909 research outputs found
Long-Run Accuracy of Variational Integrators in the Stochastic Context
This paper presents a Lie-Trotter splitting for inertial Langevin equations
(Geometric Langevin Algorithm) and analyzes its long-time statistical
properties. The splitting is defined as a composition of a variational
integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the
splitting are geometrically ergodic, the paper proves the discrete invariant
measure of the splitting approximates the invariant measure of inertial
Langevin to within the accuracy of the variational integrator in representing
the Hamiltonian. In particular, if the variational integrator admits no energy
error, then the method samples the invariant measure of inertial Langevin
without error. Numerical validation is provided using explicit variational
integrators with first, second, and fourth order accuracy.Comment: 30 page
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
A patch that imparts unconditional stability to certain explicit integrators for SDEs
This paper proposes a simple strategy to simulate stochastic differential
equations (SDE) arising in constant temperature molecular dynamics. The main
idea is to patch an explicit integrator with Metropolis accept or reject steps.
The resulting `Metropolized integrator' preserves the SDE's equilibrium
distribution and is pathwise accurate on finite time intervals. As a corollary
the integrator can be used to estimate finite-time dynamical properties along
an infinitely long solution. The paper explains how to implement the patch
(even in the presence of multiple-time-stepsizes and holonomic constraints),
how it scales with system size, and how much overhead it requires. We test the
integrator on a Lennard-Jones cluster of particles and `dumbbells' at constant
temperature.Comment: 29 pages, 5 figure
Structure preserving Stochastic Impulse Methods for stiff Langevin systems with a uniform global error of order 1 or 1/2 on position
Impulse methods are generalized to a family of integrators for Langevin
systems with quadratic stiff potentials and arbitrary soft potentials. Uniform
error bounds (independent from stiff parameters) are obtained on integrated
positions allowing for coarse integration steps. The resulting integrators are
explicit and structure preserving (quasi-symplectic for Langevin systems)
Postprocessed integrators for the high order integration of ergodic SDEs
The concept of effective order is a popular methodology in the deterministic
literature for the construction of efficient and accurate integrators for
differential equations over long times. The idea is to enhance the accuracy of
a numerical method by using an appropriate change of variables called the
processor. We show that this technique can be extended to the stochastic
context for the construction of new high order integrators for the sampling of
the invariant measure of ergodic systems. The approach is illustrated with
modifications of the stochastic -method applied to Brownian dynamics,
where postprocessors achieving order two are introduced. Numerical experiments,
including stiff ergodic systems, illustrate the efficiency and versatility of
the approach.Comment: 21 pages, to appear in SIAM J. Sci. Compu
- …