366 research outputs found
Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations
We investigate mathematically a nonlinear approximation type approach
recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to
solve high dimensional partial differential equations. We show the link between
the approach and the greedy algorithms of approximation theory studied e.g. in
[R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical
case of the Poisson equation, we show that a variational version of the
approach, based on minimization of energies, converges. On the other hand, we
show various theoretical and numerical difficulties arising with the non
variational version of the approach, consisting of simply solving the first
order optimality equations of the problem. Several unsolved issues are
indicated in order to motivate further research
A mathematical formalization of the parallel replica dynamics
The purpose of this article is to lay the mathematical foundations of a well
known numerical approach in computational statistical physics and molecular
dynamics, namely the parallel replica dynamics introduced by A.F. Voter. The
aim of the approach is to efficiently generate a coarse-grained evolution (in
terms of state-to-state dynamics) of a given stochastic process. The approach
formally consists in concurrently considering several realizations of the
stochastic process, and tracking among the realizations that which, the
soonest, undergoes an important transition. Using specific properties of the
dynamics generated, a computational speed-up is obtained. In the best cases,
this speed-up approaches the number of realizations considered. By drawing
connections with the theory of Markov processes and, in particular, exploiting
the notion of quasi-stationary distribution, we provide a mathematical setting
appropriate for assessing theoretically the performance of the approach, and
possibly improving it
Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics
This paper is an introduction to the modelling of viscoelastic fluids, with
an emphasis on micro-macro (or multiscale) models. Some elements of
mathematical and numerical analysis are provided. These notes closely follow
the lectures delivered by the second author at the Chinese Academy of Science
during the Workshop "Stress Tensor Effects on Fluid Mechanics", in January
2010
Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients
In this paper we give an affirmative answer to an open question mentioned in
[Le Bris and Lions, Comm. Partial Differential Equations 33 (2008),
1272--1317], that is, we prove the well-posedness of the Fokker-Planck type
equations with Sobolev diffusion coefficients and BV drift coefficients.Comment: 11 pages. The proof has been modifie
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
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