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