  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.
