23,719 research outputs found
The Magnus expansion and some of its applications
Approximate resolution of linear systems of differential equations with
varying coefficients is a recurrent problem shared by a number of scientific
and engineering areas, ranging from Quantum Mechanics to Control Theory. When
formulated in operator or matrix form, the Magnus expansion furnishes an
elegant setting to built up approximate exponential representations of the
solution of the system. It provides a power series expansion for the
corresponding exponent and is sometimes referred to as Time-Dependent
Exponential Perturbation Theory. Every Magnus approximant corresponds in
Perturbation Theory to a partial re-summation of infinite terms with the
important additional property of preserving at any order certain symmetries of
the exact solution. The goal of this review is threefold. First, to collect a
number of developments scattered through half a century of scientific
literature on Magnus expansion. They concern the methods for the generation of
terms in the expansion, estimates of the radius of convergence of the series,
generalizations and related non-perturbative expansions. Second, to provide a
bridge with its implementation as generator of especial purpose numerical
integration methods, a field of intense activity during the last decade. Third,
to illustrate with examples the kind of results one can expect from Magnus
expansion in comparison with those from both perturbative schemes and standard
numerical integrators. We buttress this issue with a revision of the wide range
of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its
applications to several physical problem
Structure-Preserving Discretization of Incompressible Fluids
The geometric nature of Euler fluids has been clearly identified and
extensively studied over the years, culminating with Lagrangian and Hamiltonian
descriptions of fluid dynamics where the configuration space is defined as the
volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed
as a consequence of Noether's theorem associated with the particle relabeling
symmetry of fluid mechanics. However computational approaches to fluid
mechanics have been largely derived from a numerical-analytic point of view,
and are rarely designed with structure preservation in mind, and often suffer
from spurious numerical artifacts such as energy and circulation drift. In
contrast, this paper geometrically derives discrete equations of motion for
fluid dynamics from first principles in a purely Eulerian form. Our approach
approximates the group of volume-preserving diffeomorphisms using a finite
dimensional Lie group, and associated discrete Euler equations are derived from
a variational principle with non-holonomic constraints. The resulting discrete
equations of motion yield a structure-preserving time integrator with good
long-term energy behavior and for which an exact discrete Kelvin's circulation
theorem holds
New Langevin and Gradient Thermostats for Rigid Body Dynamics
We introduce two new thermostats, one of Langevin type and one of gradient
(Brownian) type, for rigid body dynamics. We formulate rotation using the
quaternion representation of angular coordinates; both thermostats preserve the
unit length of quaternions. The Langevin thermostat also ensures that the
conjugate angular momenta stay within the tangent space of the quaternion
coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have
constructed three geometric numerical integrators for the Langevin thermostat
and one for the gradient thermostat. The numerical integrators reflect key
properties of the thermostats themselves. Namely, they all preserve the unit
length of quaternions, automatically, without the need of a projection onto the
unit sphere. The Langevin integrators also ensure that the angular momenta
remain within the tangent space of the quaternion coordinates. The Langevin
integrators are quasi-symplectic and of weak order two. The numerical method
for the gradient thermostat is of weak order one. Its construction exploits
ideas of Lie-group type integrators for differential equations on manifolds. We
numerically compare the discretization errors of the Langevin integrators, as
well as the efficiency of the gradient integrator compared to the Langevin ones
when used in the simulation of rigid TIP4P water model with smoothly truncated
electrostatic interactions. We observe that the gradient integrator is
computationally less efficient than the Langevin integrators. We also compare
the relative accuracy of the Langevin integrators in evaluating various static
quantities and give recommendations as to the choice of an appropriate
integrator.Comment: 16 pages, 4 figure
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation
The spectrum of the generator (Kolmogorov operator) of a diffusion process,
referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed
characterization of correlation functions and power spectra of stochastic
systems via decomposition formulas in terms of RP resonances. Stochastic
analysis techniques relying on the theory of Markov semigroups for the study of
the RP spectrum and a rigorous reduction method is presented in Part I. This
framework is here applied to study a stochastic Hopf bifurcation in view of
characterizing the statistical properties of nonlinear oscillators perturbed by
noise, depending on their stability. In light of the H\"ormander theorem, it is
first shown that the geometry of the unperturbed limit cycle, in particular its
isochrons, is essential to understand the effect of noise and the phenomenon of
phase diffusion. In addition, it is shown that the spectrum has a spectral gap,
even at the bifurcation point, and that correlations decay exponentially fast.
Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are
then obtained, away from the bifurcation point, based on the knowledge of the
linearized deterministic dynamics and the characteristics of the noise. These
formulas allow one to understand how the interaction of the noise with the
deterministic dynamics affect the decay of correlations. Numerical results
complement the study of the RP spectrum at the bifurcation, revealing useful
scaling laws. The analysis of the Markov semigroup for stochastic bifurcations
is thus promising in providing a complementary approach to the more geometric
random dynamical system approach. This approach is not limited to
low-dimensional systems and the reduction method presented in part I is applied
to a stochastic model relevant to climate dynamics in part III
New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)
We consider hypo-elliptic diffusion and convection-diffusion on , the quotient of the Lie group of rigid body motions SE(3) in
which group elements are equivalent if they are equal up to a rotation around
the reference axis. We show that we can derive expressions for the convolution
kernels in terms of eigenfunctions of the PDE, by extending the approach for
the SE(2) case. This goes via application of the Fourier transform of the PDE
in the spatial variables, yielding a second order differential operator. We
show that the eigenfunctions of this operator can be expressed as (generalized)
spheroidal wave functions. The same exact formulas are derived via the Fourier
transform on SE(3). We solve both the evolution itself, as well as the
time-integrated process that corresponds to the resolvent operator.
Furthermore, we have extended a standard numerical procedure from SE(2) to
SE(3) for the computation of the solution kernels that is directly related to
the exact solutions. Finally, we provide a novel analytic approximation of the
kernels that we briefly compare to the exact kernels.Comment: Revised and restructure
Gauge Theory for Finite-Dimensional Dynamical Systems
Gauge theory is a well-established concept in quantum physics,
electrodynamics, and cosmology. This theory has recently proliferated into new
areas, such as mechanics and astrodynamics. In this paper, we discuss a few
applications of gauge theory in finite-dimensional dynamical systems with
implications to numerical integration of differential equations. We distinguish
between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry
is, in essence, re-scaling of the independent variable, while descriptive gauge
symmetry is a Yang-Mills-like transformation of the velocity vector field,
adapted to finite-dimensional systems. We show that a simple gauge
transformation of multiple harmonic oscillators driven by chaotic processes can
render an apparently "disordered" flow into a regular dynamical process, and
that there exists a remarkable connection between gauge transformations and
reduction theory of ordinary differential equations. Throughout the discussion,
we demonstrate the main ideas by considering examples from diverse engineering
and scientific fields, including quantum mechanics, chemistry, rigid-body
dynamics and information theory
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