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We provide a number of extensions and further interpretations of theParameterized-Background Data-Weak (PBDW) formulation, a real-time and in-situData Assimilation (DA) framework for physical systems modeled by parametrizedPartial Differential Equations (PDEs), proposed in [Y Maday, AT Patera, JDPenn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. Given $M$ noisymeasurements 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 parameterizedmathematical 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 theadaptive formulation proposed in [T Taddei, M2AN, 51(5), 1827-1858] to generallinear observation functionals, to effectively deal with noisy observations;second, we consider an user-defined choice of the update space, to improveconvergence with respect to the number of measurements; third, we propose an\emph{a priori} error analysis for general linear functionals in the presenceof noise, to identify the different sources of state estimation error andultimately motivate the adaptive procedure. We present results for twosynthetic model problems in Acoustics, to illustrate the elements of themethodology and to prove its effectiveness. We further present results for asynthetic problem in Fluid Mechanics to demonstrate the applicability of theapproach to vector-valued fields.

Adaptive PBDW approach to state estimation: noisy observations; user-defined update spaces