The proper generalized decomposition: a powerful tool for model reduction

One of my best scientific meetings was with Paco Chinesta some years ago. This close partnership has started when he visited Grenoble. He asked me: Can you solve quickly Fokker-Planck equation describing microscopic behaviour of complex fluids?. This question was the beginning of a discovery in the world of the kinetic theory microscopic descriptions. It was also the starting point of the development of efficient numerical techniques reducing simulation costs. Smooth Particle Hydrodynamic was the first release of our attempts. POD and other derived techniques using 'a priori' model reduction was the second release. Finally a development of a powerful technique was the main achievement of these works, now known as Proper Generalized Decomposition (PGD). With Paco, we established many collaborations with highly recognized scientists, allowing the proposal of new numerical strategies. Among them I would mention Elias Cueto, Antonio Falco, Etienne Pruliere, David Ryckelynck and many others that deserve my acknowledgments. In this paper I'm going to present first the general framework related to the purpose of a fine microscopic description. Then, some details of one of the developed numerical techniques (the PGD) will be presented. Finally, many examples in the kinetic theory description will be addressed for illustrating the possibilities of this powerful technique for solving problems never until now solved.