Data Assimilation for Linear Parabolic Equations: Minimax Projection Method

Abstract

Abstract. In this paper we propose a state estimation method for linear parabolic partial differential equations (PDE) that accounts for errors in the model, truncation, and observations. 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 construct a numerical time integrator that preserves the monotonic decay of a nonstationary Lyapunov function along the solution. 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.

A Macroscopic Traffic Data-Assimilation Framework Based on the Fourier–Galerkin Method and Minimax Estimation

“…Finally, we set Y i = U obs i + η i , where η i is additive noise. The Fourier-Galerkin reduced model, (12), is solved with RK4, using the same computational time step as the Godunov scheme aforementioned. This is done in order to ensure coherence between the model (Fourier-Galerkin) and the "truth" (Godunov).…”

“…In our case, term e m addresses the projection error introduced by the Galerkin method. In other words, e m models the impact of higher order Fourier modes onto those described by the Galerkin model (12).…”

“…Now, we use the following algorithm to estimate the state of the nonlinear (12). Assume that X (1) (t) represents an estimate of the projection coefficients, X (obtained, for instance, solving the conservation law (1) for given initial and boundary conditions).…”

“…We refer the reader to [8]- [11] for basic information on the minimax framework. Discussion on robust projection methods by means of the minimax state estimation approach can be found in [12] and [13].…”

Source estimation for wave equations with uncertain parameters

“…In fact, u "lives" in the physical space and represents the approximation of the values of the displacement field u i over the set of chosen GLL quadrature points. It was recognized in [11], [12] that the approximation error of the spectral element method, as well as any Galerkin projection method, may be modelled as an uncertain but bounded input g in the following form:…”