We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, 343...

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...

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 [P. Calderoni,...

We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale...

The parareal in time algorithm allows for efficient parallel numerical simulations of time-dependent problems. It is based on a decomposition of the time interval into subintervals, and on a predictor-corrector strategy, where the propagations over each subinterval for the corrector stage are concurrently performed on the different processors that are available. In this article, we are concerned with the long time integration of Hamiltonian systems. Geometric, structure-preserving integrators are...

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case....

In order to describe a solid which deforms smoothly in some region, but
non smoothly in some other region, many multiscale methods have recently
been proposed. They aim at coupling an atomistic model (discrete
mechanics) with a macroscopic model
(continuum mechanics).
We provide here a theoretical ground for such a coupling in a
one-dimensional setting. We briefly study the general case of a convex
energy, and next concentrate on
a specific example of a nonconvex energy, the Lennard-Jones case....

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