Los puntos de Fekete y el sptimo problema de Smale Grupo
VARIDIS: Enrique Bendito, ngeles Carmona, Andrs Marcos Encinas,
Jose Manuel Gesto, Agustn Medina. Deptartamento de Matemtica
Aplicada III Outline I.1. The Fekete problem and Smales 7th
problem. I.2. The Forces Method on the 2-sphere. II. A
numerical-statistical approach to Smales 7th problem. III. The
Forces Method on W-compact sets and other extensions. Part I:
Introduction to the Fekete problem and to the Forces Method The
Fekete problem The search of the regularity: Stone models of the
five Platonic polyhedra. They date from about 2000BC and are kept
in the Ashmolean Museum in Oxford (figure extracted from a work by
Atiyah and Sutcliffe). The Fekete problem Best-packing problem:
Maximize the minimum distance between points + constraints. The
solutions of this problem define lattices that exhibite a high
degree of regularity (many equilateral triangles). (Nurmela) The
Fekete problem N. Copernicus ( ) G. Galilei ( ) J. Kepler ( ) I.
Newton ( ) The Fekete problem Ch.A. de Coulomb ( ) The Fekete
problem Potential energy: Forces field: Equilibrium positions: The
Fekete problem J.J. Thomson ( ) The plum-pudding model (1904):
Thomsons problem: The Fekete problem A. Einstein ( ) M.K. Planck (
) W. Heisenberg ( ) N. Bohr ( ) E. Schrdinger ( ) The Fekete
problem Molecular Mechanics, Electrostatics, Crystallography,
structures of viruses, proteins, bacteri, multi-electron bubbles,
microclusters of rare gases Van der Waals interaction:
Lennard-Jones energy (Bowick et al.) (Atiyah&Sutcliffe) J.D.
van der Waals ( ) The Fekete problem M. Fekete ( ) Transfinite
diameter: Logarithmic potential energy: The Fekete problem G. Polya
( ) G. Szeg ( ) O. Frostman ( ) Best-packing problem The Fekete
problem Numerical Integration: Polynomial Interpolation:
(Hesthaven) The Fekete problem Computer Aided Design: Mesh
generation: Visualization of implicitly defined surfaces:
(Witkin&Heckbert) (Shimada&Gossard) (Person&Strang) The
Fekete problem Logarithmic energy: Newtonian energy: Best-packing
problem: General case: Rieszs energies: We call the Fekete problem
that of determining the N -tuples of points, that minimize on a
compact set a potential energy functional that depends on the
relative distances between the N points. The N -tuples are called
the Fekete points. Smales 7th problem S. Smale (1930- ) Fields
medalist in Personal interests: Complexity Theory and Numerical
Analysis (polynomial time algorithms). With M. Shub, he studied the
complexity of the problem of finding the roots of a polynomial
system. The notion of condition number of a polynomial is crucial
in this study. Author of the list Mathematical problems for the
XXIth century, presented at the Fields Institute in 1997. Smales
7th problem It is possible to design an algorithm that finds a
configuration x of points on the 2-sphere satisfying the condition
in time polynomial in N ? Here represents the logarithmic potential
energy and are the Fekete points associated with this energy on the
2-sphere. It is known that Smales 7th problem State of the art
Massive multiextremality: lots of local minima with very similar
energy values. State of the art Massive multiextremality: lots of
local minima with very similar energy values. State of the art
Erber&Hockney for State of the art The energy of the global
minimum (the Fekete points) is unknown: few theoretical results.
Potential Theory: Zhou: numerical results for Rakhmanov, Saff and
Zhou: State of the art The computation of a local minimum is a
highly non-linear optimization problem with constraints: the use of
numerical methods is necessary. No general results about
convergence, stability, robustness and computational cost have been
published. Many algorithms have been used: Classic Optimization
Algorithms (Relaxation, Gradient, Conjugate Gradient, Newton,
quasi-Newton), Combinatorial Optimization Methods (Simulated
Annealing, Genetic Algorithms), ODE integrators (Runge-Kutta,
simplectic integrators). Most of the research has focused on the
case of the 2-sphere and. Recently some authors have presented
configurations for thousands of points. The spiral points:
Rakhmanov, Saff and Zhou. State of the art The Forces Method +
return algorithm The algorithm: Disequilibrium degree: The Forces
Method + return algorithm The algorithm: Convergence curve:
Numerical experiments The cost of a local minimum Cost at each
step: the logarithmic energy requires only elementary operations
for the actualization of the forces (O(N 2 ) operations), since it
is not necessary to compute the energy. The cost of a local minimum
The energy The line-search procedure: minimize the energy in the
advance direction. MareNostrum ( hours): for N=10 7, a total of 400
steps from a difficult starting position (10080 CPUs working in
parallel). Large scale experiments The cluster Clonetroop ( hours):
numerical experiments to study the properties of the Forces Method
and the first 2 10 6 data for Smales 7th problem. The FinisTerrae
challenge ( hours): I. The cost of a local minimum ( hours): -For
N=10000, a total of 1000 runs attaining an error of For N=20000, a
total of 100 runs attaining an error of 5 For N=50000, a total of
10 runs attaining an error of II. Robustness (40000 hours, 1024
CPUs working in parallel): -For N=10 6, a total of 3000 steps from
a delta starting position. III. Sample information for Smales 7th
problem ( hours): -Almost 5.1 10 7 runs for different N between 300
and 1000. Large scale experiments The FinisTerrae challenge
MareNostrum
