#### Conference abstracts

Session A6 - Mathematical Foundations of Data Assimilation and Inverse Problems

July 10, 15:00 ~ 15:30

## Reduced Basis' Acquisition by a Learning Process for Rapid On-line Approximation of Solution to PDE's : laminar flow past a backstep

### Yvon Maday

### Université Pierre et Marie Curie, Laboratoire J.-L. Lions, France - maday@ann.jussieu.fr

Reduced Basis Methods for the approximation to parameter dependent Partial Differential Equations are now well developed and start to be used in industrial framework. The classical implementation of the Reduced Basis Method goes through two stages : in the the first one, offline and time consuming, from standard approximation methods a reduced basis is constructed, then in a second stage, online and very cheap, a small problem, of the size of the reduced basis, is solved.

The offline stage is a learning one from which the online stage can proceed efficiently. In this presentation we propose to complement the offline stage with some statistical learning ingredients in order to build more knowledge in the process so as to tackle either different classes of problems or even speed more the online approximation. The method is presented on a simple flow problem governed by the Navier Stokes equation and illustrated on the test case of a flow pas a backward step.

Joint work with P. Gallinari and O. Schwander (Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7606, LIP6),, Y. Maday (Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universtaire de France, Division of Applied Mathematics, Brown University),, M. Sangnier (Sorbonne Universités, UPMC Univ Paris 06, F-75005, Paris, France LSTA ), and T. Taddei (Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7598, Laboratoire Jacques-Louis Lions)..