The Generalized Empirical Interpolation Method

Abstract

The empirical interpolation method is an interpolation scheme with problem dependent basis functions and interpolation nodes, originally developed for parameter dependent functions. It was developed in connection with the reduced basis framework for fast evaluation of output from parameterized partial differential equations, but the procedure may be applicable to a variety of problems, such as image and pattern recognition, numerical integration and data compression. We present the theoretical background and implementation of the method, and give examples to verify exponential convergence for analytic problems. An extension of the method was proposed recently, denoted as the generalized empirical interpolation method (GEIM). The GEIM considers a parametric manifold of functions, with a set of linear functionals. Further, we explore how the interpolation points can be used as measurement points in the estimation of parameters from noisy data. We present the statistical framework, and we show how we can identify a set of parameter values that are consistent with our measurements.