Abstract
We present an Adaptive Parametrized-Background Data-Weak (APBDW) approach to the steady-state variational data assimilation (state estimation) problem for systems modeled by partial differential equations. The variational formulation is based on the Tikhonov regularization of the PBDW formulation [Y. Maday, A.T. Patera, J.D. Penn and M. Yano, Int. J. Numer. Meth. Eng. 102 (2015) 933–965] for pointwise noisy measurements. We propose an adaptive procedure based on a posteriori estimates of the L 2 state-estimation error to improve performance. We also present a priori estimates for the L 2 state-estimation error that motivate the approach and guide the adaptive procedure. We provide numerical experiments for a synthetic acoustic problem to illustrate the different elements of the methodology, and we consider an experimental thermal patch configuration to demonstrate the applicability of our approach to real physical systems.
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