827 research outputs found
On the nonlinear stability of symplectic integrators
The modified Hamiltonian is used to study the nonlinear stability of symplectic integrators, especially
for nonlinear oscillators. We give conditions under which an initial condition on a compact
energy surface will remain bounded for exponentially long times for sufficiently small time steps.
For example, the implicit midpoint rule achieves this for the critical energy surface of the H´enon-
Heiles system, while the leapfrog method does not. We construct explicit methods which are
nonlinearly stable for all simple mechanical systems for exponentially long times. We also address
questions of topological stability, finding conditions under which the original and modified energy
surfaces are topologically equivalent
An asynchronous leapfrog method II
A second order explicit one-step numerical method for the initial value
problem of the general ordinary differential equation is proposed. It is
obtained by natural modifications of the well-known leapfrog method, which is a
second order, two-step, explicit method. According to the latter method, the
input data for an integration step are two system states, which refer to
different times. The usage of two states instead of a single one can be seen as
the reason for the robustness of the method. Since the time step size thus is
part of the step input data, it is complicated to change this size during the
computation of a discrete trajectory. This is a serious drawback when one needs
to implement automatic time step control.
The proposed modification transforms one of the two input states into a
velocity and thus gets rid of the time step dependency in the step input data.
For these new step input data, the leapfrog method gives a unique prescription
how to evolve them stepwise.
The stability properties of this modified method are the same as for the
original one: the set of absolute stability is the interval [-i,+i] on the
imaginary axis. This implies exponential growth of trajectories in situations
where the exact trajectory has an asymptote.
By considering new evolution steps that are composed of two consecutive old
evolution steps we can average over the velocities of the sub-steps and get an
integrator with a much larger set of absolute stability, which is immune to the
asymptote problem.
The method is exemplified with the equation of motion of a one-dimensional
non-linear oscillator describing the radial motion in the Kepler problem.Comment: 41 pages, 25 figure
Reversible Architectures for Arbitrarily Deep Residual Neural Networks
Recently, deep residual networks have been successfully applied in many
computer vision and natural language processing tasks, pushing the
state-of-the-art performance with deeper and wider architectures. In this work,
we interpret deep residual networks as ordinary differential equations (ODEs),
which have long been studied in mathematics and physics with rich theoretical
and empirical success. From this interpretation, we develop a theoretical
framework on stability and reversibility of deep neural networks, and derive
three reversible neural network architectures that can go arbitrarily deep in
theory. The reversibility property allows a memory-efficient implementation,
which does not need to store the activations for most hidden layers. Together
with the stability of our architectures, this enables training deeper networks
using only modest computational resources. We provide both theoretical analyses
and empirical results. Experimental results demonstrate the efficacy of our
architectures against several strong baselines on CIFAR-10, CIFAR-100 and
STL-10 with superior or on-par state-of-the-art performance. Furthermore, we
show our architectures yield superior results when trained using fewer training
data.Comment: Accepted at AAAI 201
Linear multistep methods for integrating reversible differential equations
This paper studies multistep methods for the integration of reversible
dynamical systems, with particular emphasis on the planar Kepler problem. It
has previously been shown by Cano & Sanz-Serna that reversible linear
multisteps for first-order differential equations are generally unstable. Here,
we report on a subset of these methods -- the zero-growth methods -- that evade
these instabilities. We provide an algorithm for identifying these rare
methods. We find and study all zero-growth, reversible multisteps with six or
fewer steps. This select group includes two well-known second-order multisteps
(the trapezoidal and explicit midpoint methods), as well as three new
fourth-order multisteps -- one of which is explicit. Variable timesteps can be
readily implemented without spoiling the reversibility. Tests on Keplerian
orbits show that these new reversible multisteps work well on orbits with low
or moderate eccentricity, although at least 100 steps/radian are required for
stability.Comment: 31 pages, 9 figures, in press at The Astronomical Journa
Symplectic integrators with adaptive time steps
In recent decades, there have been many attempts to construct symplectic
integrators with variable time steps, with rather disappointing results. In
this paper we identify the causes for this lack of performance, and find that
they fall into two categories. In the first, the time step is considered a
function of time alone, \Delta=\Delta(t). In this case, backwards error
analysis shows that while the algorithms remain symplectic, parametric
instabilities arise because of resonance between oscillations of \Delta(t) and
the orbital motion. In the second category the time step is a function of phase
space variables \Delta=\Delta(q,p). In this case, the system of equations to be
solved is analyzed by introducing a new time variable \tau with dt=\Delta(q,p)
d\tau. The transformed equations are no longer in Hamiltonian form, and thus
are not guaranteed to be stable even when integrated using a method which is
symplectic for constant \Delta. We analyze two methods for integrating the
transformed equations which do, however, preserve the structure of the original
equations. The first is an extended phase space method, which has been
successfully used in previous studies of adaptive time step symplectic
integrators. The second, novel, method is based on a non-canonical
mixed-variable generating function. Numerical trials for both of these methods
show good results, without parametric instabilities or spurious growth or
damping. It is then shown how to adapt the time step to an error estimate found
by backward error analysis, in order to optimize the time-stepping scheme.
Numerical results are obtained using this formulation and compared with other
time-stepping schemes for the extended phase space symplectic method.Comment: 23 pages, 9 figures, submitted to Plasma Phys. Control. Fusio
Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories
Symplectic integrators offer many advantages for the numerical solution of
Hamiltonian differential equations, including bounded energy error and the
preservation of invariant sets. Two of the central Hamiltonian systems
encountered in plasma physics --- the flow of magnetic field lines and the
guiding center motion of magnetized charged particles --- resist symplectic
integration by conventional means because the dynamics are most naturally
formulated in non-canonical coordinates, i.e., coordinates lacking the familiar
partitioning. Recent efforts made progress toward non-canonical
symplectic integration of these systems by appealing to the variational
integration framework; however, those integrators were multistep methods and
later found to be numerically unstable due to parasitic mode instabilities.
This work eliminates the multistep character and, therefore, the parasitic mode
instabilities via an adaptation of the variational integration formalism that
we deem ``degenerate variational integration''. Both the magnetic field line
and guiding center Lagrangians are degenerate in the sense that their resultant
Euler-Lagrange equations are systems of first-order ODEs. We show that
retaining the same degree of degeneracy when constructing a discrete Lagrangian
yields one-step variational integrators preserving a non-canonical symplectic
structure on the original Hamiltonian phase space. The advantages of the new
algorithms are demonstrated via numerical examples, demonstrating superior
stability compared to existing variational integrators for these systems and
superior qualitative behavior compared to non-conservative algorithms
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