  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.
