Retrieval of initial condition for Burgers' equation using reduced-order EnKF via POD-based sparse observations

Abstract

<p style='text-indent:20px;'>The data assimilation (DA) is a popular method to solve uncertainty quantification problem which has attracted more and more attention in disaster assessment and climate change research. However, the DA usually faces the computational problem associated with "the curse of dimensionality'' when the high-dimensional inverse problem is involved. For this, the usual strategy is to combine the dimensionality reduction method with the DA to mitigate the computational demand in practical application. In this paper, we develop a reduced-order model (ROM)-based EnKF used for retrieving the initial condition, where the prediction model and the observation model are all POD-dependent. In addition to the reduced-order model (ROM) being prediction model, the idea of compressed sensing motivates the acquisition of the observation model that allows an efficient combination of the EnKF with QD algorithm to obtain the optimal sparse observation locations. In this way, computational acceleration is gained in retrieving the initial condition. The effectiveness of this algorithm is demonstrated through retrieving the initial condition of an one-dimensional (1-D) Burgers' equation. Experimental results show that a satisfactory retrieval result is obtained using the current ROM-based EnKF with the computational (CPU) time, <inline-formula><tex-math id="M1">\begin{document}$ 42.64\rm{s} $\end{document}</tex-math></inline-formula>, nearly 25 times faster, and the error result <inline-formula><tex-math id="M2">\begin{document}$ 7.47\times 10^{-3} $\end{document}</tex-math></inline-formula>, two order of magnitude <inline-formula><tex-math id="M3">\begin{document}$ O(10^2) $\end{document}</tex-math></inline-formula> more accurate, than the implementation of the traditional full-order model (FOM)-based EnKF where the computational instability is observed at each iteration though in the same condition as set in the ROM-based EnKF, leading to a poor result (the CPU time <inline-formula><tex-math id="M4">\begin{document}$ 1057.97\rm{s} $\end{document}</tex-math></inline-formula>, and the error accuracy <inline-formula><tex-math id="M5">\begin{document}$ 4.06\times 10^{-1}) $\end{document}</tex-math></inline-formula>. The present study extends the application of ROM to efficiently dealing with ill-posed problem as a regularization strategy through replacing the FOM with the ROM such that the low-order truncation of the POD basis leads to the ROM with as few degrees of freedom as possible, which can efficiently inhibit the higher frequency errors, and hence achieve stability of the proposed approach.</p>