16,567 research outputs found
Kernel methods for detecting coherent structures in dynamical data
We illustrate relationships between classical kernel-based dimensionality
reduction techniques and eigendecompositions of empirical estimates of
reproducing kernel Hilbert space (RKHS) operators associated with dynamical
systems. In particular, we show that kernel canonical correlation analysis
(CCA) can be interpreted in terms of kernel transfer operators and that it can
be obtained by optimizing the variational approach for Markov processes (VAMP)
score. As a result, we show that coherent sets of particle trajectories can be
computed by kernel CCA. We demonstrate the efficiency of this approach with
several examples, namely the well-known Bickley jet, ocean drifter data, and a
molecular dynamics problem with a time-dependent potential. Finally, we propose
a straightforward generalization of dynamic mode decomposition (DMD) called
coherent mode decomposition (CMD). Our results provide a generic machine
learning approach to the computation of coherent sets with an objective score
that can be used for cross-validation and the comparison of different methods
Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas
We employ Navier-Stokes granular hydrodynamics to investigate the long-time
behavior of clustering instability in a freely cooling dilute granular gas in
two dimensions. We find that, in circular containers, the homogeneous cooling
state (HCS) of the gas loses its stability via a sub-critical pitchfork
bifurcation. There are no time-independent solutions for the gas density in the
supercritical region, and we present analytical and numerical evidence that the
gas develops thermal collapse unarrested by heat diffusion. To get more
insight, we switch to a simpler geometry of a narrow-sector-shaped container.
Here the HCS loses its stability via a transcritical bifurcation. For some
initial conditions a time-independent inhomogeneous density profile sets in,
qualitatively similar to that previously found in a narrow-channel geometry.
For other initial conditions, however, the dilute gas develops thermal collapse
unarrested by heat diffusion. We determine the dynamic scalings of the flow
close to collapse analytically and verify them in hydrodynamic simulations. The
results of this work imply that, in dimension higher than one, Navier-Stokes
hydrodynamics of a dilute granular gas is prone to finite-time density blowups.
This provides a natural explanation to the formation of densely packed clusters
of particles in a variety of initially dilute granular flows.Comment: 18 pages, 19 figure
Geometric Finite Element Discretization of Maxwell Equations in Primal and Dual Spaces
Based on a geometric discretization scheme for Maxwell equations, we unveil a
mathematical\textit{\}transformation between the electric field intensity
and the magnetic field intensity , denoted as Galerkin duality. Using
Galerkin duality and discrete Hodge operators, we construct two system
matrices, (primal formulation) and (dual
formulation) respectively, that discretize the second-order vector wave
equations. We show that the primal formulation recovers the conventional
(edge-element) finite element method (FEM) and suggests a geometric foundation
for it. On the other hand, the dual formulation suggests a new (dual) type of
FEM. Although both formulations give identical dynamical physical solutions,
the dimensions of the null spaces are different.Comment: 22 pages and 4 figure
A stability condition for turbulence model: From EMMS model to EMMS-based turbulence model
The closure problem of turbulence is still a challenging issue in turbulence
modeling. In this work, a stability condition is used to close turbulence.
Specifically, we regard single-phase flow as a mixture of turbulent and
non-turbulent fluids, separating the structure of turbulence. Subsequently,
according to the picture of the turbulent eddy cascade, the energy contained in
turbulent flow is decomposed into different parts and then quantified. A
turbulence stability condition, similar to the principle of the
energy-minimization multi-scale (EMMS) model for gas-solid systems, is
formulated to close the dynamic constraint equations of turbulence, allowing
the heterogeneous structural parameters of turbulence to be optimized. We call
this model the `EMMS-based turbulence model', and use it to construct the
corresponding turbulent viscosity coefficient. To validate the EMMS-based
turbulence model, it is used to simulate two classical benchmark problems,
lid-driven cavity flow and turbulent flow with forced convection in an empty
room. The numerical results show that the EMMS-based turbulence model improves
the accuracy of turbulence modeling due to it considers the principle of
compromise in competition between viscosity and inertia.Comment: 26 pages, 13 figures, 2 table
Frequency-domain algorithm for the Lorenz-gauge gravitational self-force
State-of-the-art computations of the gravitational self-force (GSF) on
massive particles in black hole spacetimes involve numerical evolution of the
metric perturbation equations in the time-domain, which is computationally very
costly. We present here a new strategy, based on a frequency-domain treatment
of the perturbation equations, which offers considerable computational saving.
The essential ingredients of our method are (i) a Fourier-harmonic
decomposition of the Lorenz-gauge metric perturbation equations and a numerical
solution of the resulting coupled set of ordinary equations with suitable
boundary conditions; (ii) a generalized version of the method of extended
homogeneous solutions [Phys. Rev. D {\bf 78}, 084021 (2008)] used to circumvent
the Gibbs phenomenon that would otherwise hamper the convergence of the Fourier
mode-sum at the particle's location; and (iii) standard mode-sum
regularization, which finally yields the physical GSF as a sum over regularized
modal contributions. We present a working code that implements this strategy to
calculate the Lorenz-gauge GSF along eccentric geodesic orbits around a
Schwarzschild black hole. The code is far more efficient than existing
time-domain methods; the gain in computation speed (at a given precision) is
about an order of magnitude at an eccentricity of 0.2, and up to three orders
of magnitude for circular or nearly circular orbits. This increased efficiency
was crucial in enabling the recently reported calculation of the long-term
orbital evolution of an extreme mass ratio inspiral [Phys. Rev. D {\bf 85},
061501(R) (2012)]. Here we provide full technical details of our method to
complement the above report.Comment: 27 pages, 4 figure
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