## The GEIM to Inter-play with Data and Numerical Simulations for Real-Time Decisions

- Oct. 27, 2014
- 1 p.m.
- LeConte 312

## Abstract

The extension of classical lagrangian interpolation leads to the search of new methods that use interpolating basis functions of not necessarily polynomial nature. The Empirical Interpolation Method (EIM, see [1]) is a contribution in this directions that has been proposed for the approximation of functions of a compact set *F* of small Kolmogorov *n*-width in the continuous functions. The interpolating spaces *X _{n}* together with the suitable interpolating points are simultaneously given by a constructive greedy algorithm. A recent generalization of this procedure consists in replacing the point evaluations by a class of linear forms. This technique (called GEIM as for "generalized EIM", see [2]) allows to extend the concept of interpolation from the set of continuous functions to any Banach space

*X*. In this framework, we propose in this talk an overview of GEIM. After explaining some theoretical issues like the well-posedness of the interpolation process, the concept of Lebesgue constant in a Banach space ([2], [4]) and some results of convergence decay rates ([3]), we will focus on the potential applications of GEIM. Some examples will be shown in order to illustrate how the method can reconstruct in real-time a physical or industrial process by combining measurements from the experiment and mathematical models (via parameter dependent PDE's). Relying on these ideas, we will also outline how the method could be used to build an adaptive tool for the supervision of experiments that could distinguish between normal and accidental conditions. We believe that this tool could help in taking real-time decisions about the security of the process.

This work has been done in collaboration with Yvon Maday, Anthony Patera, Gabriel Turinici and Masayuki Yano.

**REFERENCES**

[1] Barrault, M.; Maday, Y.; Nguyen, N.C. and Patera, A.T., An empirical interpolation method, *C. R. Acad. Sci. Paris, Serie I.,* vol. 339, 667-672, 2004.

[2] Maday, Y. and Mula, O., GEIM: application of reduced basis techniques to data assimilation, *Analysis and Numerics of Partial Differential Equations*, vol. XIII, 221-236, 2013.

[3] Maday, Y., Mula, O. and Turinici, G., Convergence analysis of GEIM, *Submitted*.

[4] Maday, Y., Mula, O., Patera A.T. and Yano, M. The generalized EIM: stability theory on Banach spaces with an application to the Stokes equation, *Submitted*.