We provide a number of extensions and further interpretations of the Parameterized-Background Data-Weak (PBDW) formulation, a real-time and in-situ Data Assimilation (DA) framework for physical systems modeled by parametrized Partial Differential Equations (PDEs), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. Given $M$ noisy measurements of the state, PBDW seeks an approximation of the form $u^{star} = z^{star} + eta^{star}$, where the emph{background} $z^{star}$ belongs to a $N$-dimensional emph{background space} informed by a parameterized mathematical model, and the emph{update} $eta^{star}$ belongs to a $M$-dimensional emph{update space} informed by the experimental observations. The contributions of the present work are threefold: first, we extend the adaptive formulation proposed in [T Taddei, M2AN, 51(5), 1827-1858] to general linear observation functionals, to effectively deal with noisy observations; second, we consider an user-defined choice of the update space, to improve convergence with respect to the number of measurements; third, we propose an emph{a priori} error analysis for general linear functionals in the presence of noise, to identify the different sources of state estimation error and ultimately motivate the adaptive procedure. We present results for two synthetic model problems in Acoustics, to illustrate the elements of the methodology and to prove its effectiveness. We further present results for a synthetic problem in Fluid Mechanics to demonstrate the applicability of the approach to vector-valued fields.
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