State Estimation for Linear Parabolic Equations: Minimax Projection Method

In this paper we propose a state estimation approach for linear parabolic Partial Differential Equations (PDE) with uncertain parameters. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means of a Differential-Algebraic Equation (DAE). The original estimation problem for the PDE is then recast as a state estimation problem for the constructed DAE using a linear continuous minimax filter. We develop a discretization method of order p to go from continuous to discrete time to realize a numerical simulation. The time discretisation method preserves quadratic invariants, thus ensuring that the state estimation error, which is proved to hold for the continuous filter, is preserved for the discrete filter during the numerical simulation. To conclude we demonstrate the efficacy of the proposed method by applying it to the tracking of a discharged pollutant slick in a 2D fluid.