36 research outputs found
New entropy estimates for the Oldroyd-B model, and related models
This short note presents the derivation of a new {\it a priori} estimate for
the Oldroyd-B model. Such an estimate may provide useful information when
investigating the long-time behaviour of macro-macro models, and the stability
of numerical schemes. We show how this estimate can be used as a guideline to
derive new estimates for other macroscopic models, like the FENE-P model
Pathwise estimates for an effective dynamics
Starting from the overdamped Langevin dynamics in , we consider a scalar Markov
process which approximates the dynamics of the first component .
In the previous work [F. Legoll, T. Lelievre, Nonlinearity 2010], the fact that
is a good approximation of is proven in
terms of time marginals, under assumptions quantifying the timescale separation
between the first component and the other components of . Here, we prove
an upper bound on the trajectorial error , for any , under a similar set
of assumptions. We also show that the technique of proof can be used to obtain
quantitative averaging results
A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations
We introduce a micro-macro parareal algorithm for the time-parallel
integration of multiscale-in-time systems. The algorithm first computes a
cheap, but inaccurate, solution using a coarse propagator (simulating an
approximate slow macroscopic model), which is iteratively corrected using a
fine-scale propagator (accurately simulating the full microscopic dynamics).
This correction is done in parallel over many subintervals, thereby reducing
the wall-clock time needed to obtain the solution, compared to the integration
of the full microscopic model. We provide a numerical analysis of the algorithm
for a prototypical example of a micro-macro model, namely singularly perturbed
ordinary differential equations. We show that the computed solution converges
to the full microscopic solution (when the parareal iterations proceed) only if
special care is taken during the coupling of the microscopic and macroscopic
levels of description. The convergence rate depends on the modeling error of
the approximate macroscopic model. We illustrate these results with numerical
experiments
A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells
We propose a numerical procedure to study closure approximations for FENE
dumbbells in terms of chosen macroscopic state variables, enabling to test
straightforwardly which macroscopic state variables should be included to build
good closures. The method involves the reconstruction of a polymer distribution
related to the conditional equilibrium of a microscopic Monte Carlo simulation,
conditioned upon the desired macroscopic state. We describe the procedure in
detail, give numerical results for several strategies to define the set of
macroscopic state variables, and show that the resulting closures are related
to those obtained by a so-called quasi-equilibrium approximation
\cite{Ilg:2002p10825}
Greedy algorithms for high-dimensional non-symmetric linear problems
In this article, we present a family of numerical approaches to solve
high-dimensional linear non-symmetric problems. The principle of these methods
is to approximate a function which depends on a large number of variates by a
sum of tensor product functions, each term of which is iteratively computed via
a greedy algorithm. There exists a good theoretical framework for these methods
in the case of (linear and nonlinear) symmetric elliptic problems. However, the
convergence results are not valid any more as soon as the problems considered
are not symmetric. We present here a review of the main algorithms proposed in
the literature to circumvent this difficulty, together with some new
approaches. The theoretical convergence results and the practical
implementation of these algorithms are discussed. Their behaviors are
illustrated through some numerical examples.Comment: 57 pages, 9 figure
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
Convergence of a greedy algorithm for high-dimensional convex nonlinear problems
In this article, we present a greedy algorithm based on a tensor product
decomposition, whose aim is to compute the global minimum of a strongly convex
energy functional. We prove the convergence of our method provided that the
gradient of the energy is Lipschitz on bounded sets. The main interest of this
method is that it can be used for high-dimensional nonlinear convex problems.
We illustrate this method on a prototypical example for uncertainty propagation
on the obstacle problem.Comment: 36 pages, 9 figures, accepted in Mathematical Models and Methods for
Applied Science
Analysis of Adaptive Multilevel Splitting algorithms in an idealized case
The Adaptive Multilevel Splitting algorithm is a very powerful and versatile
method to estimate rare events probabilities. It is an iterative procedure on
an interacting particle system, where at each step, the less well-adapted
particles among are killed while new better adapted particles are
resampled according to a conditional law. We analyze the algorithm in the
idealized setting of an exact resampling and prove that the estimator of the
rare event probability is unbiased whatever . We also obtain a precise
asymptotic expansion for the variance of the estimator and the cost of the
algorithm in the large limit, for a fixed