los puntos de fekete y el séptimo problema de smale
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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.
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Part I: Introduction to the Fekete problem and to the Forces Method
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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).
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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)
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The Fekete problem
N. Copernicus(1473-1543)
G. Galilei(1564-1642)
J. Kepler(1571-1630)
I. Newton(1643-1727)
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The Fekete problem
Ch.A. de Coulomb(1736-1806)
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The Fekete problemPotential energy:
Forces field:
Equilibrium positions:
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The Fekete problem
J.J. Thomson(1856-1940)
The “plum-pudding” model (1904):
Thomson’s problem:
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The Fekete problem
A. Einstein(1879-1955)
M.K. Planck(1858-1947)
W. Heisenberg(1901-1979)
N. Bohr(1885-1962)
E. Schrödinger(1887-1961)
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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)
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The Fekete problem
M. Fekete(1886-1957)
Transfinite diameter:
Logarithmic potential energy:
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The Fekete problem
G. Polya(1887-1985)
G. Szegö(1895-1985)
O. Frostman(1907-1977)
Best-packing problem
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The Fekete problemNumerical Integration:
Polynomial Interpolation:
(Hesthaven)
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The Fekete problemComputer Aided Design:
Mesh generation:
Visualization of implicitly defined surfaces:
(Witkin&Heckbert)
(Shimada&Gossard)
(Person&Strang)
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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.
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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.
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Smale’s 7th problem
It is known that
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Smale’s 7th problem
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Smale’s 7th problem
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Smale’s 7th problem
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Smale’s 7th problem
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Smale’s 7th problem
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State of the art
Massive multiextremality: lots of local minimawith very similar energy values.
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State of the art
Massive multiextremality: lots of local minimawith very similar energy values.
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State of the art
Erber&Hockney
for
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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:
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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.
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The spiral points: Rakhmanov, Saff and Zhou.
State of the art
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The Forces Method
+ return algorithm
The algorithm:
Disequilibrium degree:
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The Forces Method
+ return algorithm
The algorithm:
Convergence curve:
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Numerical experiments
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Numerical experiments
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Numerical experiments
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Numerical experiments
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Numerical experiments
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Numerical experiments
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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.
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The cost of a local minimum
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The cost of a local minimum
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The energy
The line-search procedure:minimize the energy in the advance direction.
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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.
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Large scale experiments
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Large scale experiments
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The FinisTerrae challenge
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The FinisTerrae challenge
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The FinisTerrae challenge
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The FinisTerrae challenge
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The FinisTerrae challenge
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MareNostrum