The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

The Generalized Empirical Interpolation Method (GEIM) is an extension first presented by Maday and Mula in Maday and Mula (2013) in 2013 of the classical empirical interpolation method (presented in 2004 by Barrault, Maday, Nguyen and Patera in Barrault et al. (2004)) where the evaluation at interpolating points is replaced by the more practical evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in Maday and Mula (2013), this allows to relax the continuity constraint in the target functions and expand both the application domain and the stability of the approach. In this paper, we present a thorough analysis of the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it to an inf–sup problem in the case of Hilbert spaces. In the second part of the paper, it will be explained how GEIM can be employed to monitor in real time physical experiments by providing an online accurate approximation of the phenomenon that is computed by combining the acquisition of a minimal number, optimally placed, measurements from the processes with their mathematical models (parameter-dependent PDEs). This idea is illustrated through a parameter dependent Stokes problem in which it is shown that the pressure and velocity fields can efficiently be reconstructed with a relatively low-dimensional interpolation space.