26,572 research outputs found
A Study on Rotation Invariance in Differential Evolution
The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Epistasis is the correlation between the variables of a function and is a challenge often posed by real-world optimisation problems. Synthetic benchmark problems simulate a highly epistatic problem by performing a so-called problem's rotation.
Mutation in Differential Evolution (DE) is inherently rotational invariant since it simultaneously perturbs all the variables. On the other hand, crossover, albeit fundamental for achieving a good performance, retains some of the variables, thus being inadequate to tackle highly epistatic problems.
This article proposes an extensive study on rotational invariant crossovers in DE. We propose an analysis of the literature, a taxonomy of the proposed method and an experimental setup where each problem is addressed in both its non-rotated and rotated version. Our experimental study includes problems over five different levels of dimensionality and nine algorithms.
Numerical results show that 1) for a fixed quota of transferred design variables, the exponential crossover displays a better performance, on both rotated and non-rotated problems, in high dimensions while the binomial crossover seems to be preferable in low dimensions; 2) the rotational invariant mutation DE/current-to-rand is not competitive with standard DE implementations throughout the entire set of experiments we have presented; 3) DE crossovers that perform a change of coordinates to distribute the moves over the components of the offspring offer high-performance results on some problems. However, on average the standard DE/rand/1/exp appears to achieve the best performance on both rotated and non-rotated testbeds
Nonlinear Evolution Equations Invariant Under Schroedinger Group in three-dimensional Space-time
A classification of all possible realizations of the Galilei,
Galilei-similitude and Schroedinger Lie algebras in three-dimensional
space-time in terms of vector fields under the action of the group of local
diffeomorphisms of the space \R^3\times\C is presented. Using this result a
variety of general second order evolution equations invariant under the
corresponding groups are constructed and their physical significance are
discussed
Relativistic diffusion of elementary particles with spin
We obtain a generalization of the relativistic diffusion of Schay and Dudley
for particles with spin. The diffusion equation is a classical version of an
equation for the Wigner function of an elementary particle. The elementary
particle is described by a unitary irreducible representation of the Poincare
group realized in the Hilbert space of wave functions in the momentum space.
The arbitrariness of the Wigner rotation appears as a gauge freedom of the
diffusion equation. The spin is described as a connection of a fiber bundle
over the momentum hyperbolic space (the mass-shell). Motion in an
electromagnetic field, transport equations and equilibrium states are
discussed.Comment: 21 pages,minor changes,the version published in Journ.Phys.
Unstable manifolds and Schroedinger dynamics of Ginzburg-Landau vortices
The time evolution of several interacting Ginzburg-Landau vortices according
to an equation of Schroedinger type is approximated by motion on a
finite-dimensional manifold. That manifold is defined as an unstable manifold
of an auxiliary dynamical system, namely the gradient flow of the
Ginzburg-Landau energy functional. For two vortices the relevant unstable
manifold is constructed numerically and the induced dynamics is computed. The
resulting model provides a complete picture of the vortex motion for arbitrary
vortex separation, including well-separated and nearly coincident vortices.Comment: 23 pages amslatex, 5 eps figures, minor typos correcte
The shear dynamo problem for small magnetic Reynolds numbers
We study large-scale dynamo action due to turbulence in the presence of a
linear shear flow, in the low conductivity limit. Our treatment is
nonperturbative in the shear strength and makes systematic use of both the
shearing coordinate transformation and the Galilean invariance of the linear
shear flow. The velocity fluctuations are assumed to have low magnetic Reynolds
number (Rm) but could have arbitrary fluid Reynolds number. The magnetic
fluctuations are determined to lowest order in Rm by explicit calculation of
the resistive Green's function for the linear shear flow. The mean
electromotive force is calculated and an integro-differential equation is
derived for the time evolution of the mean magnetic field. In this equation,
velocity fluctuations contribute to two different kinds of terms, the C and D
terms, in which first and second spatial derivatives of the mean magnetic
field, respectively, appear inside the spacetime integrals. The contribution of
the D terms is such that the time evolution of the cross-shear components of
the mean field do not depend on any other components excepting themselves.
Therefore, to lowest order in Rm but to all orders in the shear strength, the D
terms cannot give rise to a shear-current assisted dynamo effect. Casting the
integro-differential equation in Fourier space, we show that the normal modes
of the theory are a set of shearing waves, labelled by their sheared
wavevectors. The integral kernels are expressed in terms of the velocity
spectrum tensor, which is the fundamental quantity that needs to be specified
to complete the integro-differential equation description of the time evolution
of the mean magnetic field.Comment: Near-final version; Accepted for publication in the Journal of Fluid
Mechanics; References added; 22 pages, 2 figure
Experimental Comparisons of Derivative Free Optimization Algorithms
In this paper, the performances of the quasi-Newton BFGS algorithm, the
NEWUOA derivative free optimizer, the Covariance Matrix Adaptation Evolution
Strategy (CMA-ES), the Differential Evolution (DE) algorithm and Particle Swarm
Optimizers (PSO) are compared experimentally on benchmark functions reflecting
important challenges encountered in real-world optimization problems.
Dependence of the performances in the conditioning of the problem and
rotational invariance of the algorithms are in particular investigated.Comment: 8th International Symposium on Experimental Algorithms, Dortmund :
Germany (2009
About Symmetries in Physics
The goal of this introduction to symmetries is to present some general ideas,
to outline the fundamental concepts and results of the subject and to situate a
bit the following lectures of this school. [These notes represent the write-up
of a lecture presented at the fifth ``Seminaire Rhodanien de Physique: Sur les
Symetries en Physique" held at Dolomieu (France), 17-21 March 1997. Up to the
appendix and the graphics, it is to be published in "Symmetries in Physics",
F.Gieres, M.Kibler,C.Lucchesi and O.Piguet, eds. (Editions Frontieres, 1998).]Comment: Latex, 42 pages, 4 figure
Fuzzy Fluid Mechanics in Three Dimensions
We introduce a rotation invariant short distance cut-off in the theory of an
ideal fluid in three space dimensions, by requiring momenta to take values in a
sphere. This leads to an algebra of functions in position space is
non-commutative. Nevertheless it is possible to find appropriate analogues of
the Euler equations of an ideal fluid. The system still has a hamiltonian
structure. It is hoped that this will be useful in the study of possible
singularities in the evolution of Euler (or Navier-Stokes) equations in three
dimensions.Comment: Additional reference
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