In this talk we will discuss recent advances in data assimilation using  finite element methods. Two  cases will be considered: (1) integration of geometry data and (2) integration of measurements of the solution. We will consider integration of geometry data that is given implicitly, for instance in the form of the contour line of a distance function (level set). Recent techniques that allow for computations on meshes that are not fitted to the geometry will be presented and their robustness and accuracy discussed. Then, in the second case, we will consider the case where instead of boundary or initial data we have access to some measurements in an interior subdomain. The notion of ill-posed problems and standard Tikhonov regularisation techniques will be briefly discussed. Then we will propose a new framework for data assimilation based on weakly consistent regularisation of the discrete equations. We show how in this framework the stability estimates available for inverse problems may be used to derive error estimates accounting simultaneously for discretisation errors and perturbations in data. Some computational examples will illustrate the theory in both cases.

Erik Burman defended his thesis at Chalmers University of Technology in 1998. He spent two years as a post doc at Ecole Polytechnique, Palaiseau working with Alexandre Ern and Vincent Giovangigli. Then he spent two years as a post doc at Ecole Polytechnique Fédérale de Lausanne (EPFL), working in the group of Jacques Rappaz. After that he got a permanent position in the group of Alfio Quarteroni (EPFL) where he stayed until 2007. From 2007 to 2012 he was Professor of Mathematics at the University of Sussex and also served as Head of Department. Finally from 2013 he has held the Chair of Computational Mathematics at University College London.
His research expertise lies in the fields of stabilised finite element methods for fluid and continuum mechanics, unfitted finite element methods for multiphysics interface problems and weakly consistent regularization methods for inverse problems.