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Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method

Fabien CasenaveAlexandre ErnTony Lelièvre — 2014

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

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off...

Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process

Benjamin JourdainTony LelièvreRaphaël Roux — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We study a free energy computation procedure, introduced in [Darve and Pohorille, (2001) 9169–9183; Hénin and Chipot, (2004) 2904–2914], which relies on the long-time behavior of a nonlinear stochastic differential equation. This nonlinearity comes from a conditional expectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions to this equation has been proved in [Lelièvre , (2008) 1155–1181],...

Free-energy-dissipative schemes for the Oldroyd-B model

Sébastien BoyavalTony LelièvreClaude Mangoubi — 2009

ESAIM: Mathematical Modelling and Numerical Analysis

In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the of the Oldroyd-B system proposed by Fattal and Kupferman in [ (2004) 281–285],...

Derivation of Langevin dynamics in a nonzero background flow field

Matthew DobsonFrédéric LegollTony LelièvreGabriel Stoltz — 2013

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

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,...

Diffusion Monte Carlo method: Numerical Analysis in a Simple Case

Mohamed El MakriniBenjamin JourdainTony Lelièvre — 2007

ESAIM: Mathematical Modelling and Numerical Analysis

The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a fixed number of random walkers evolving according to a stochastic differential equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove...

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