Los puntos de Fekete y el séptimo problema de Smale
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Transcript of Los puntos de Fekete y el séptimo problema de Smale
Los puntos de Fekete y el séptimo problema de Smale
Grupo VARIDIS: Enrique Bendito, Ángeles Carmona,
Andrés Marcos Encinas, Jose Manuel Gesto, Agustín Medina.
Deptartamento de Matemática Aplicada III
Outline
I.1. The Fekete problem and Smale’s 7th problem.
I.2. The Forces Method on the 2-sphere.
II. A numerical-statistical approach to
Smale’s 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 arekept 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(1473-1543)
G. Galilei(1564-1642)
J. Kepler(1571-1630)
I. Newton(1643-1727)
The Fekete problem
Ch.A. de Coulomb(1736-1806)
The Fekete problemPotential energy:
Forces field:
Equilibrium positions:
The Fekete problem
J.J. Thomson(1856-1940)
The “plum-pudding” model (1904):
Thomson’s problem:
The Fekete problem
A. Einstein(1879-1955)
M.K. Planck(1858-1947)
W. Heisenberg(1901-1979)
N. Bohr(1885-1962)
E. Schrödinger(1887-1961)
The Fekete problemMolecular 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(1837-1923)
The Fekete problem
M. Fekete(1886-1957)
Transfinite diameter:
Logarithmic potential energy:
The Fekete problem
G. Polya(1887-1985)
G. Szegö(1895-1985)
O. Frostman(1907-1977)
Best-packing problem
The Fekete problemNumerical Integration:
Polynomial Interpolation:
(Hesthaven)
The Fekete problemComputer 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:
Riesz’s 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.
Smale’s 7th problem
S. Smale(1930- )
Fields medalist in 1966.
Personal interests: Complexity Theory and Numerical Analysis (polynomial time algorithms).
With M. Shub, he studied the complexity of theproblem 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 forthe XXIth century”, presented at the Fields Institute in 1997.
Smale’s 7th problem
It is known that
Smale’s 7th problem
Smale’s 7th problem
Smale’s 7th problem
Smale’s 7th problem
Smale’s 7th problem
State of the art
Massive multiextremality: lots of local minimawith very similar energy values.
State of the art
Massive multiextremality: lots of local minimawith 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 artThe 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
Numerical experiments
Numerical experiments
Numerical experiments
Numerical experiments
Numerical experiments
The cost of a local minimumCost at each step: the logarithmic energy requires only elementary operations for the actualization of the forces (O(N2) operations), since it is not necessary tocompute the energy.
The cost of a local minimum
The cost of a local minimum
The energy
The line-search procedure:minimize the energy in the advance direction.
MareNostrum (485000 hours): for N=107, a total of 400 steps from a difficult starting position (10080 CPUs working in parallel).
Large scale experiments
The cluster Clonetroop (100000 hours): numerical experiments to study the properties of the Forces Method and the first 2·106 data for Smale’s 7th problem.
The FinisTerrae challenge (350000 hours):I. The cost of a local minimum (150000 hours): -For N=10000, a total of 1000 runs attaining an error of 10-9 . -For N=20000, a total of 100 runs attaining an error of 5·10-10 . -For N=50000, a total of 10 runs attaining an error of 10-10 .II. Robustness (40000 hours, 1024 CPUs working in parallel): -For N=106, a total of 3000 steps from a delta starting position.III. Sample information for Smale’s 7th problem (160000 hours): -Almost 5.1·107 runs for different N between 300 and 1000.
Large scale experiments
Large scale experiments
The FinisTerrae challenge
The FinisTerrae challenge
The FinisTerrae challenge
The FinisTerrae challenge
The FinisTerrae challenge
MareNostrum