237 research outputs found
A numerical approach related to defect-type theories for some weakly random problems in homogenization
We present in this paper an approach for computing the homogenized behavior
of a medium that is a small random perturbation of a periodic reference
material. The random perturbation we consider is, in a sense made precise in
our work, a rare event at the microscopic level. It however affects the
macroscopic properties of the material, and we indeed provide a method to
compute the first and second-order corrections. To this end, we formally
establish an asymptotic expansion of the macroscopic properties. Our
perturbative approach shares common features with a defect-type theory of solid
state physics. The computational efficiency of the approach is demonstrated
Periodic long-time behaviour for an approximate model of nematic polymers
We study the long-time behaviour of a nonlinear Fokker-Planck equation, which
models the evolution of rigid polymers in a given flow, after a closure
approximation. The aim of this work is twofold: first, we propose a microscopic
derivation of the classical Doi closure, at the level of the kinetic equation ;
second, we prove the convergence of the solution to the Fokker-Planck equation
to periodic solutions in the long-time limit
Special Quasirandom Structures: a selection approach for stochastic homogenization
We adapt and study a variance reduction approach for the homogenization of
elliptic equations in divergence form. The approach, borrowed from atomistic
simulations and solid-state science [von Pezold et al, Physical Review B 2010;
Wei et al, Physical Review B 1990; Zunger et al, Physical Review Letters 1990],
consists in selecting random realizations that best satisfy some statistical
properties (such as the volume fraction of each phase in a composite material)
usually only obtained asymptotically.
We study the approach theoretically in some simplified settings
(one-dimensional setting, perturbative setting in higher dimensions), and
numerically demonstrate its efficiency in more general cases
Adaptive low-rank approximation and denoised Monte-Carlo approach for high-dimensional Lindblad equations
We present a twofold contribution to the numerical simulation of Lindblad
equations. First, an adaptive numerical approach to approximate Lindblad
equations using low-rank dynamics is described: a deterministic low-rank
approximation of the density operator is computed, and its rank is adjusted
dynamically, using an on-the-fly estimator of the error committed when reducing
the dimension. On the other hand, when the intrinsic dimension of the Lindblad
equation is too high to allow for such a deterministic approximation, we
combine classical ensemble averages of quantum Monte Carlo trajectories and a
denoising technique. Specifically, a variance reduction method based upon the
consideration of a low-rank dynamics as a control variate is developed.
Numerical tests for quantum collapse and revivals show the efficiency of each
approach, along with the complementarity of the two approaches.Comment: 5 pages, 3 figures, Submitte
Some variance reduction methods for numerical stochastic homogenization
We overview a series of recent works devoted to variance reduction techniques
for numerical stochastic homogenization. Numerical homogenization requires
solving a set of problems at the micro scale, the so-called corrector problems.
In a random environment, these problems are stochastic and therefore need to be
repeatedly solved, for several configurations of the medium considered. An
empirical average over all configurations is then performed using the
Monte-Carlo approach, so as to approximate the effective coefficients necessary
to determine the macroscopic behavior. Variance severely affects the accuracy
and the cost of such computations. Variance reduction approaches, borrowed from
other contexts of the engineering sciences, can be useful. Some of these
variance reduction techniques are presented, studied and tested here
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