A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics

Abstract

We present a parameterized-background data-weak (PBDW) formulation of the variational data assimilation (state estimation) problem for systems modeled by partial differential equations. The main contributions are a constrained optimization weak framework informed by the notion of experimentally observable spaces; a priori and a posteriori error estimates for the field and associated linear-functional outputs; weak greedy construction of prior (background) spaces associated with an underlying potentially high-dimensional parametric manifold; stability-informed choice of observation functionals and related sensor locations; and finally, output prediction from the optimality saddle in O(M[superscript 3) operations, where M is the number of experimental observations. We present results for a synthetic Helmholtz acoustics model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory. To conclude, we consider a physical raised-box acoustic resonator chamber: we integrate the PBDW methodology and a Robotic Observation Platform to achieve real-time in situ state estimation of the time-harmonic pressure field; we demonstrate the considerable improvement in prediction provided by the integration of a best-knowledge model and experimental observations; we extract, even from these results with real data, the numerical trends indicated by the theoretical convergence and stability analyses.