1,240 research outputs found
Automating warm-up length estimation
There are two key issues in assuring the accuracy of estimates of performance obtained from a simulation model. The first is the removal of any initialisation bias, the second is ensuring that enough output data is produced to obtain an accurate estimate of performance. This paper is concerned with the first issue, and more specifically warm-up estimation. Our aim is to produce an automated procedure, for inclusion into commercial simulation software, for estimating the length of warm-up and hence removing initialisation bias from simulation output data. This paper describes the extensive literature search that was carried out in order to find and assess the various existing warm-up methods, the process of short-listing and testing of candidate methods. In particular it details the extensive testing of the warm-up MSER-5 method. © 2010 Operational Research Society Ltd. All rights reserved
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen
On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain
The continuous and discrete symmetries of the Kuramoto-Sivashinsky system
restricted to a spatially periodic domain play a prominent role in shaping the
invariant sets of its chaotic dynamics. The continuous spatial translation
symmetry leads to relative equilibrium (traveling wave) and relative periodic
orbit (modulated traveling wave) solutions. The discrete symmetries lead to
existence of equilibrium and periodic orbit solutions, induce decomposition of
state space into invariant subspaces, and enforce certain structurally stable
heteroclinic connections between equilibria. We show, on the example of a
particular small-cell Kuramoto-Sivashinsky system, how the geometry of its
dynamical state space is organized by a rigid `cage' built by heteroclinic
connections between equilibria, and demonstrate the preponderance of unstable
relative periodic orbits and their likely role as the skeleton underpinning
spatiotemporal turbulence in systems with continuous symmetries. We also offer
novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space
flow through projections onto low-dimensional, PDE representation independent,
dynamically invariant intrinsic coordinate frames, as well as in terms of the
physical, symmetry invariant energy transfer rates.Comment: 31 pages, 17 figures; added references, corrected typos. Due to file
size restrictions some figures in this preprint are of low quality. A high
quality copy may be obtained from
http://www.cns.gatech.edu/~predrag/papers/preprints.html#rp
Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks
We introduce a data-driven forecasting method for high-dimensional chaotic
systems using long short-term memory (LSTM) recurrent neural networks. The
proposed LSTM neural networks perform inference of high-dimensional dynamical
systems in their reduced order space and are shown to be an effective set of
nonlinear approximators of their attractor. We demonstrate the forecasting
performance of the LSTM and compare it with Gaussian processes (GPs) in time
series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation
and a prototype climate model. The LSTM networks outperform the GPs in
short-term forecasting accuracy in all applications considered. A hybrid
architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is
proposed to ensure convergence to the invariant measure. This novel hybrid
method is fully data-driven and extends the forecasting capabilities of LSTM
networks.Comment: 31 page
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