Block Diagonally Dominant Positive Definite Sub-optimal Filters and Smoothers

We examine stochastic dynamical systems where the transition matrix, $\mathrm{\Phi }$, and the system noise, $\mathbf{\Gamma }\mathbf{Q}{\mathbf{\Gamma }}^{\mathbf{T}}$, covariance are nearly block diagonal. When ${\mathbf{H}}^{\mathbf{T}}{\mathbf{R}}^{-\mathbf{1}}\mathbf{H}$ is also nearly block diagonal, where $\mathbf{R}$ is the observation noise covariance and $\mathbf{H}$ 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.