Hybrid domain decomposition filters for parabolic spatially distributed processes

This paper proposes a domain decomposition scheme as a means for low order modeling of estimators for systems governed by partial differential equations. This entails the decomposition of the state estimator defined over the entire spatial domain into two separate estimators defined over two non-overlapping subdomains and coupled through the transmission conditions at their boundary. Each estimator is a copy of the original process defined over its subdomain and having an output injection term which is weighted by a filter kernel. Each of the filter kernels is related to the filter kernel of the estimator of the process defined over the entire spatial domain. The sensor providing process information is in the interior of the inner subdomain. Using a hybrid version of the domain decomposition method wherein the kernel of the outer domain is nullified, a significant computational savings can be obtained and is viewed as an alternate for low order approximation of estimators for PDEs. This decoupling facilitates a multi-grid implementation whereby the inner subdomain uses a refined grid as a means of increasing spatial resolution and the outer subdomain where the value of information is minimal, uses a coarse grid. The well-posedness of the proposed hybrid domain decomposition estimator is established and numerical studies of an advection-diffusion PDE over a rectangular domain are presented to provide an appreciation of the domain decomposition methods as a means of low order modeling of estimators for PDEs.