2,157 research outputs found
Symplectic Model Reduction of Hamiltonian Systems
In this paper, a symplectic model reduction technique, proper symplectic
decomposition (PSD) with symplectic Galerkin projection, is proposed to save
the computational cost for the simplification of large-scale Hamiltonian
systems while preserving the symplectic structure. As an analogy to the
classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is
designed to build a symplectic subspace to fit empirical data, while the
symplectic Galerkin projection constructs a reduced Hamiltonian system on the
symplectic subspace. For practical use, we introduce three algorithms for PSD,
which are based upon: the cotangent lift, complex singular value decomposition,
and nonlinear programming. The proposed technique has been proven to preserve
system energy and stability. Moreover, PSD can be combined with the discrete
empirical interpolation method to reduce the computational cost for nonlinear
Hamiltonian systems. Owing to these properties, the proposed technique is
better suited than the classical POD-Galerkin approach for model reduction of
Hamiltonian systems, especially when long-time integration is required. The
stability, accuracy, and efficiency of the proposed technique are illustrated
through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure
The instanton method and its numerical implementation in fluid mechanics
A precise characterization of structures occurring in turbulent fluid flows
at high Reynolds numbers is one of the last open problems of classical physics.
In this review we discuss recent developments related to the application of
instanton methods to turbulence. Instantons are saddle point configurations of
the underlying path integrals. They are equivalent to minimizers of the related
Freidlin-Wentzell action and known to be able to characterize rare events in
such systems. While there is an impressive body of work concerning their
analytical description, this review focuses on the question on how to compute
these minimizers numerically. In a short introduction we present the relevant
mathematical and physical background before we discuss the stochastic Burgers
equation in detail. We present algorithms to compute instantons numerically by
an efficient solution of the corresponding Euler-Lagrange equations. A second
focus is the discussion of a recently developed numerical filtering technique
that allows to extract instantons from direct numerical simulations. In the
following we present modifications of the algorithms to make them efficient
when applied to two- or three-dimensional fluid dynamical problems. We
illustrate these ideas using the two-dimensional Burgers equation and the
three-dimensional Navier-Stokes equations
Burgers Turbulence
The last decades witnessed a renewal of interest in the Burgers equation.
Much activities focused on extensions of the original one-dimensional
pressureless model introduced in the thirties by the Dutch scientist J.M.
Burgers, and more precisely on the problem of Burgers turbulence, that is the
study of the solutions to the one- or multi-dimensional Burgers equation with
random initial conditions or random forcing. Such work was frequently motivated
by new emerging applications of Burgers model to statistical physics,
cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the
simplest instances of a nonlinear system out of equilibrium. The study of
random Lagrangian systems, of stochastic partial differential equations and
their invariant measures, the theory of dynamical systems, the applications of
field theory to the understanding of dissipative anomalies and of multiscaling
in hydrodynamic turbulence have benefited significantly from progress in
Burgers turbulence. The aim of this review is to give a unified view of
selected work stemming from these rather diverse disciplines.Comment: Review Article, 49 pages, 43 figure
Achieving precise mechanical control in intrinsically noisy systems
How can precise control be realized in intrinsically noisy systems? Here, we develop a general theoretical framework that provides a way of achieving precise control in signal-dependent noisy environments. When the control signal has Poisson or supra-Poisson noise, precise control is not possible. If, however, the control signal has sub-Poisson noise, then precise control is possible. For this case, the precise control solution is not a function, but a rapidly varying random process that must be averaged with respect to a governing probability density functional. Our theoretical approach is applied to the control of straight-trajectory arm movement. Sub-Poisson noise in the control signal is shown to be capable of leading to precise control. Intriguingly, the control signal for this system has a natural counterpart, namely the bursting pulses of neurons-trains of Dirac-delta functions-in biological systems to achieve precise control performance
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