206 research outputs found
Free Energy Methods for Bayesian Inference: Efficient Exploration of Univariate Gaussian Mixture Posteriors
Because of their multimodality, mixture posterior distributions are difficult
to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a
strategy to enhance the sampling of MCMC in this context, using a biasing
procedure which originates from computational Statistical Physics. The
principle is first to choose a "reaction coordinate", that is, a "direction" in
which the target distribution is multimodal. In a second step, the marginal
log-density of the reaction coordinate with respect to the posterior
distribution is estimated; minus this quantity is called "free energy" in the
computational Statistical Physics literature. To this end, we use adaptive
biasing Markov chain algorithms which adapt their targeted invariant
distribution on the fly, in order to overcome sampling barriers along the
chosen reaction coordinate. Finally, we perform an importance sampling step in
order to remove the bias and recover the true posterior. The efficiency factor
of the importance sampling step can easily be estimated \emph{a priori} once
the bias is known, and appears to be rather large for the test cases we
considered. A crucial point is the choice of the reaction coordinate. One
standard choice (used for example in the classical Wang-Landau algorithm) is
minus the log-posterior density. We discuss other choices. We show in
particular that the hyper-parameter that determines the order of magnitude of
the variance of each component is both a convenient and an efficient reaction
coordinate. We also show how to adapt the method to compute the evidence
(marginal likelihood) of a mixture model. We illustrate our approach by
analyzing two real data sets
The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint
The purpose of this paper is to provide a mathematical analysis of the
Adler-Wiser formula relating the macroscopic relative permittivity tensor to
the microscopic structure of the crystal at the atomic level. The technical
level of the presentation is kept at its minimum to emphasize the mathematical
structure of the results. We also briefly review some models describing the
electronic structure of finite systems, focusing on density operator based
formulations, as well as the Hartree model for perfect crystals or crystals
with a defect.Comment: Proceedings of the Workshop "Numerical Analysis of Multiscale
Computations" at Banff International Research Station, December 200
Error Analysis of Modified Langevin Dynamics
We consider Langevin dynamics associated with a modified kinetic energy
vanishing for small momenta. This allows us to freeze slow particles, and hence
avoid the re-computation of inter-particle forces, which leads to computational
gains. On the other hand, the statistical error may increase since there are a
priori more correlations in time. The aim of this work is first to prove the
ergodicity of the modified Langevin dynamics (which fails to be hypoelliptic),
and next to analyze how the asymptotic variance on ergodic averages depends on
the parameters of the modified kinetic energy. Numerical results illustrate the
approach, both for low-dimensional systems where we resort to a Galerkin
approximation of the generator, and for more realistic systems using Monte
Carlo simulations
A mathematical analysis of the GW0 method for computing electronic excited energies of molecules
This paper analyses the GW method for finite electronic systems. In a first
step, we provide a mathematical framework for the usual one-body operators that
appear naturally in many-body perturbation theory. We then discuss the GW
equations which construct an approximation of the one-body Green's function,
and give a rigorous mathematical formulation of these equations. Finally, we
study the well-posedness of the GW0 equations, proving the existence of a
unique solution to these equations in a perturbative regime
Error estimates and variance reduction for nonequilibrium stochastic dynamics
Equilibrium properties in statistical physics are obtained by computing
averages with respect to Boltzmann-Gibbs measures, sampled in practice using
ergodic dynamics such as the Langevin dynamics. Some quantities however cannot
be computed by simply sampling the Boltzmann-Gibbs measure, in particular
transport coefficients, which relate the current of some physical quantity of
interest to the forcing needed to induce it. For instance, a temperature
difference induces an energy current, the proportionality factor between these
two quantities being the thermal conductivity. From an abstract point of view,
transport coefficients can also be considered as some form of sensitivity
analysis with respect to an added forcing to the baseline dynamics. There are
various numerical techniques to estimate transport coefficients, which all
suffer from large errors, in particular large statistical errors. This
