448 research outputs found
Dynamic isoperimetry and the geometry of Lagrangian coherent structures
The study of transport and mixing processes in dynamical systems is
particularly important for the analysis of mathematical models of physical
systems. We propose a novel, direct geometric method to identify subsets of
phase space that remain strongly coherent over a finite time duration. This new
method is based on a dynamic extension of classical (static) isoperimetric
problems; the latter are concerned with identifying submanifolds with the
smallest boundary size relative to their volume.
The present work introduces \emph{dynamic} isoperimetric problems; the study
of sets with small boundary size relative to volume \emph{as they are evolved
by a general dynamical system}. We formulate and prove dynamic versions of the
fundamental (static) isoperimetric (in)equalities; a dynamic Federer-Fleming
theorem and a dynamic Cheeger inequality. We introduce a new dynamic Laplacian
operator and describe a computational method to identify coherent sets based on
eigenfunctions of the dynamic Laplacian.
Our results include formal mathematical statements concerning geometric
properties of finite-time coherent sets, whose boundaries can be regarded as
Lagrangian coherent structures. The computational advantages of our new
approach are a well-separated spectrum for the dynamic Laplacian, and
flexibility in appropriate numerical approximation methods. Finally, we
demonstrate that the dynamic Laplacian operator can be realised as a
zero-diffusion limit of a newly advanced probabilistic transfer operator method
(Froyland, 2013) for finding coherent sets, which is based on small diffusion.
Thus, the present approach sits naturally alongside the probabilistic approach
(Froyland, 2013), and adds a formal geometric interpretation
Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach
The long-term distributions of trajectories of a flow are described by
invariant densities, i.e. fixed points of an associated transfer operator. In
addition, global slowly mixing structures, such as almost-invariant sets, which
partition phase space into regions that are almost dynamically disconnected,
can also be identified by certain eigenfunctions of this operator. Indeed,
these structures are often hard to obtain by brute-force trajectory-based
analyses. In a wide variety of applications, transfer operators have proven to
be very efficient tools for an analysis of the global behavior of a dynamical
system.
The computationally most expensive step in the construction of an approximate
transfer operator is the numerical integration of many short term trajectories.
In this paper, we propose to directly work with the infinitesimal generator
instead of the operator, completely avoiding trajectory integration. We propose
two different discretization schemes; a cell based discretization and a
spectral collocation approach. Convergence can be shown in certain
circumstances. We demonstrate numerically that our approach is much more
efficient than the operator approach, sometimes by several orders of magnitude
Transport in time-dependent dynamical systems: Finite-time coherent sets
We study the transport properties of nonautonomous chaotic dynamical systems
over a finite time duration. We are particularly interested in those regions
that remain coherent and relatively non-dispersive over finite periods of time,
despite the chaotic nature of the system. We develop a novel probabilistic
methodology based upon transfer operators that automatically detects maximally
coherent sets. The approach is very simple to implement, requiring only
singular vector computations of a matrix of transitions induced by the
dynamics. We illustrate our new methodology on an idealized stratospheric flow
and in two and three dimensional analyses of European Centre for Medium Range
Weather Forecasting (ECMWF) reanalysis data
Optimal mixing enhancement
We introduce a general-purpose method for optimising the mixing rate of
advective fluid flows. An existing velocity field is perturbed in a
neighborhood to maximize the mixing rate for flows generated by velocity fields
in this neighborhood. Our numerical approach is based on the infinitesimal
generator of the flow and is solved by standard linear programming methods. The
perturbed flow may be easily constrained to preserve the same steady state
distribution as the original flow, and various natural geometric constraints
can also be simply applied. The same technique can also be used to optimize the
mixing rate of advection-diffusion flow models by manipulating the drift term
in a small neighborhood
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