## MPI Colloquia Series: Prof. Dr. Albert Cohen, Université Pierre et Marie Curie, Paris

# MPI Colloquia Series: Prof. Dr. Albert Cohen: Reduced Model Estimation from Data Measurements

- Date: May 31, 2018
- Time: 16:00 - 18:00
- Speaker: Prof. Dr. Albert Cohen
- Université Pierre et Marie Curie, Paris; Laboratoire Jacques-Louis Lions
- Location: Lukasklause (Otto-von-Guericke-Zentrum), Schleinufer 1, 39104 Magdeburg
- Host: jointly organized by the Faculty of Mathematics at Otto von Guericke University and Max Planck Institute Magdeburg
- Contact: sek-csc@mpi-magdeburg.mpg.de

The Max Planck Institute Magdeburg and the Faculty of Mathematics at the Otto von Guericke University Magdeburg invite you to its series of colloquia. Top-class scientists, invited by the Max Planck Institute Magdeburg, give a survey of their research work. Everybody who is interested, is invited to attend.

Afterwards, the Otto von Guericke University invites you to the inaugural lecture of Prof. Dr. Alexandra Carpentier.

#### Reduced Model Estimation from Data Measurements

#### Abstract

One typical scenario in data assimilation is the following: one observes m linear measurements of a function u which is solution to a partial differential equation where certain parameters are unknown. The measurement functionals are picked

from a certain dictionary D, for example when placing sensors at m chosen locations. The state estimation problem then consists in recovering u from these measurements.

One possible approach to this problem exploits the fact that the family of solution for all potential parameter values is well approximated by linear spaces of moderate dimension n. Such spaces are typically obtained by reduced model techniques, such as reduced bases, proper orthogonal polynomial expansions in the parametric variable.

The numerical method achieves a reconstruction which has the accuracy of the best approximation from the n-dimensional space to the unknown solution u, up to a multiplicative constant which takes the form of an inverse inf-sup constant between the approximation space and the spacegenerated by the Riesz representers of the linear forms giving rise to the measurements.

One issue discussed in this talk is how to select the measurement functionals within D to maintain this constant of reasonable size, with m as small as possible. In particular, we present a greedy algorithm allowing for a stepwise selection process of reasonable computational cost, and we analyze its properties.