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 ingredient for a successful reduced order solution to inverse problems is the consistency of the reduced order Karush-KuhnTucker conditions with respect to the full optimality conditions. In particular, accurate reduced order approximations are needed for both the forward dynamical model and for the adjoint model. New bases selection strategies 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 findings. In case of Petrov-Galerkin reduced data assimilation stabilization strategies must be considered for the reduced order models. The new hybrid tensorial POD/DEIM shallow water ROM data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.