This paper introduces score-based sequential Langevin sampling (SSLS), a novel approach to nonlinear data assimilation within a recursive Bayesian filtering framework. The proposed method decomposes the assimilation process into alternating prediction and update steps, using dynamic models for state prediction and incorporating observational data via score-based Langevin Monte Carlo during the updates. To overcome inherent challenges in highly non-log-concave posterior sampling, we integrate an annealing strategy into the update mechanism. Theoretically, we establish convergence guarantees for SSLS in total variation (TV) distance, yielding concrete insights into the algorithm's error behavior with respect to key hyperparameters. Crucially, our derived error bounds demonstrate the asymptotic stability of SSLS, guaranteeing that local posterior sampling errors do not accumulate indefinitely over time. Extensive numerical experiments across challenging scenarios, including high-dimensional systems, strong nonlinearity, and sparse observations, highlight the robust performance of the proposed method. Furthermore, SSLS effectively quantifies the uncertainty associated with state estimates, rendering it particularly valuable for reliable error calibration.

翻译：本文提出了基于分数的顺序朗之万采样（SSLS），一种在递归贝叶斯滤波框架内进行非线性数据同化的新方法。该方法将同化过程分解为交替的预测和更新步骤，使用动态模型进行状态预测，并在更新过程中通过基于分数的朗之万蒙特卡洛方法融入观测数据。为克服高度非对数凹后验采样中的固有挑战，我们在更新机制中集成了退火策略。理论上，我们建立了SSLS在总变差（TV）距离下的收敛保证，从而对算法关于关键超参数的误差行为提供了具体见解。关键的是，我们推导的误差界证明了SSLS的渐近稳定性，确保局部后验采样误差不会随时间无限累积。在包括高维系统、强非线性和稀疏观测等具有挑战性场景中的大量数值实验，突出了所提方法的鲁棒性能。此外，SSLS有效量化了与状态估计相关的不确定性，使其在可靠误差校准方面尤为有价值。