The Empirical Interpolation Method

Abstract

In this thesis we look at the Empirical Interpolation Method (EIM) and how it can be used in different applications. We propose a new formulation of EIM to make it easier to perform analytical operations like differentiation and integration of the basis functions as well as to apply EIM to a variety of problems. The new formulation is used to develop quadrature rules for the circle and semicircle, as well as for arbitrary simple polygons. The new formulation is also used to solve partial differential equations using a collocation approach on various domains including the circle, semicircle and triangle. The framework is briefly applied to compression of 3D animation in addition to recognition of images and sound.Several of the methods show great potential, with exponential convergence for quadrature and collocation for regular problems. However, there are also serious issues that must be addressed if the methods are to be developed further. These issues are related to making the methods more robust and stable.