Reduced basis methods for non linear problems with an incursion in a general theory for interpolation set of points.

The extension of the reduced basis technique to nonlinear partial differential equations has led us to introduce an ``empirical Lagrangian interpolation'' method on a finite dimensional vectorial space spanned by functions that actually can be of any type. The efficiency that this approach, together with the simplicity of its implementation, has revealed in the applications, has led us to deepen its analysis. Lagrangian interpolation is a classical way to approximate general functions by finite sums of well chosen, pre-definite, linearly independent basis functions which is much more simple to implement than determining the best fits with respect to some Banach (or even Hilbert) norms. In addition, only partial knowledge is required (here values on some set of points). The problem of the definition of the best sample of points is nevertheless rather complex and is not solved in the general case. We propose a way to derive such sets of points. We do not claim that!
the points resulting from the construction explained here are optimal in any sense, nevertheless, the process is very general, simple to implement and, compared to situations where the best behavior is known, it is relatively competitive. The application to the approximation of nonlinear PDE will be presented.