Domain decomposition methods for the state estimation of parabolic PDEs in 2D rectangular domains: well-posedness and convergence

Abstract

This paper provides the necessary system-theoretic conditions for the establishment of the well-posedness and convergence of filters of parabolic PDEs in 2D rectangular domains. Motivated by computational savings considerations, both a Domain Decomposition and a hybrid Domain Decomposition (DD) filters for such PDEs are presented. By decomposing the spatial domain into an inner subdomain that includes the sensing device(s) and an outer subdomain that does not include any sensor(s), the resulting state estimators can employ different numerical grids to compute the associated filter gains. The ultimate goal is to have variable spatial resolution of the filters that is dependent on the sensor location. Different from multi-grid methods, the proposed DD filters provide additional flexibility on the numerical implementation of the proposed filters and the eventual code parallelization.