Nonlinear Assimilation via Score-based Sequential Langevin Sampling

Abstract

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.

Figures (19)

• Figure 1: An illustrative schematic of the state-space model. The latent states $(\mathbf{X}_{k})_{k\in{\mathbb{N}}}$ are unobservable and evolves according to known transition densities $(\rho_{k})_{k\in{\mathbb{N}}}$, which are specified by a dynamics model \ref{['eq:dynamic']}. The observations $(\mathbf{Y}_{k})_{k\in{\mathbb{N}}}$ are linked with states by a known likelihood $g_{k}$ characterized by the measurement model \ref{['eq:measurement']}.
• Figure 2: Schematic representation of score-based sequential Langevin sampling. (Left) The prediction step involves sampling from the approximated prediction distribution and estimating the prediction score. (Right) The posterior score is then obtained by combining the prediction score with the gradient of the log-likelihood. The update step samples from the posterior distribution using ALMC. Combining these two phases characterizes a recursion from the previous posterior to the current posterior.
• Figure 3: Schematic comparison of vanilla and annealed Langevin algorithms. (Top) The vanilla Langevin algorithm samples from the target posterior distribution, using the prediction distribution as initialization. (Bottom) The annealed Langevin algorithm employs a sequence of interpolations that smoothly transition from the prediction distribution to the target posterior distribution.
• Figure 4: The organization of theoretical results. The definition of constants are given in Theorem \ref{['theorem:section:convergence']}.
• Figure 5: Results of assimilation for Langevin diffusion with a double-well potential \ref{['eq:dw:dynamics']} with a linear measurement model \ref{['eq:dw:m1']}. The ensemble mean of SSLS, APF, and EnKF at each time steps are shown in the figure.

Theorems & Definitions (87)

• Remark 2.1: Inflation
• Remark 2.2: An alternative annealing strategy
• Remark 2.3: Computational cost reduction
• Definition 3.1: Total variation distance
• Definition 3.2: Chi-squared divergence
• Example 3.3: Gaussian distribution
• Example 3.4: Gaussian mixture
• Example 3.5: Gaussian convolution
• Proposition 3.6
• Remark 3.7: Condition number