5,294 research outputs found
Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents
High-dimensional chaotic dynamical systems can exhibit strongly transient
features. These are often associated with instabilities that have finite-time
duration. Because of the finite-time character of these transient events, their
detection through infinite-time methods, e.g. long term averages, Lyapunov
exponents or information about the statistical steady-state, is not possible.
Here we utilize a recently developed framework, the Optimally Time-Dependent
(OTD) modes, to extract a time-dependent subspace that spans the modes
associated with transient features associated with finite-time instabilities.
As the main result, we prove that the OTD modes, under appropriate conditions,
converge exponentially fast to the eigendirections of the Cauchy--Green tensor
associated with the most intense finite-time instabilities. Based on this
observation, we develop a reduced-order method for the computation of
finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems,
the computational cost of the reduced-order method is orders of magnitude lower
than the full FTLE computation. We demonstrate the validity of the theoretical
findings on two numerical examples
Kernel Analog Forecasting: Multiscale Test Problems
Data-driven prediction is becoming increasingly widespread as the volume of
data available grows and as algorithmic development matches this growth. The
nature of the predictions made, and the manner in which they should be
interpreted, depends crucially on the extent to which the variables chosen for
prediction are Markovian, or approximately Markovian. Multiscale systems
provide a framework in which this issue can be analyzed. In this work kernel
analog forecasting methods are studied from the perspective of data generated
by multiscale dynamical systems. The problems chosen exhibit a variety of
different Markovian closures, using both averaging and homogenization;
furthermore, settings where scale-separation is not present and the predicted
variables are non-Markovian, are also considered. The studies provide guidance
for the interpretation of data-driven prediction methods when used in practice.Comment: 30 pages, 14 figures; clarified several ambiguous parts, added
references, and a comparison with Lorenz' original method (Sec. 4.5
Nonparametric Uncertainty Quantification for Stochastic Gradient Flows
This paper presents a nonparametric statistical modeling method for
quantifying uncertainty in stochastic gradient systems with isotropic
diffusion. The central idea is to apply the diffusion maps algorithm to a
training data set to produce a stochastic matrix whose generator is a discrete
approximation to the backward Kolmogorov operator of the underlying dynamics.
The eigenvectors of this stochastic matrix, which we will refer to as the
diffusion coordinates, are discrete approximations to the eigenfunctions of the
Kolmogorov operator and form an orthonormal basis for functions defined on the
data set. Using this basis, we consider the projection of three uncertainty
quantification (UQ) problems (prediction, filtering, and response) into the
diffusion coordinates. In these coordinates, the nonlinear prediction and
response problems reduce to solving systems of infinite-dimensional linear
ordinary differential equations. Similarly, the continuous-time nonlinear
filtering problem reduces to solving a system of infinite-dimensional linear
stochastic differential equations. Solving the UQ problems then reduces to
solving the corresponding truncated linear systems in finitely many diffusion
coordinates. By solving these systems we give a model-free algorithm for UQ on
gradient flow systems with isotropic diffusion. We numerically verify these
algorithms on a 1-dimensional linear gradient flow system where the analytic
solutions of the UQ problems are known. We also apply the algorithm to a
chaotically forced nonlinear gradient flow system which is known to be well
approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11
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