Compactness of linearized kinetic operators

International audienceThis article reviews various results on the compactness of the linearized Boltzmann operator and of its generalization to mixtures of non-reactive monatomic gases

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Hot new directions for quasi-Monte Carlo research in step with applications

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications

Uncertainty quantification for kinetic models in socio-economic and life sciences

Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of the modeling parameters. For this reason the presence of random inputs characterizing the parameters uncertainty should be considered as an essential feature in the modeling process. In this survey we introduce several examples of such kinetic models, that are mathematically described by nonlinear Vlasov and Fokker--Planck equations, and present different numerical approaches for uncertainty quantification which preserve the main features of the kinetic solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic Equations

Lattice Boltzmann simulations of soft matter systems

This article concerns numerical simulations of the dynamics of particles immersed in a continuum solvent. As prototypical systems, we consider colloidal dispersions of spherical particles and solutions of uncharged polymers. After a brief explanation of the concept of hydrodynamic interactions, we give a general overview over the various simulation methods that have been developed to cope with the resulting computational problems. We then focus on the approach we have developed, which couples a system of particles to a lattice Boltzmann model representing the solvent degrees of freedom. The standard D3Q19 lattice Boltzmann model is derived and explained in depth, followed by a detailed discussion of complementary methods for the coupling of solvent and solute. Colloidal dispersions are best described in terms of extended particles with appropriate boundary conditions at the surfaces, while particles with internal degrees of freedom are easier to simulate as an arrangement of mass points with frictional coupling to the solvent. In both cases, particular care has been taken to simulate thermal fluctuations in a consistent way. The usefulness of this methodology is illustrated by studies from our own research, where the dynamics of colloidal and polymeric systems has been investigated in both equilibrium and nonequilibrium situations.Comment: Review article, submitted to Advances in Polymer Science. 16 figures, 76 page

The performance of stochastic designs in wellbore drilling operations

© 2018, The Author(s). Wellbore drilling operations frequently entail the combination of a wide range of variables. This is underpinned by the numerous factors that must be considered in order to ensure safety and productivity. The heterogeneity and sometimes unpredictable behaviour of underground systems increases the sensitivity of drilling activities. Quite often the operating parameters are set to certify effective and efficient working processes. However, failings in the management of drilling and operating conditions sometimes result in catastrophes such as well collapse or fluid loss. This study investigates the hypothesis that optimising drilling parameters, for instance mud pressure, is crucial if the margin of safe operating conditions is to be properly defined. This was conducted via two main stages: first a deterministic analysis—where the operating conditions are predicted by conventional modelling procedures—and then a probabilistic analysis via stochastic simulations—where a window of optimised operation conditions can be obtained. The outcome of additional stochastic analyses can be used to improve results derived from deterministic models. The incorporation of stochastic techniques in the evaluation of wellbore instability indicates that margins of the safe mud weight window are adjustable and can be extended considerably beyond the limits of deterministic predictions. The safe mud window is influenced and hence can also be amended based on the degree of uncertainty and the permissible level of confidence. The refinement of results from deterministic analyses by additional stochastic simulations is vital if a more accurate and reliable representation of safe in situ and operating conditions is to be obtained during wellbore operations.Published versio

An implicit Monte arlo method for rarefied gas dynamics - 1. The space homogeneous case

For the space homogeneous Boltzmann equation, we formulate a hybrid Monte Carlo method that is robust in the fluid dynamic limit. This method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution (a generalized Wild expansion). A class of implicit, yet explicitly implementable, numerical schemes is obtained by substituting a Maxwellian distribution in place of the high order terms in the expansion. The numerical solution is represented as a convex combination of a non-equilibrium particle distribution and a Maxwellian. The hybrid distribution is then evolved by Monte Carlo using the implicit formulation for the time evolution. Computational simulations of spatially homogeneous problems by our method are presented here for the Kac model and for the variable hard sphere model (including Maxwell molecules). Comparison to exact solutions and to direct simulation Monte Carlo (DSMC) computations shows the robustness and the efficiency of the new method

Non-local Scattering kernel and Hydrodynamic Limit

In this paper we study the interaction of a fluid with a wall in the framework of the kinetic theory. We consider the possibility that the fluid molecules can penetrate the wall to be reflected by the inner layers of the wall. This results in a scattering kernel which is a non-local generalization of the classical Maxwell scattering kernel. The proposed scattering kernel satisfies a global mass conservation law and a generalized reciprocity relation. We study the hydrodynamic limit performing a Knudsen layer analysis, and derive a new class of (weakly) nonlocal boundary conditions to be imposed to the Navier-Stokes equations