30,763 research outputs found
Computing probabilities of very rare events for Langevin processes: a new method based on importance sampling
Langevin equations are used to model many processes of physical interest,
including low-energy nuclear collisions. In this paper we develop a general
method for computing probabilities of very rare events (e.g. small fusion
cross-sections) for processes described by Langevin dynamics. As we demonstrate
with numerical examples as well as an exactly solvable model, our method can
converge to the desired answer at a rate which is orders of magnitude faster
than that achieved with direct simulations of the process in question.Comment: 18 pages + 7 figures, to appear in Nucl.Phys.
Path integrals and symmetry breaking for optimal control theory
This paper considers linear-quadratic control of a non-linear dynamical
system subject to arbitrary cost. I show that for this class of stochastic
control problems the non-linear Hamilton-Jacobi-Bellman equation can be
transformed into a linear equation. The transformation is similar to the
transformation used to relate the classical Hamilton-Jacobi equation to the
Schr\"odinger equation. As a result of the linearity, the usual backward
computation can be replaced by a forward diffusion process, that can be
computed by stochastic integration or by the evaluation of a path integral. It
is shown, how in the deterministic limit the PMP formalism is recovered. The
significance of the path integral approach is that it forms the basis for a
number of efficient computational methods, such as MC sampling, the Laplace
approximation and the variational approximation. We show the effectiveness of
the first two methods in number of examples. Examples are given that show the
qualitative difference between stochastic and deterministic control and the
occurrence of symmetry breaking as a function of the noise.Comment: 21 pages, 6 figures, submitted to JSTA
A linear theory for control of non-linear stochastic systems
We address the role of noise and the issue of efficient computation in
stochastic optimal control problems. We consider a class of non-linear control
problems that can be formulated as a path integral and where the noise plays
the role of temperature. The path integral displays symmetry breaking and there
exist a critical noise value that separates regimes where optimal control
yields qualitatively different solutions. The path integral can be computed
efficiently by Monte Carlo integration or by Laplace approximation, and can
therefore be used to solve high dimensional stochastic control problems.Comment: 5 pages, 3 figures. Accepted to PR
A primer on noise-induced transitions in applied dynamical systems
Noise plays a fundamental role in a wide variety of physical and biological
dynamical systems. It can arise from an external forcing or due to random
dynamics internal to the system. It is well established that even weak noise
can result in large behavioral changes such as transitions between or escapes
from quasi-stable states. These transitions can correspond to critical events
such as failures or extinctions that make them essential phenomena to
understand and quantify, despite the fact that their occurrence is rare. This
article will provide an overview of the theory underlying the dynamics of rare
events for stochastic models along with some example applications
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