1,959 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
Hamiltonian dynamics and geometry of phase transitions in classical XY models
The Hamiltonian dynamics associated to classical, planar, Heisenberg XY
models is investigated for two- and three-dimensional lattices. Besides the
conventional signatures of phase transitions, here obtained through time
averages of thermodynamical observables in place of ensemble averages,
qualitatively new information is derived from the temperature dependence of
Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests
to consider other observables of geometric meaning tightly related with the
largest Lyapunov exponent. The numerical computation of these observables -
unusual in the study of phase transitions - sheds a new light on the
microscopic dynamical counterpart of thermodynamics also pointing to the
existence of some major change in the geometry of the mechanical manifolds at
the thermodynamical transition. Through the microcanonical definition of the
entropy, a relationship between thermodynamics and the extrinsic geometry of
the constant energy surfaces of phase space can be naturally
established. In this framework, an approximate formula is worked out,
determining a highly non-trivial relationship between temperature and topology
of the . Whence it can be understood that the appearance of a phase
transition must be tightly related to a suitable major topology change of the
. This contributes to the understanding of the origin of phase
transitions in the microcanonical ensemble.Comment: in press on Physical Review E, 43 pages, LaTeX (uses revtex), 22
PostScript figure
Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise
The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail
Implicit Langevin Algorithms for Sampling From Log-concave Densities
For sampling from a log-concave density, we study implicit integrators
resulting from -method discretization of the overdamped Langevin
diffusion stochastic differential equation. Theoretical and algorithmic
properties of the resulting sampling methods for and a
range of step sizes are established. Our results generalize and extend prior
works in several directions. In particular, for , we prove
geometric ergodicity and stability of the resulting methods for all step sizes.
We show that obtaining subsequent samples amounts to solving a strongly-convex
optimization problem, which is readily achievable using one of numerous
existing methods. Numerical examples supporting our theoretical analysis are
also presented
Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories
In this work, we consider an inhomogeneous (discrete time) Markov chain and
are interested in its long time behavior. We provide sufficient conditions to
ensure that some of its asymptotic properties can be related to the ones of a
homogeneous (continuous time) Markov process. Renowned examples such as a
bandit algorithms, weighted random walks or decreasing step Euler schemes are
included in our framework. Our results are related to functional limit
theorems, but the approach differs from the standard "Tightness/Identification"
argument; our method is unified and based on the notion of pseudotrajectories
on the space of probability measures
- …