POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush–Kuhn–Tucker conditions acc...

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Bibliographic Details
Published in:Journal of computational physics 2015-08, Vol.295, p.569-595
Main Authors: Ştefănescu, R., Sandu, A., Navon, I.M.
Format: Article
Language:English
Subjects:
APPROXIMATIONS
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Computation
Data assimilation
Discrete empirical interpolation method (DEIM)
FINITE DIFFERENCE METHOD
Finite difference methods
FLOW MODELS
FOUR-DIMENSIONAL CALCULATIONS
Galerkin methods
Geophysics
INTERPOLATION
Inverse problems
KINETIC EQUATIONS
Mathematical models
NONLINEAR PROBLEMS
Projection
Proper orthogonal decomposition
Reduced order
Reduced-order models (ROMs)
Shallow water equations
SIMULATION
Strategy
TWO-DIMENSIONAL CALCULATIONS
VARIATIONAL METHODS
WATER
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Summary:This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush–Kuhn–Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for proper orthogonal decomposition (POD) ROM data assimilation using both Galerkin and Petrov–Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov–Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2015.04.030