Accelerating inverse inference of ensemble Kalman filter via reduced-order model trained using adaptive sparse observations

Abstract

The implementation of data assimilation often struggles with low computational efficiency due to the complexity of high-fidelity numerical modeling and the large size of the observation vectors. To address this challenge, it is necessary to employ reduced-order strategies for both the full-order dynamical model and observation operator. Proper orthogonal decomposition (POD) based reduced-order model (ROM) and discrete empirical interpolation method (DEIM) based sparse observation identification are often considered as primary solutions. However, their effective combination requires the determination of high-quality POD reduced-order basis vectors. The current research focuses on incorporating the ROM and DEIM into a Kalman filter framework, resulting in a reduced-order Kalman filter with sparse observations. The key aspect is identifying the optimal POD basis, which can be achieved iteratively by introducing a relative root-mean-square-error (RMSE) for a near-optimal exploration of both the desired low-order POD basis and observation locations. The singular value decomposition (SVD) of the snapshots is performed in a low-dimensional subspace, leading to significant computational savings. This improved POD basis not only enhances the reconstruction of the ROM, but also helps build the DEIM observation matrix, resulting in simultaneous dimensionality reduction of the state and observation space. The effectiveness of this algorithm is demonstrated through the retrieval of initial conditions for one-dimensional tsunami wave equations with real data scenarios. Experiments are conducted for both smooth and nonsmoothed initial conditions. The results show that when the observation data is available at 3 and 6 spatial observation locations, our method only requires fewer vectors with the first 45 and 90 dominant modes derived using the RMSE, compared to as many as 210 and 252 derived from the relative energy criterion. Our method enables a low-modal ROM to effectively reconstruct the wave height field, leading to considerable prediction capability for potential long-wave dynamics with comparable accuracy. The CPU time savings achieved through our method is at least one to two orders of magnitude compared to the CPU time required by the full-order model.