Hybrid domain decomposition filters for advection–diffusion PDEs with mobile sensors

Abstract

This paper examines one aspect of computational control, namely the reduction of computational load for the real-time estimation of distributed parameter systems over large spatial domains. One way to improve the performance of the state estimator is to employ a small number of mobile devices that are free to move within the spatial domain while collecting process information. Another way is to use numerical methods to reduce the computational load as dictated by the discretization index of the finite dimensional representation of the filter. This is done via the use of domain decomposition methods and the domain decomposition filter is presented here for the first time along with its well-posedness. In the subdomain surrounding the sensor a refined grid is used as it improves the spatial resolution, and in the subdomain excluding the vicinity of the sensing device, a naïve observer with coarse grid is used which results in a significant reduction of the computational cost. However, due to the transmission conditions that couple the information between the inner subdomain surrounding the sensor and the outer subdomain excluding the sensor, the filters in the two subdomains implement an information exchange protocol rendering them consensus filters. The two approaches are combined thus resulting in a state estimation scheme where the mobile sensor residing in the inner subdomain carries its own refined grid. Not only is the sensor mobile, but also the refined grid surrounding the sensor is moving along with the sensor. A further reduction in computational costs is presented in the form of a hybrid DD filter wherein the filter kernel is rendered sparse thus ensuring that no residual measurement signals enter as disturbance signals in the naïve observer defined over the outer subdomain. Extensive simulation studies of a 2D PDE with realistic parameters are included to demonstrate the proposed hybrid DD filter.