•Notícia

Mathematicians from the Technical University of Catalonia (UPC), Enrique Bendito, Ángeles Carmona, Andrés M. Encinas and José Manuel Gesto, have solved a century-old mathematical problem known as the "Fekete points problem". The group developed an algorithm which has applications in biology and chemistry. The scope of this tool's validity was proven thanks to the calculating power of the FinisTerrae supercomputer and the way is now open to solving even more complex mathematical problems.

"The Fekete points problem consists of determining the position of a certain number of points on an object such that the potential energy produced by the interaction of these points is minimum", said Enrique Bendito. That is, if we have a number of particles that affect other particles, for example, electrically-charged particles, the idea is to distribute them on the surface of an object so that they interfere with each other as little as possible, the interactions between them are compensated and they reach a state of equilibrium.

Finding an appropriate configuration for the points becomes more complicated as the number of particles increases and the geometry of the object on which they are to be placed becomes more irregular.

This group, from the Department of Applied mathematics III has developed an algorithm that provides solutions for a wide range of geometries and for different types of interaction between the particles.

This algorithm goes far beyond any that have yet been created to solve the problem: it does not require as much calculation time to obtain the configuration of the points and is not limited to spheres but can work on objects with far more complex geometries, such as bananas, apples or polyhedra. Furthermore, it is able to situate far more points in equilibrium on the object than previous algorithms.

The validity of this algorithm had already been shown using conventional computers, as reported in 2007 in the Journal of Computational Physics. Thanks to the FinisTerrae supercomputer at the Computational Centre of Galicia (CESGA), the group was able to prove the validity of the algorithm with a far higher number of points.

Until now, algorithms approximating the problem have been able to situate a few thousand points on a sphere. Working with FinisTerrae, the UPC researchers have managed to find equilibrium configurations on a sphere for up to 50,000 points. Furthermore, in order to test the capacity of the algorithm, they approached the problem with up to a million points. "Using FinisTerrae, we have clearly shown that our algorithm is robust, versatile and efficient", said José Manuel Gesto.

The CESGA set the challenge of solving the Fekete points problem with the help of the supercomputer in order to test the calculating ability of FinisTerrae during its test period. The work of the supercomputer, carried out during two weeks in February, required 35,000 hours of calculation time. If only one of the FinisTerrae CPUs had been used, it would have required 40 years to perform the calculation. When the calculation was performed for a million points, 1024 CPUs worked in parallel for a day and a half.

The UPC group has been working for five years with this algorithm, which can be applied to "any research involving the interaction of particles under the laws of classical mechanics". The basic research carried out provides a tool that will be useful in biology and chemistry studies. For example, the algorithm has applications "in studies on the shape of molecules and crystalline structures, gases, viruses, proteins and bacteria", said Bendito.

In fact, the behavior of biological elements is affected by natural forces (attraction, repulsion, etc.) that are governed by the laws of mechanics. Therefore, the algorithm will help to model the behavior of biological elements.

The search for Fekete points is a step forward in solving another key problem of modern mathematics: Smale's problem 7. At the end of the 20th century, the International Mathematical Union asked what were the main problems remaining to be solved in the 21st century and the prestigious mathematician, Stephen Smale, provided a list of 18 problems; the seventh of these is closely linked to Fekete points. Problem 7 posits the possibility of finding

configurations, sufficiently close to the optimum configurations, on a 2-sphere. These configurations would be used to solve specific systems of equations. Thanks to the new algorithm and FinisTerrae, the group has obtained more than 50 million ways of distributing points on a sphere and this is "the largest sample ever obtained on Smale's problem 7", said Gesto