A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics
Author(s)Maday, Yvon; Patera, Anthony T.; Yano, Masayuki; Penn, James Douglass
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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.
DepartmentMassachusetts Institute of Technology. Department of Mechanical Engineering
International Journal for Numerical Methods in Engineering
Maday, Yvon, Anthony T. Patera, James D. Penn, and Masayuki Yano. “A Parameterized-Background Data-Weak Approach to Variational Data Assimilation: Formulation, Analysis, and Application to Acoustics.” Int. J. Numer. Meth. Engng 102, no. 5 (August 15, 2014): 933–965.