study, that the proposed interacting-particle method considerably outperforms a simple-Monte Carlo method, when the probability to estimate is small.Résumé. Dans lesétudes de protection en neutronique, celles fondées sur des codes Monte-Carlo posent d'importants problèmes d'estimation de faibles probabilités. La particularité de cesétudes est que les faibles probabilités sont exprimées en termes de lois sur des chaines de Markov, contrairement a des lois sur des vecteurs aléatoires dans les cas les plus classiques. Ainsi, les méthodes classiques d'estimation de faibles probabilités, portant sur des vecteurs aléatoires, ne peuvent s'utiliser telles qu'elles, pour ces problèmes neutroniques. Un méthode récente d'estimation de faibles probabilités, par système de particules en intéraction, reposant sur l'algorithme de Hastings-Metropolis, est présentée. Il est alors montré comment adapter l'algorithme de Hastings-Metropolis au cas des chaines de Markov. Un résultat de convergence est ainsi prouvé. Ensuite, il est expliqué en détail comment appliquer la méthode obtenueà uneétude de protection par Monte-Carlo. Finalement, pour cetteétude, il est montré que la méthode par système de particules en intéraction est considérablement plus efficace qu'une méthode par Monte Carlo classique, lorsque la probabilitéà estimer est faible.
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