Block Diagonally Dominant Positive Definite Sub-optimal Filters and Smoothers

We examine stochastic dynamical systems where the transition matrix, $\Phi$, and the system noise, $\bf{\Gamma}\bf{Q}\bf{\Gamma}^T$, covariance are nearly block diagonal. When $\bf{H}^T \bf{R}^{-1} \bf{H}$ is also nearly block diagonal, where $\bf{R}$ is the observation noise covariance and $\bf{H}$ is the observation matrix, our suboptimal filter/smoothers are always positive semi-definite, and have improved numerical properties. Applications for distributed dynamical systems with time dependent pixel imaging are discussed.

💡 Research Summary

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The paper addresses the problem of filtering and smoothing for stochastic dynamical systems whose transition matrix, system‑noise covariance, and measurement‑information matrix are nearly block diagonal (N.B.D.). The authors assume that the transition matrix Φ(i+1,i), the system‑noise term Γ Q Γᵀ, the initial covariance P(0|0), and the information matrix Jᵢ = Hᵀᵢ R⁻¹ᵢ Hᵢ can each be written as a block‑diagonal dominant matrix plus small off‑diagonal perturbations scaled by a formal coupling parameter ε.

Matrix expansion and stabilizing transformations
A N.B.D. matrix P(ε) is expanded as
 P(ε) = P⁽⁰⁾ + ε P⁽¹⁾ + ε² P⁽²⁾(ε),
where P⁽⁰⁾ is strictly block diagonal and the higher‑order terms contain the off‑diagonal couplings. Because truncating this series can destroy positive‑definiteness, the authors introduce three stabilizing transformations:

• T₁: T₁