EU Regional School - Maday Seminar- Part2
Prof. Yvon Maday - An Empirical Interpolation Procedure: The Magic Points
Université Pierre et Marie Curie, Paris
Lagrangian interpolation is a classical way to approximate general functions by ﬁnite sums of well chosen, pre-deﬁnite, linearly independent generating functions; it is much simpler 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 deﬁning the best sample of points is nevertheless rather complex and is in general not solved. In this talk we describe a way to derive such sets of points. We do not claim that the points resulting from this construction are optimal in any sense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, the process is very general and simple to implement, and compared to situations where the best behavior is known, it is relatively competitive. The application of this interpolation will be presented that allows to treat efficiently non linearities in PDE when discretization with reduced basis methods is performed.