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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. Outline.
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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
Outline I.1. TheFeketeproblem and Smale’s 7th problem. I.2. TheForcesMethodonthe 2-sphere. II. A numerical-statisticalapproachto Smale’s 7th problem. III. TheForcesMethodonW-compact sets and otherextensions.
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. (Nurmela) The solutions of this problem define lattices that exhibite a high degree of “regularity” (many equilateral triangles).
The Fekete problem I. Newton (1643-1727) N. Copernicus (1473-1543) G. Galilei (1564-1642) J. Kepler (1571-1630)
The Fekete problem Ch.A. de Coulomb (1736-1806)
The Fekete problem Potential energy: Forces field: Equilibrium positions:
The Fekete problem The “plum-pudding” model (1904): J.J. Thomson (1856-1940) Thomson’s problem:
The Fekete problem A. Einstein (1879-1955) M.K. Planck (1858-1947) N. Bohr (1885-1962) E. Schrödinger (1887-1961) W. Heisenberg (1901-1979)
The Fekete problem Molecular Mechanics, Electrostatics, Crystallography, structures of viruses, proteins, bacteri, multi-electron bubbles, microclusters of rare gases… (Atiyah&Sutcliffe) (Bowick et al.) Van der Waals interaction: Lennard-Jones energy J.D. van der Waals (1837-1923)
The Fekete problem Transfinite diameter: Logarithmic potential energy: M. Fekete (1886-1957)
The Fekete problem G. Szegö (1895-1985) O. Frostman (1907-1977) G. Polya (1887-1985) Best-packing problem
The Fekete problem Numerical Integration: Polynomial Interpolation: (Hesthaven)
The Fekete problem Computer Aided Design: Mesh generation: (Person&Strang) (Shimada&Gossard) Visualization of implicitly defined surfaces: (Witkin&Heckbert)
The Fekete problem 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. Logarithmic energy: Riesz’s energies: General case: Newtonian energy: Best-packing problem:
Smale’s 7th problem Fields medalist in 1966. 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. S. Smale (1930- )
Smale’s 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 ? Hererepresentsthelogarithmicpotentialenergy and are the Feketepointsassociatedwiththisenergyonthe 2-sphere. It is known that
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: Rakhmanov, Saff and Zhou: Zhou: numerical results for
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. 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. No general results about convergence, stability, robustness and computational cost have been published.
State of the art The spiral points: Rakhmanov, Saff and Zhou.
The Forces Method The algorithm: + return algorithm Disequilibrium degree:
The Forces Method The algorithm: + return algorithm Convergence curve:
The cost of a local minimum Cost at each step: the logarithmic energy requires only elementary operations for the actualization of the forces (O(N2) operations), since it is not necessary to compute the energy.
The energy The line-search procedure: minimize the energy in the advance direction.
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. MareNostrum (485000 hours): for N=107, a total of 400 steps from a difficult starting position (10080 CPUs working in parallel).
