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A domain decomposition analysis for a two-scale linear transport problem

François Golse, Shi Jin, C. David Levermore (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, i.e. up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusive region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating...

A Domain Decomposition Analysis for a Two-Scale Linear Transport Problem

François Golse, Shi Jin, C. David Levermore (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, i.e. up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusive region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating...

A Metropolis adjusted Nosé-Hoover thermostat

Benedict Leimkuhler, Sebastian Reich (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a Monte Carlo technique for sampling from the canonical distribution in molecular dynamics. The method is built upon the Nosé-Hoover constant temperature formulation and the generalized hybrid Monte Carlo method. In contrast to standard hybrid Monte Carlo methods only the thermostat degree of freedom is stochastically resampled during a Monte Carlo step.

A new exponentially fitted triangular finite element method for the continuity equations in the drift-diffusion model of semiconductor devices

Song Wang (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we present a novel exponentially fitted finite element method with triangular elements for the decoupled continuity equations in the drift-diffusion model of semiconductor devices. The continuous problem is first formulated as a variational problem using a weighted inner product. A Bubnov-Galerkin finite element method with a set of piecewise exponential basis functions is then proposed. The method is shown to be stable and can be regarded as an extension to two dimensions of the...

Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems

Markos A. Katsoulakis, Petr Plecháč, Luc Rey-Bellet, Dimitrios K. Tsagkarogiannis (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the...

Combining stochastic and deterministic approaches within high efficiency molecular simulations

Bruno Escribano, Elena Akhmatskaya, Jon Mujika (2013)

Open Mathematics

Generalized Shadow Hybrid Monte Carlo (GSHMC) is a method for molecular simulations that rigorously alternates Monte Carlo sampling from a canonical ensemble with integration of trajectories using Molecular Dynamics (MD). While conventional hybrid Monte Carlo methods completely re-sample particle’s velocities between MD trajectories, our method suggests a partial velocity update procedure which keeps a part of the dynamic information throughout the simulation. We use shadow (modified) Hamiltonians,...

Low-variance direct Monte Carlo simulations using importance weights

Husain A. Al-Mohssen, Nicolas G. Hadjiconstantinou (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present an efficient approach for reducing the statistical uncertainty associated with direct Monte Carlo simulations of the Boltzmann equation. As with previous variance-reduction approaches, the resulting relative statistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by the characteristic value of quantity of interest) is small and independent of the magnitude of the deviation from equilibrium, making the simulation of arbitrarily small deviations from equilibrium possible....

Numerical study of the systematic error in Monte Carlo schemes for semiconductors

Orazio Muscato, Wolfgang Wagner, Vincenza Di Stefano (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The paper studies the convergence behavior of Monte Carlo schemes for semiconductors. A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially one-dimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field...

Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics

Mireille Bossy, Nicolas Champagnat, Sylvain Maire, Denis Talay (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of ℝ d . This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator...

Sampling the Fermi statistics and other conditional product measures

A. Gaudillière, J. Reygner (2011)

Annales de l'I.H.P. Probabilités et statistiques

Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the temperature as well as the energy values and degeneracies of the energy levels we give an explicit upper bound with leading term km ln k for the mixing time of the dynamics. We obtain such construction and upper bound as a special case of a general result on (non-homogeneous)...

Theoretical and numerical comparison of some sampling methods for molecular dynamics

Eric Cancès, Frédéric Legoll, Gabriel Stoltz (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

The purpose of the present article is to compare different phase-space sampling methods, such as purely stochastic methods (Rejection method, Metropolized independence sampler, Importance Sampling), stochastically perturbed Molecular Dynamics methods (Hybrid Monte Carlo, Langevin Dynamics, Biased Random Walk), and purely deterministic methods (Nosé-Hoover chains, Nosé-Poincaré and Recursive Multiple Thermostats (RMT) methods). After recalling some theoretical convergence properties for the...

Une méthode nodale appliquée à un problème de diffusion à coefficients généralisés

Abdelkader Laazizi, Nagib Guessous (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we consider second order neutrons diffusion problem with coefficients in L∞(Ω). Nodal method of the lowest order is applied to approximate the problem's solution. The approximation uses special basis functions [1] in which the coefficients appear. The rate of convergence obtained is O(h2) in L2(Ω), with a free rectangular triangulation.

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