Data assimilation for linear parabolic equations: minimax projection method

Sergiy Zhuk (1) , Frank Jason (2) , Isabelle Herlin (3) , Robert Shorten (1, 4)

1 IBM Research - Ireland
2 Universiteit Utrecht / Utrecht University [Utrecht]
3 Clime - Coupling environmental data and simulation models for software integration
4 UCD - University College Dublin [Dublin]

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 two-dimensional fluid.

Mots clés: projection; structure preservation; PDEs; DAEs; state estimation; minimax

Domaines: Modélisation et simulation

Sergiy Zhuk, Frank Jason, Isabelle Herlin, Robert Shorten. Data assimilation for linear parabolic equations: minimax projection method. SIAM Journal on Scientific Computing, 2015, 37 (3), pp.A1174-A1196. ⟨10.1137/13094709X⟩. ⟨hal-01174081⟩