Adjoint-based optimization with quantized local reduced-order models for spatiotemporally chaotic systems

Abstract

We introduce a computationally efficient and accurate reduced order modelling approach for the optimization of spatiotemporally chaotic systems. The proposed method combines quantized local reduced order modelling with adjoint-based optimization. We employ the methodology in a variational data assimilation problem for the chaotic Kuramoto-Sivashinsky equation and show that it successfully reconstructs the full trajectory for up to 0.25 Lyapunov times given full state measurements at the final time. The proposed algorithm provides 3.5 times speed-up when compared to the full order model. The proposed method opens up new possibilities for the reduced order modelling of spatiotemporally chaotic systems.

Figures (3)

• Figure 1: Overview of adjoint-based optimization with quantized local reduced-order models (ql-ROMs): (1) data collection; (2) phase-space quantization and local basis/model construction; (3) ql-ROM adjoint integrated backward in time with jump/coordinate-transformation at cluster switches; (4) variational data assimilation using the ql-ROM direct--adjoint loop to update the initial condition.
• Figure 2: Accuracy of the adjoint sensitivity computed using the quantized local reduced model in comparison to the full order model. (a) the $\ell_2$-norm of the error between these two gradient vectors, denoted by $\epsilon$, and (b) the cosine similarity between these two gradient vectors, denoted by $\gamma$. The error bars indicate mean and $\pm 1$ standard deviation.
• Figure 3: Variational data assimilation on the Kuramoto-Sivashinsky equation using the adjoint of the quantized local reduced order model. The method reconstructs the true trajectory from measurements at the final time $T = 0.25$ LT following the convergence of the optimization objective, $\mathcal{J}$.