4,286 research outputs found
D R A F T Smooth Conditional Transition Paths in Dynamical Gaussian Networks
Abstract. We propose an algorithm for determining optimal transition paths between given configurations of systems consisting of many objects. It is based on the Principle of Least Action and variational equations for Freidlin-Wentzell action functionals in Gaussian networks set-up. We use our method to construct a system controlling motion and redeployment between unit's formations. Another application of the algorithm allows a realistic transformation between two sequences of character animations in a virtual environment. The efficiency of the algorithm has been evaluated in a simple sandbox environment implemented with the use of the NVIDIA CUDA technology
Emulating dynamic non-linear simulators using Gaussian processes
The dynamic emulation of non-linear deterministic computer codes where the
output is a time series, possibly multivariate, is examined. Such computer
models simulate the evolution of some real-world phenomenon over time, for
example models of the climate or the functioning of the human brain. The models
we are interested in are highly non-linear and exhibit tipping points,
bifurcations and chaotic behaviour. However, each simulation run could be too
time-consuming to perform analyses that require many runs, including
quantifying the variation in model output with respect to changes in the
inputs. Therefore, Gaussian process emulators are used to approximate the
output of the code. To do this, the flow map of the system under study is
emulated over a short time period. Then, it is used in an iterative way to
predict the whole time series. A number of ways are proposed to take into
account the uncertainty of inputs to the emulators, after fixed initial
conditions, and the correlation between them through the time series. The
methodology is illustrated with two examples: the highly non-linear dynamical
systems described by the Lorenz and Van der Pol equations. In both cases, the
predictive performance is relatively high and the measure of uncertainty
provided by the method reflects the extent of predictability in each system
State and parameter estimation using Monte Carlo evaluation of path integrals
Transferring information from observations of a dynamical system to estimate
the fixed parameters and unobserved states of a system model can be formulated
as the evaluation of a discrete time path integral in model state space. The
observations serve as a guiding potential working with the dynamical rules of
the model to direct system orbits in state space. The path integral
representation permits direct numerical evaluation of the conditional mean path
through the state space as well as conditional moments about this mean. Using a
Monte Carlo method for selecting paths through state space we show how these
moments can be evaluated and demonstrate in an interesting model system the
explicit influence of the role of transfer of information from the
observations. We address the question of how many observations are required to
estimate the unobserved state variables, and we examine the assumptions of
Gaussianity of the underlying conditional probability.Comment: Submitted to the Quarterly Journal of the Royal Meteorological
Society, 19 pages, 5 figure
Invariant Manifolds and Rate Constants in Driven Chemical Reactions
Reaction rates of chemical reactions under nonequilibrium conditions can be
determined through the construction of the normally hyperbolic invariant
manifold (NHIM) [and moving dividing surface (DS)] associated with the
transition state trajectory. Here, we extend our recent methods by constructing
points on the NHIM accurately even for multidimensional cases. We also advance
the implementation of machine learning approaches to construct smooth versions
of the NHIM from a known high-accuracy set of its points. That is, we expand on
our earlier use of neural nets, and introduce the use of Gaussian process
regression for the determination of the NHIM. Finally, we compare and contrast
all of these methods for a challenging two-dimensional model barrier case so as
to illustrate their accuracy and general applicability.Comment: 28 pages, 13 figures, table of contents figur
Stable resonances and signal propagation in a chaotic network of coupled units
We apply the linear response theory developed in \cite{Ruelle} to analyze how
a periodic signal of weak amplitude, superimposed upon a chaotic background, is
transmitted in a network of non linearly interacting units. We numerically
compute the complex susceptibility and show the existence of specific poles
(stable resonances) corresponding to the response to perturbations transverse
to the attractor. Contrary to the poles of correlation functions they depend on
the pair emitting/receiving units. This dynamic differentiation, induced by non
linearities, exhibits the different ability that units have to transmit a
signal in this network.Comment: 10 pages, 3 figures, to appear in Phys. rev.
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