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 查看全文>>