The four-dimensional variational data assimilation (4D-Var) has emerged as an important methodology, widely used in numerical weather prediction, oceanographic modeling, and climate forecasting. Classical unconstrained gradient-based algorithms often struggle with local minima, making their numerical performance highly sensitive to the initial guess. In this study, we exploit the separable structure of the 4D-Var problem to propose a practical variant of the alternating direction method of multipliers (ADMM), referred to as the linearized multi-block ADMM with regularization. Unlike classical first-order optimization methods that primarily focus on initial conditions, our approach derives the Euler-Lagrange equation for the entire dynamical system, enabling more comprehensive and effective utilization of observational data. When the initial condition is poorly chosen, the arg min operation steers the iteration towards the observational data, thereby reducing sensitivity to the initial guess. The quadratic subproblems further simplify the solution process, while the parallel structure enhances computational efficiency, especially when utilizing modern hardware. To validate our approach, we demonstrate its superior performance using the Lorenz system, even in the presence of noisy observational data. Furthermore, we showcase the effectiveness of the linearized multi-block ADMM with regularization in solving the 4D-Var problems for the viscous Burgers' equation, across various numerical schemes, including finite difference, finite element, and spectral methods. Finally, we illustrate the recovery of dynamics under noisy observational data in a 2D turbulence scenario, particularly focusing on vorticity concentration, highlighting the robustness of our algorithm in handling complex physical phenomena.

翻译：四维变分数据同化（4D-Var）已成为数值天气预报、海洋建模和气候预测等领域广泛采用的重要方法。经典的无约束梯度算法常陷入局部极小值困境，其数值性能对初始猜测高度敏感。本研究通过挖掘4D-Var问题的可分离结构，提出了乘子交替方向法（ADMM）的一种实用变体——正则化线性化多块ADMM。与主要关注初始条件的经典一阶优化方法不同，本方法推导了整个动力系统的欧拉-拉格朗日方程，实现了对观测数据更全面有效的利用。当初始条件选择不当时，arg min运算会将迭代导向观测数据，从而降低对初始猜测的敏感性。二次子问题进一步简化求解过程，而并行结构则提升了计算效率，尤其在利用现代硬件时表现显著。为验证本方法，我们以Lorenz系统为例展示了其优越性能，即使在含噪声观测数据条件下仍保持稳定。此外，我们通过粘性Burgers方程的4D-Var问题求解，在有限差分法、有限元法和谱方法等多种数值格式中，验证了正则化线性化多块ADMM的有效性。最后，我们在二维湍流场景中演示了含噪声观测数据下的动力学重构，特别聚焦涡量集中现象，凸显了本算法处理复杂物理现象的鲁棒性。