81 research outputs found
Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model
The aim of this work is the quantification and prediction of rare events
characterized by extreme intensity in nonlinear waves with broad spectra. We
consider a one-dimensional non- linear model with deep-water waves dispersion
relation, the Majda-McLaughlin-Tabak (MMT) model, in a dynamical regime that is
characterized by broadband spectrum and strong non- linear energy transfers
during the development of intermittent events with finite-lifetime. To
understand the energy transfers that occur during the development of an extreme
event we perform a spatially localized analysis of the energy distribution
along different wavenumbers by means of the Gabor transform. A stochastic
analysis of the Gabor coefficients reveals i) the low-dimensionality of the
intermittent structures, ii) the interplay between non-Gaussian statis- tical
properties and nonlinear energy transfers between modes, as well as iii) the
critical scales (or critical Gabor coefficients) where a critical amount of
energy can trigger the formation of an extreme event. We analyze the unstable
character of these special localized modes directly through the system equation
and show that these intermittent events are due to the interplay of the system
nonlinearity, the wave dispersion, and the wave dissipation which mimics wave
breaking. These localized instabilities are triggered by random localizations
of energy in space, created by the dispersive propagation of low-amplitude
waves with random phase. Based on these properties, we design low-dimensional
functionals of these Gabor coefficients that allow for the prediction of the
extreme event well before the nonlinear interactions begin to occur.Comment: 21 pages, 14 figure
A variational approach to probing extreme events in turbulent dynamical systems
Extreme events are ubiquitous in a wide range of dynamical systems, including
turbulent fluid flows, nonlinear waves, large scale networks and biological
systems. Here, we propose a variational framework for probing conditions that
trigger intermittent extreme events in high-dimensional nonlinear dynamical
systems. We seek the triggers as the probabilistically feasible solutions of an
appropriately constrained optimization problem, where the function to be
maximized is a system observable exhibiting intermittent extreme bursts. The
constraints are imposed to ensure the physical admissibility of the optimal
solutions, i.e., significant probability for their occurrence under the natural
flow of the dynamical system. We apply the method to a body-forced
incompressible Navier--Stokes equation, known as the Kolmogorov flow. We find
that the intermittent bursts of the energy dissipation are independent of the
external forcing and are instead caused by the spontaneous transfer of energy
from large scales to the mean flow via nonlinear triad interactions. The global
maximizer of the corresponding variational problem identifies the responsible
triad, hence providing a precursor for the occurrence of extreme dissipation
events. Specifically, monitoring the energy transfers within this triad, allows
us to develop a data-driven short-term predictor for the intermittent bursts of
energy dissipation. We assess the performance of this predictor through direct
numerical simulations.Comment: Minor revisions, generalized the constraints in Eq. (2
Model order reduction for stochastic dynamical systems with continuous symmetries
Stochastic dynamical systems with continuous symmetries arise commonly in
nature and often give rise to coherent spatio-temporal patterns. However,
because of their random locations, these patterns are not well captured by
current order reduction techniques and a large number of modes is typically
necessary for an accurate solution. In this work, we introduce a new
methodology for efficient order reduction of such systems by combining (i) the
method of slices, a symmetry reduction tool, with (ii) any standard order
reduction technique, resulting in efficient mixed symmetry-dimensionality
reduction schemes. In particular, using the Dynamically Orthogonal (DO)
equations in the second step, we obtain a novel nonlinear Symmetry-reduced
Dynamically Orthogonal (SDO) scheme. We demonstrate the performance of the SDO
scheme on stochastic solutions of the 1D Korteweg-de Vries and 2D Navier-Stokes
equations.Comment: Minor revision
A sequential sampling strategy for extreme event statistics in nonlinear dynamical systems
We develop a method for the evaluation of extreme event statistics associated
with nonlinear dynamical systems, using a small number of samples. From an
initial dataset of design points, we formulate a sequential strategy that
provides the 'next-best' data point (set of parameters) that when evaluated
results in improved estimates of the probability density function (pdf) for a
scalar quantity of interest. The approach utilizes Gaussian process regression
to perform Bayesian inference on the parameter-to-observation map describing
the quantity of interest. We then approximate the desired pdf along with
uncertainty bounds utilizing the posterior distribution of the inferred map.
The 'next-best' design point is sequentially determined through an optimization
procedure that selects the point in parameter space that maximally reduces
uncertainty between the estimated bounds of the pdf prediction. Since the
optimization process utilizes only information from the inferred map it has
minimal computational cost. Moreover, the special form of the metric emphasizes
the tails of the pdf. The method is practical for systems where the
dimensionality of the parameter space is of moderate size, i.e. order O(10). We
apply the method to estimate the extreme event statistics for a very
high-dimensional system with millions of degrees of freedom: an offshore
platform subjected to three-dimensional irregular waves. It is demonstrated
that the developed approach can accurately determine the extreme event
statistics using limited number of samples
Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation
We develop performance criteria for the objective comparison of different
classes of single-degree-of-freedom oscillators under stochastic excitation.
For each family of oscillators, these objective criteria take into account the
maximum possible energy harvested for a given response level, which is a
quantity that is directly connected to the size of the harvesting
configuration. We prove that the derived criteria are invariant with respect to
magnitude or temporal rescaling of the input spectrum and they depend only on
the relative distribution of energy across different harmonics of the
excitation. We then compare three different classes of linear and nonlinear
oscillators and using stochastic analysis tools we illustrate that in all cases
of excitation spectra (monochromatic, broadband, white-noise) the optimal
performance of all designs cannot exceed the performance of the linear design.
Subsequently, we study the robustness of this optimal performance to small
perturbations of the input spectrum and illustrate the advantages of nonlinear
designs relative to linear ones.Comment: 24 pages, 12 figure
Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes
In this work, we consider systems that are subjected to intermittent
instabilities due to external stochastic excitation. These intermittent
instabilities, though rare, have a large impact on the probabilistic response
of the system and give rise to heavy-tailed probability distributions. By
making appropriate assumptions on the form of these instabilities, which are
valid for a broad range of systems, we formulate a method for the analytical
approximation of the probability distribution function (pdf) of the system
response (both the main probability mass and the heavy-tail structure). In
particular, this method relies on conditioning the probability density of the
response on the occurrence of an instability and the separate analysis of the
two states of the system, the unstable and stable state. In the stable regime
we employ steady state assumptions, which lead to the derivation of the
conditional response pdf using standard methods for random dynamical systems.
The unstable regime is inherently transient and in order to analyze this regime
we characterize the statistics under the assumption of an exponential growth
phase and a subsequent decay phase until the system is brought back to the
stable attractor. The method we present allows us to capture the statistics
associated with the dynamics that give rise to heavy-tails in the system
response and the analytical approximations compare favorably with direct Monte
Carlo simulations, which we illustrate for two prototype intermittent systems:
an intermittently unstable mechanical oscillator excited by correlated
multiplicative noise and a complex mode in a turbulent signal with fixed
frequency, where multiplicative stochastic damping and additive noise model
interactions between various modes.Comment: 29 pages, 15 figure
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