Résumé (EN)

We present an error analysis and further numerical investigations of the Parameterized-Background Data-Weak (PBDW) formulation to variational Data Assimilation (state estimation), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. The PBDW algorithm is a state estimation method involving reduced models. It aims at approximating an unknown function utrue living in a high-dimensional Hilbert space from M measurement observations given in the form ym=ℓm(utrue),m=1,…,M, where ℓm are linear functionals. The method approximates utrue with u^=z^+η^. The \emph{background} z^ belongs to an N-dimensional linear space ZN built from reduced modelling of a parameterized mathematical model, and the \emph{update} η^ belongs to the space UM spanned by the Riesz representers of (ℓ1,…,ℓM). When the measurements are noisy {--- i.e., ym=ℓm(utrue)+ϵm with ϵm being a noise term --- } the classical PBDW formulation is not robust in the sense that, if N increases, the reconstruction accuracy degrades. In this paper, we propose to address this issue with an extension of the classical formulation, {which consists in} searching for the background z^ either on the whole ZN in the noise-free case, or on a well-chosen subset KN⊂ZN in presence of noise. The restriction to KN makes the reconstruction be nonlinear and is the key to make the algorithm significantly more robust against noise. We {further} present an \emph{a priori} error and stability analysis, and we illustrate the efficiency of the approach on several numerical examples.