Lagrangian Grid-based Estimation of Nonlinear Systems with Invertible Dynamics

This paper deals with the state estimation of non-linear and non-Gaussian systems with an emphasis on the numerical solution to the Bayesian recursive relations. In particular, this paper builds upon the Lagrangian grid-based filter (GbF) recently-developed for linear systems and extends it for systems with nonlinear dynamics that are invertible. The proposed nonlinear Lagrangian GbF reduces the computational complexity of the standard GbFs from quadratic to log-linear, while preserving all the strengths of the original GbF such as robustness, accuracy, and deterministic behaviour. The proposed filter is compared with the particle filter in several numerical studies using the publicly available MATLAB\textregistered\ implementation\footnote{https://github.com/pesslovany/Matlab-LagrangianPMF}.

💡 Research Summary

This paper addresses the challenging problem of Bayesian state estimation for discrete‑time nonlinear, non‑Gaussian systems by extending the Lagrangian grid‑based filter (LGbF) to models whose state transition function is analytically invertible. Traditional grid‑based filters (often called Eulerian GbFs) place a fixed or adaptively reshaped grid in the state space and evaluate the Chapman‑Kolmogorov prediction integral by summing over all pairs of grid points. Consequently, the prediction step has quadratic computational complexity O(N²), where N is the total number of grid points, which quickly becomes prohibitive as the state dimension grows.

The key contribution of the work is a single‑grid Lagrangian prediction scheme that reduces this complexity to O(N log N) while preserving the deterministic nature and robustness of grid‑based methods. The approach relies on three main ideas:

1. Invertible dynamics – If the transition map fₖ(x) is analytically invertible, a new prediction grid Ξₖ₊₁ can be constructed and then back‑propagated through the inverse map fₖ⁻¹. This creates a one‑to‑one correspondence between the current filtering grid Ξₖ and the future grid, eliminating the need to evaluate the transition density for every pair of points.

2. Moment‑based grid design – Using the current point‑mass density (PMD) the filter computes the posterior mean and covariance. These moments are fed into a local filter (e.g., EKF or UKF) to obtain predicted moments. A rectangular, equidistant grid is then placed around the predicted mean with a scaling factor κ that determines its extent. Because the grid is equidistant, fast convolution via the discrete Fourier transform (DFT) becomes possible.

3. Advection‑diffusion split – The deterministic part of the dynamics (advection) is handled by moving each grid point forward with fₖ. The stochastic part (diffusion) is treated by convolving the advected PMD with the process‑noise density p(wₖ). The convolution is performed in the frequency domain using the DFT, which yields the O(N log N) cost.

The algorithm proceeds as follows: (i) after a measurement update the filter computes the posterior mean ˆxₖ|ₖ and covariance Cₖ|ₖ from the PMD; (ii) a local filter predicts ˆxₖ₊₁|ₖ and Cₖ₊₁|ₖ, and a new equidistant grid Ξₖ₊₁ is generated around these moments; (iii) Ξₖ₊₁ is back‑propagated through fₖ⁻¹ to obtain a non‑uniform “back‑propagated” grid, on which the current PMD is interpolated; (iv) advection moves the interpolated PMD forward; (v) diffusion is applied via DFT‑based convolution.

The authors emphasize that this single‑grid scheme avoids the fragmentation of the conditional PDF that can occur in previous two‑grid approaches for highly nonlinear dynamics, and it also sidesteps the memory explosion caused by dense fine grids. The method is applicable to a broad class of invertible nonlinear models, including coordinated‑turn models with unknown turn rate, Dubins vehicle models, and chaotic maps such as the Hénon or Chirikov–Taylor maps.

Numerical experiments compare the proposed LGbF against a standard particle filter (PF) on several benchmark problems: (a) a 2‑D coordinated‑turn tracking scenario, (b) a 4‑D nonlinear tracking problem, and (c) a chaotic map estimation task. Performance metrics include root‑mean‑square error (RMSE) and runtime. Results show that the LGbF achieves estimation accuracy comparable to or better than the PF while requiring significantly less computation—especially in higher dimensions where the PF’s particle count must increase dramatically to maintain accuracy. Moreover, because the LGbF is deterministic, repeated runs with the same seed produce identical outputs, a desirable property for safety‑critical applications.

The paper also discusses limitations. The invertibility assumption excludes many practical systems with non‑invertible or multi‑valued dynamics. The choice of the scaling factor κ and handling of grid boundaries can affect robustness, and there is no adaptive mechanism to refine grid density where the posterior becomes highly concentrated. Future work suggested includes (i) extending the method to locally invertible or piecewise‑invertible dynamics, (ii) developing adaptive grid refinement strategies, and (iii) exploiting GPU‑accelerated FFTs to further reduce runtime for real‑time deployment.

In summary, the authors present a mathematically rigorous and computationally efficient extension of Lagrangian grid‑based filtering to nonlinear invertible systems. By leveraging the inverse dynamics to maintain a single, equidistant prediction grid and by exploiting FFT‑based convolution, the method bridges the gap between the high accuracy of global grid filters and the low computational burden of particle filters, making it a promising tool for high‑dimensional, nonlinear estimation problems in navigation, tracking, economics, and chaotic system analysis.