When parameter estimation is solved in a high-dimensional space, the dimensionality reduction strategy becomes the primary consideration for alleviating the tremendous computational cost. In the present study, the discrete empirical interpolation method (DEIM) is explored to retrieve the initial condition (IC) by combining the polynomial chaos (PC) based ensemble Kalman filter (i.e. PC-EnKF), where a non-intrusive PC expansion is considered as a surrogate model in place of the forward model in the prediction step of the ensemble Kalman filter, resulting in fewer forward model integrations but with a comparable accuracy as Monte Carlo-based approaches. The DEIM acts as a hyper-reduction tool to provide the low-dimensional input for the high-dimensional initial field, which can be reconstructed using the information on the sparse interpolation grid points that is adaptively obtained through PC-EnKF data assimilation method. Thus an innovative framework to reconstruct the IC is developed. The detailed procedure at each assimilation iteration includes: the determination of the spatial interpolation points, the estimation of the initial values on the interpolation locations using the optimal observations, and the reconstruction of IC in the full space. The current study uses the reconstruction field of initial conditions of the Navier-Stokes equations as an example to illustrate the efficacy of our method. The experimental results demonstrate the proposed algorithm achieves a satisfactory reconstruction for the initial field. The proposed method helps to extend the applicable area of DEIM in solving inverse problems.

当在高维空间中解决参数估计时，降维策略成为减轻巨大计算成本的主要考虑因素。在本研究中，通过结合基于多项式混沌（PC）的集成卡尔曼滤波器（即PC-EnKF），探索了离散经验内插法（DEIM）来检索初始条件（IC），其中非侵入式PC扩展为在集成卡尔曼滤波器的预测步骤中，该模型被视为替代前向模型的替代模型，从而导致前向模型集成较少，但准确性与基于Monte Carlo的方法相当。DEIM用作超简化工具，可为高维初始场提供低维输入，可以使用通过PC-EnKF数据同化方法自适应获得的稀疏插值网格点上的信息进行重构。因此，开发了用于重构IC的创新框架。每个同化迭代的详细过程包括：确定空间插值点，使用最佳观测值估计插值位置上的初始值以及在整个空间中重建IC。当前的研究以Navier-Stokes方程初始条件的重构场为例来说明我们方法的有效性。实验结果表明，该算法对初始场实现了令人满意的重建。