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

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 framework for physical systems modeled by parameterized PDEs, proposed in [Y. Maday et al., Int. J. Numer. Methods Engrg., 102 (2015), pp. 933{965]. Given M noisy measurements of the state, PBDW seeks an approximation of the form u*=z+\eta, where the background z* belongs to an N-dimensional background space informed by a parameterized mathematical model and the update belongs to an M-dimensional 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, ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 1827{1858] to general linear observation functionals to effectively deal with noisy observations; second, we consider a user-defined choice of the update space to improve convergence with respect to the number of measurements. Third, we propose an a priori error analysis for general linear functionals in the presence of noise to identify the di erent 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.