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The development of efficient numerical methods for kinetic equations with stochastic parameters is a challenge due to the high dimensionality of the problem. Recently we introduced a multiscale control variate strategy which is capable of considerably accelerating the slow convergence of standard Monte Carlo methods for uncertainty quantification. Here we generalize this class of methods to the case of multiple control variates. We show that the additional degrees of freedom can be used to further improve the variance reduction properties of multiscale control variate methods. [ABSTRACT FROM AUTHOR]

• Uncertainty quantification for the Boltzmann equation. • Variance reduction of standard Monte Carlo sampling methods. • Control variate methods based on reduced models obtained in a multi-scale setting. • Asymptotic behavior of the control variate methods and fluid-dynamic limit. • Error estimates show a strong acceleration of the convergence with respect to standard Monte Carlo. Kinetic equations play a major rule in modeling large systems of interacting particles. Uncertainties may be due to various reasons, like lack of knowledge on the microscopic interaction details or incomplete informations at the boundaries. These uncertainties, however, contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge. In this paper we consider the construction of novel multi-scale methods for such problems which, thanks to a control variate approach, are capable to reduce the variance of standard Monte Carlo techniques. [ABSTRACT FROM AUTHOR]

Abstract: In the present work, we present a novel numerical algorithm to couple the Direct Simulation Monte Carlo method (DSMC) for the solution of the Boltzmann equation with a finite volume like method for the solution of the Euler equations. Recently we presented in different methodologies which permit to solve fluid dynamics problems with localized regions of departure from thermodynamical equilibrium. The methods rely on the introduction of buffer zones which realize a smooth transition between the kinetic and the fluid regions. In this paper we extend the idea of buffer zones and dynamic coupling to the case of the Monte Carlo methods. To facilitate the coupling and avoid the onset of spurious oscillations in the fluid regions which are consequences of the coupling with a stochastic numerical scheme, we use a new technique which permits to reduce the variance of the particle methods . In addition, the use of this method permits to obtain estimations of the breakdowns of the fluid models less affected by fluctuations and consequently to reduce the kinetic regions and optimize the coupling. In the last part of the paper several numerical examples are presented to validate the method and measure its computational performances. [Copyright &y& Elsevier]

In this work we consider the development of a new family of hybrid numerical methods for the solution of kinetic equations which involves different scales. The basic idea is to couple macroscopic and microscopic models in all cases in which the macroscopic model does not provide correct results. The key aspect in the development of the algorithms is the choice of a suitable hybrid representation of the solution and a merging of Monte Carlo methods in nonequilibrium regimes with deterministic methods in equilibrium ones. This approach permits us to treat efficiently both the microscopic and the macroscopic scales. Applications to the Boltzmann-BGK equation are presented to show the performance of the new methods. [ABSTRACT FROM AUTHOR]

• The proposed method merges Monte Carlo approach and a finite volume method. • The statistical noise diminishes when the scaling parameter decreases. • The method is uniformly stable wrt the scaling parameter and the space mesh size. • No need of artificial transitions from the microscopic part to the macroscopic one. • Allows to run numerical tests on complex problems including full 3 dimensional ones. In this work, we develop a new class of numerical schemes for collisional kinetic equations in the diffusive regime. The first step consists in reformulating the problem by decomposing the solution in the time evolution of an equilibrium state plus a perturbation. Then, the scheme combines a Monte Carlo solver for the perturbation with an Eulerian method for the equilibrium part, and is designed in such a way to be uniformly stable with respect to the diffusive scaling and to be consistent with the asymptotic diffusion equation. Moreover, since particles are only used to describe the perturbation part of the solution, the scheme becomes computationally less expensive – and is thus an asymptotically complexity diminishing scheme (ACDS) – as the solution approaches the equilibrium state due to the fact that the number of particles diminishes accordingly. This contrasts with standard methods for kinetic equations where the computational cost increases (or at least does not decrease) with the number of interactions. At the same time, the statistical error due to the Monte Carlo part of the solution decreases as the system approaches the equilibrium state: the method automatically degenerates to a solution of the macroscopic diffusion equation in the limit of infinite number of interactions. After a detailed description of the method, we perform several numerical tests and compare this new approach with classical numerical methods on various problems up to the full three dimensional case. [ABSTRACT FROM AUTHOR]