14,799 research outputs found
Global bifurcation for the Whitham equation
We prove the existence of a global bifurcation branch of -periodic,
smooth, traveling-wave solutions of the Whitham equation. It is shown that any
subset of solutions in the global branch contains a sequence which converges
uniformly to some solution of H\"older class , . Bifurcation formulas are given, as well as some properties along
the global bifurcation branch. In addition, a spectral scheme for computing
approximations to those waves is put forward, and several numerical results
along the global bifurcation branch are presented, including the presence of a
turning point and a `highest', cusped wave. Both analytic and numerical results
are compared to traveling-wave solutions of the KdV equation
Functional factor analysis for periodic remote sensing data
We present a new approach to factor rotation for functional data. This is
achieved by rotating the functional principal components toward a predefined
space of periodic functions designed to decompose the total variation into
components that are nearly-periodic and nearly-aperiodic with a predefined
period. We show that the factor rotation can be obtained by calculation of
canonical correlations between appropriate spaces which make the methodology
computationally efficient. Moreover, we demonstrate that our proposed rotations
provide stable and interpretable results in the presence of highly complex
covariance. This work is motivated by the goal of finding interpretable sources
of variability in gridded time series of vegetation index measurements obtained
from remote sensing, and we demonstrate our methodology through an application
of factor rotation of this data.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS518 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Deep learning as closure for irreversible processes: A data-driven generalized Langevin equation
The ultimate goal of physics is finding a unique equation capable of
describing the evolution of any observable quantity in a self-consistent way.
Within the field of statistical physics, such an equation is known as the
generalized Langevin equation (GLE). Nevertheless, the formal and exact GLE is
not particularly useful, since it depends on the complete history of the
observable at hand, and on hidden degrees of freedom typically inaccessible
from a theoretical point of view. In this work, we propose the use of deep
neural networks as a new avenue for learning the intricacies of the unknowns
mentioned above. By using machine learning to eliminate the unknowns from GLEs,
our methodology outperforms previous approaches (in terms of efficiency and
robustness) where general fitting functions were postulated. Finally, our work
is tested against several prototypical examples, from a colloidal systems and
particle chains immersed in a thermal bath, to climatology and financial
models. In all cases, our methodology exhibits an excellent agreement with the
actual dynamics of the observables under consideration
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