contribution reviews the most popular methods, namely the Green-Kubo approach
where the transport coefficient is expressed as some time-integrated
correlation function, and the approach based on longtime averages of the
stochastic dynamics perturbed by an external driving (so-called nonequilibrium
molecular dynamics). In each case, the various sources of errors are made
precise, in particular the bias related to the time discretization of the
underlying continuous dynamics, and the variance of the associated Monte Carlo
estimators. Some recent alternative techniques to estimate transport
coefficients are also discussed
Local density dependent potential for compressible mesoparticles
We focus on finding a coarse grained description able to reproduce the
thermodynamic behavior of a molecular system by using mesoparticles
representing several molecules. Interactions between mesoparticles are modelled
by an interparticle potential, and an additional internal equation of state is
used to account for the thermic contribution of coarse grained internal degrees
of freedom. Moreover, as strong non-equilibrium situations over a wide range of
pressure and density are targeted, the internal compressibility of these
mesoparticles has to be considered. This is done by introducing a dependence of
the potential on the local environment of the mesoparticles, either by defining
a spherical local density or by means of a Voronoi tessellation. As an example,
a local density dependent potential is fitted to reproduce the Hugoniot curve
of a model of nitromethane, where each mesoparticle represents one thousand
molecules
Derivation of Langevin Dynamics in a Nonzero Background Flow Field
We propose a derivation of a nonequilibrium Langevin dynamics for a large
particle immersed in a background flow field. A single large particle is placed
in an ideal gas heat bath composed of point particles that are distributed
consistently with the background flow field and that interact with the large
particle through elastic collisions. In the limit of small bath atom mass, the
large particle dynamics converges in law to a stochastic dynamics. This
derivation follows the ideas of [D. D\"urr, S. Goldstein, and J. L. Lebowitz,
1981 and 1983; P. Calderoni, D. D\"urr, and S. Kusuoka, 1989] and provides
extensions to handle the nonzero background flow. The derived nonequilibrium
Langevin dynamics is similar to the dynamics in [M. McPhie, et al., 2001]. Some
numerical experiments illustrate the use of the obtained dynamic to simulate
homogeneous liquid materials under flow.Comment: Minor revisions, refined discussion of the laminar bath approach and
non-Hamiltonian dynamics approach in Section 2. 41 pages, 8 figure
Long-time convergence of an Adaptive Biasing Force method
We propose a proof of convergence of an adaptive method used in molecular
dynamics to compute free energy profiles. Mathematically, it amounts to
studying the long-time behavior of a stochastic process which satisfies a
non-linear stochastic differential equation, where the drift depends on
conditional expectations of some functionals of the process. We use entropy
techniques to prove exponential convergence to the stationary state
Robust determination of maximally-localized Wannier functions
We propose an algorithm to determine Maximally Localized Wannier Functions
(MLWFs). This algorithm, based on recent theoretical developments, does not
require any physical input such as initial guesses for the Wannier functions,
unlike popular schemes based on the projection method. We discuss how the
projection method can fail on fine grids when the initial guesses are too far
from MLWFs. We demonstrate that our algorithm is able to find localized Wannier
functions through tests on two-dimensional systems, simplified models of
semiconductors, and realistic DFT systems by interfacing with the Wannier90
code. We also test our algorithm on the Haldane and Kane-Mele models to examine
how it fails in the presence of topological obstructions
An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach
We present a numerical method to accurately simulate particle size
distributions within the formalism of rate equation cluster dynamics. This
method is based on a discretization of the associated Fokker-Planck equation.
We show that particular care has to be taken to discretize the advection part
of the Fokker-Planck equation, in order to avoid distortions of the
distribution due to numerical diffusion. For this purpose we use the
Kurganov-Noelle-Petrova scheme coupled with the monotonicity-preserving
reconstruction MP5, which leads to very accurate results. The interest of the
method is highlighted on the case of loop coarsening in aluminum. We show that
the choice of the models to describe the energetics of loops does not
significantly change the normalized loop distribution, while the choice of the
models for the absorption coefficients seems to have a significant impact on
it
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