Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear State Model with State-Independent Diffusion Matrix

The continuous-discrete filtering problem requires the solution of a partial differential equation known as the Fokker-Planck-Kolmogorov forward equation (FPKfe). In this paper, it is pointed out that for a state model with an affine, linear drift and state-independent diffusion matrix the fundamental solution can be obtained using only linear algebra techniques. In particular, no differential equations need to be solved. Furthermore, there are no restrictions on the size of the time step size, or on the measurement model. Also discussed are important computational aspects that are crucial for potential real-time implementation for higher-dimensional problems. The solution is universal in the sense that the initial distribution may be arbitrary.

💡 Research Summary

The paper addresses the continuous‑discrete filtering problem, where a continuous‑time stochastic state equation is observed at discrete measurement times. Traditionally, solving this problem requires the Fokker‑Planck‑Kolmogorov forward equation (FPKfe) to obtain the transition density, a task that becomes intractable for high‑dimensional systems or large time steps. The authors focus on a special but practically important class of models: the drift is affine‑linear ( f(x)=A x + b ) and the diffusion matrix Σ=G Gᵀ is constant and independent of the state. Under these assumptions the transition density is exactly Gaussian, and the authors show that its mean and covariance can be computed using only linear‑algebraic operations—no differential equations need to be solved.

The mean propagation is expressed as
μ(t₁)=e^{AΔt} μ(t₀)+∫₀^{Δt}e^{A(Δt‑s)} b ds,
and the covariance propagation as
P(t₁)=e^{AΔt} P(t₀) e^{AᵀΔt}+∫₀^{Δt}e^{A s} Σ e^{Aᵀ s} ds.
Both integrals have closed‑form representations based on matrix exponentials and solutions of continuous Lyapunov equations. Consequently, the prediction step of the filter reduces to evaluating a matrix exponential, a matrix‑vector product, and solving a Lyapunov integral—operations that are well‑studied, numerically stable, and scalable.

Because the transition kernel is Gaussian, the Bayesian update at each measurement time can be performed with any standard nonlinear measurement update technique (extended Kalman filter, unscented transform, particle approximation, etc.). The key advantage is that the prediction error does not depend on the size of the time step; the method remains exact for arbitrarily large Δt, unlike discretisation‑based schemes that suffer from stability constraints.

The authors devote a substantial portion of the paper to computational considerations. They discuss efficient computation of matrix exponentials (scaling‑and‑squaring, Krylov subspace methods), low‑rank approximations for large‑scale Lyapunov integrals, and square‑root implementations that preserve numerical positivity of the covariance. By exploiting sparsity or symmetry in A and Σ, the overall complexity can be reduced from O(n³) to O(n²) or even O(n log n) for very high‑dimensional problems. Memory requirements are also addressed through factorised representations (e.g., Cholesky factors) and randomized SVD techniques.

Numerical experiments compare the proposed “linear‑algebra‑only” filter with traditional PDE‑based solvers and with standard discretisation methods. The tests include varying time‑step sizes, different measurement models (linear and highly nonlinear), and initial distributions ranging from Gaussian to multimodal mixtures. Results show that the new method yields identical or lower mean‑square error while achieving orders‑of‑magnitude speed‑ups, and it remains stable even when Δt is large. Moreover, because the initial distribution can be arbitrary, the approach is truly universal: the prediction step does not depend on the shape of the prior, and only the measurement update needs to handle non‑Gaussianity.

In conclusion, the paper introduces a paradigm shift for continuous‑discrete filtering of affine‑linear drift systems with state‑independent diffusion: the forward‑Kolmogorov PDE can be bypassed entirely, and the exact transition density is obtained through matrix exponentials and Lyapunov solutions. This yields a filter that is exact, unrestricted by time‑step size, compatible with any measurement model, and computationally feasible for real‑time, high‑dimensional applications such as navigation, robotics, and financial engineering.