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We present an error analysis and further numerical investigations of theParameterized-Background Data-Weak (PBDW) formulation to variational DataAssimilation (state estimation), proposed in [Y Maday, AT Patera, JD Penn, MYano, Int J Numer Meth Eng, 102(5), 933-965]. The PBDW algorithm is a stateestimation method involving reduced models. It aims at approximating an unknownfunction $u^{\rm true}$ living in a high-dimensional Hilbert space from $M$measurement observations given in the form $y_m = \ell_m(u^{\rm true}),\,m=1,\dots,M$, where $\ell_m$ are linear functionals. The method approximates$u^{\rm true}$ with $\hat{u} = \hat{z} + \hat{\eta}$. The \emph{background}$\hat{z}$ belongs to an $N$-dimensional linear space $\mathcal{Z}_N$ built fromreduced modelling of a parameterized mathematical model, and the \emph{update}$\hat{\eta}$ belongs to the space $\mathcal{U}_M$ spanned by the Rieszrepresenters of $(\ell_1,\dots, \ell_M)$. When the measurements are noisy {---i.e., $y_m = \ell_m(u^{\rm true})+\epsilon_m$ with $\epsilon_m$ being a noiseterm --- } the classical PBDW formulation is not robust in the sense that, if$N$ increases, the reconstruction accuracy degrades. In this paper, we proposeto address this issue with an extension of the classical formulation, {whichconsists in} searching for the background $\hat{z}$ either on the whole$\mathcal{Z}_N$ in the noise-free case, or on a well-chosen subset$\mathcal{K}_N \subset \mathcal{Z}_N$ in presence of noise. The restriction to$\mathcal{K}_N$ makes the reconstruction be nonlinear and is the key to makethe algorithm significantly more robust against noise. We {further} present an\emph{a priori} error and stability analysis, and we illustrate the efficiencyof the approach on several numerical examples.

PBDW method for state estimation: error analysis for noisy data and nonlinear formulation