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This work studies reduced order modeling (ROM) approaches to speed up thesolution of variational data assimilation problems with large scale nonlineardynamical models. It is shown that a key requirement for a successful reducedorder solution is that reduced order Karush-Kuhn-Tucker conditions accuratelyrepresent their full order counterparts. In particular, accurate reduced orderapproximations are needed for the forward and adjoint dynamical models, as wellas for the reduced gradient. New strategies to construct reduced order basedare developed for Proper Orthogonal Decomposition (POD) ROM data assimilationusing both Galerkin and Petrov-Galerkin projections. For the first time POD,tensorial POD, and discrete empirical interpolation method (DEIM) are employedto develop reduced data assimilation systems for a geophysical flow model,namely, the two dimensional shallow water equations. Numerical experimentsconfirm the theoretical framework for Galerkin projection. In the case ofPetrov-Galerkin projection, stabilization strategies must be considered for thereduced order models. The new reduced order shallow water data assimilationsystem provides analyses similar to those produced by the full resolution dataassimilation system in one tenth of the computational time.

POD/DEIM Reduced-Order Strategies for Efficient Four Dimensional Variational Data Assimilation