The generalized Empirical Interpolation Method: stability theory on Hilbert spaces with an application to the Stokes equation
Résumé
The Generalized Empirical Interpolation Method (GEIM) is an extension first presented in [1] of the classical empirical interpolation method (see [2], [3], [4]) where the evaluation at interpolating points is replaced by the evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in [1], this allows to relax the continuity constraint in the target functions and expand the application domain. A special effort has been made in this paper to understand the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it in the first part of the paper to an inf-sup problem in the case of Hilbert spaces. In the second part, it will be explained how GEIM can be employed to monitor in real time physical experiments by combining the acquisition of measurements from the processes with their mathematical models (parameter-dependent PDE's). This idea will be illustrated through a parameter dependent Stokes problem in which it will be shown that the pressure and velocity fields can efficiently be reconstructed with a relatively low dimension of the interpolating spaces.
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