89 research outputs found
Sampling from Rough Energy Landscapes
We examine challenges to sampling from Boltzmann distributions associated
with multiscale energy landscapes. The multiscale features, or "roughness,"
corresponds to highly oscillatory, but bounded, perturbations of a smooth
landscape. Through a combination of numerical experiments and analysis we
demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can
be severely attenuated as the roughness increases. In contrast, we prove that
Random Walk Metropolis is insensitive to such roughness. We also formulate two
alternative sampling strategies that incorporate large scale features of the
energy landscape, while resisting the impact of fine scale roughness; these
also outperform Random Walk Metropolis. Numerical experiments on these
landscapes are presented that confirm our predictions. Open questions and
numerical challenges are also highlighted.Comment: 34 pages, first revisio
Numerical Analysis of Parallel Replica Dynamics
Parallel replica dynamics is a method for accelerating the computation of
processes characterized by a sequence of infrequent events. In this work, the
processes are governed by the overdamped Langevin equation. Such processes
spend much of their time about the minima of the underlying potential,
occasionally transitioning into different basins of attraction. The essential
idea of parallel replica dynamics is that the exit time distribution from a
given well for a single process can be approximated by the minimum of the exit
time distributions of independent identical processes, each run for only
1/N-th the amount of time.
While promising, this leads to a series of numerical analysis questions about
the accuracy of the exit distributions. Building upon the recent work in Le
Bris et al., we prove a unified error estimate on the exit distributions of the
algorithm against an unaccelerated process. Furthermore, we study a dephasing
mechanism, and prove that it will successfully complete.Comment: 37 pages, 4 figures, revised and new estimates from the previous
versio
The parallel replica method for simulating long trajectories of Markov chains
The parallel replica dynamics, originally developed by A.F. Voter,
efficiently simulates very long trajectories of metastable Langevin dynamics.
We present an analogous algorithm for discrete time Markov processes. Such
Markov processes naturally arise, for example, from the time discretization of
a continuous time stochastic dynamics. Appealing to properties of
quasistationary distributions, we show that our algorithm reproduces exactly
(in some limiting regime) the law of the original trajectory, coarsened over
the metastable states.Comment: 13 pages, 6 figure
Focusing Singularity in a Derivative Nonlinear Schr\"odinger Equation
We present a numerical study of a derivative nonlinear Schr\"odinger equation
with a general power nonlinearity, . In the
-supercritical regime, , our simulations indicate that there is
a finite time singularity. We obtain a precise description of the local
structure of the solution in terms of blowup rate and asymptotic profile, in a
form similar to that of the nonlinear Schr\"odinger equation with supercritical
power law nonlinearity.Comment: 24 pages, 17 figure
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