Domain decomposition restricted Kalman filters for parabolic PDEs with moving sensors

This paper uses mobile sensors to consider the state estimation of systems governed by parabolic partial differential equations. These equations are often used to model the dispersion of a species’ concentration in the environment, which is generated by moving sources that release contaminants. Using an abstract framework theory, the partial differential equation is written as an evolution equation in a functional space forced by an exosystem that describes the dynamics of a moving platform carrying the contaminant release with the mass release rate capturing the temporal signal of the disturbance. Using an optimal filter to reconstruct the concentration state provides computational challenges as one must solve a large-scale covariance equation (differential Riccati equation) in real-time. To address this challenge, a domain decomposition filter considers the Kalman filter in a subdomain with a naive observer implemented in the remainder of the spatial domain. To further the computational savings, a restricted Kalman filter is implemented in the inner domain, thus providing the filter kernel’s sought-after sparsity. Extensive simulation studies are presented over a large spatial domain and demonstrate the implementability of the real-time restricted Kalman filter for partial differential equations with a performance-based guided moving sensor and an unknown moving source.