We study different approaches to implementing sparse-in-time observations into the the Azouani-Olson-Titi data assimilation algorithm. We propose a new method which introduces a "data assimilation window" separate from the observational time interval. We show that by making this window as small as possible, we can drastically increase the strength of the nudging parameter without losing stability. Previous methods used old data to nudge the solution until a new observation was made. In contrast, our method stops nudging the system almost immediately after an observation is made, allowing the system relax to the correct physics. We show that this leads to an order-of-magnitude improvement in the time to convergence in our 3D Navier-Stokes simulations. Moreover, our simulations indicate that our approach converges at nearly the same rate as the idealized method of direct replacement of low Fourier modes proposed by Hayden, Olson, and Titi (HOT). However, our approach can be readily adapted to non-idealized settings, such as finite element methods, finite difference methods, etc., since there is no need to access Fourier modes as our method works for general interpolants. It is in this sense that we think of our approach as ``second best;'' that is, the ``best'' method would be the direct replacement of Fourier modes as in HOT, but this idealized approach is typically not feasible in physically realistic settings. While our method has a convergence rate that is slightly sub-optimal compared to the idealized method, it is directly compatible with real-world applications. Moreover, we prove analytically that these new algorithms are globally well-posed, and converge to the true solution exponentially fast in time. In addition, we provide the first 3D computational validation of HOT algorithm.

翻译：本研究探讨了在Azouani-Olson-Titi数据同化算法中实现稀疏时间观测的不同策略。我们提出了一种新方法，通过引入独立于观测时间间隔的"数据同化窗口"来实现。研究表明，将该窗口缩至最小可大幅增强 nudging 参数强度而不损失稳定性。传统方法利用历史数据持续 nudging 解直到新观测数据产生，而我们的方法在获得观测数据后几乎立即停止 nudging，使系统自然松弛至正确物理状态。三维纳维-斯托克斯模拟显示，该方法可将收敛时间提升一个数量级。此外，数值实验表明，本方法收敛速率接近Hayden、Olson与Titi（HOT）提出的低傅里叶模态直接替换理想化方法。由于本算法适用于通用插值函数而无需访问傅里叶模态，因此可便捷适配有限元法、有限差分法等非理想化场景。正是基于此，我们将该方法视为"次优"方案——即最优方案应为HOT的傅里叶模态直接替换法，但该理想化方法在物理真实场景中通常不可行。尽管本方法收敛速率略逊于理想化方案，但其可直接兼容实际应用。我们通过解析证明，这些新算法具有全局适定性，且解随时间呈指数级快速收敛于真实解。此外，我们首次提供了HOT算法的三维计算验证。