1,066 research outputs found
Direct N-body Simulations
Special high-accuracy direct force summation N-body algorithms and their
relevance for the simulation of the dynamical evolution of star clusters and
other gravitating N-body systems in astrophysics are presented, explained and
compared with other methods. Other methods means here approximate physical
models based on the Fokker-Planck equation as well as other, approximate
algorithms to compute the gravitational potential in N-body systems. Questions
regarding the parallel implementation of direct ``brute force'' N-body codes
are discussed. The astrophysical application of the models to the theory of
relaxing rotating and non-rotating collisional star clusters is presented,
briefly mentioning the questions of the validity of the Fokker-Planck
approximation, the existence of gravothermal oscillations and of rotation and
primordial binaries.Comment: 32 pages, 13 figures, in press in Riffert, H., Werner K. (eds),
Computational Astrophysics, The Journal of Computational and Applied
Mathematics (JCAM), Elsevier Press, Amsterdam, 199
Probability density adjoint for sensitivity analysis of the Mean of Chaos
Sensitivity analysis, especially adjoint based sensitivity analysis, is a
powerful tool for engineering design which allows for the efficient computation
of sensitivities with respect to many parameters. However, these methods break
down when used to compute sensitivities of long-time averaged quantities in
chaotic dynamical systems.
The following paper presents a new method for sensitivity analysis of {\em
ergodic} chaotic dynamical systems, the density adjoint method. The method
involves solving the governing equations for the system's invariant measure and
its adjoint on the system's attractor manifold rather than in phase-space. This
new approach is derived for and demonstrated on one-dimensional chaotic maps
and the three-dimensional Lorenz system. It is found that the density adjoint
computes very finely detailed adjoint distributions and accurate sensitivities,
but suffers from large computational costs.Comment: 29 pages, 27 figure
Nonlinear stochastic modeling with Langevin regression
Many physical systems characterized by nonlinear multiscale interactions can
be effectively modeled by treating unresolved degrees of freedom as random
fluctuations. However, even when the microscopic governing equations and
qualitative macroscopic behavior are known, it is often difficult to derive a
stochastic model that is consistent with observations. This is especially true
for systems such as turbulence where the perturbations do not behave like
Gaussian white noise, introducing non-Markovian behavior to the dynamics. We
address these challenges with a framework for identifying interpretable
stochastic nonlinear dynamics from experimental data, using both forward and
adjoint Fokker-Planck equations to enforce statistical consistency. If the form
of the Langevin equation is unknown, a simple sparsifying procedure can provide
an appropriate functional form. We demonstrate that this method can effectively
learn stochastic models in two artificial examples: recovering a nonlinear
Langevin equation forced by colored noise and approximating the second-order
dynamics of a particle in a double-well potential with the corresponding
first-order bifurcation normal form. Finally, we apply the proposed method to
experimental measurements of a turbulent bluff body wake and show that the
statistical behavior of the center of pressure can be described by the dynamics
of the corresponding laminar flow driven by nonlinear state-dependent noise.Comment: 30 pages, 13 figure
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