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

This work studies reduced order modeling (ROM) approaches to speed up thesolution of variational data assimilation problems with large scale nonlinear dynamicalmodels. It is shown that a key ingredient for a successful reduced ordersolution to inverse problems is the consistency of the reduced order Karush-Kuhn-Tucker conditions with respect to the full optimality conditions. In particular,accurate reduced order approximations are needed for both the forward dynamicalmodel and for the adjoint model. New bases selection strategies are developedfor Proper Orthogonal Decomposition (POD) ROM data assimilation using bothGalerkin and Petrov-Galerkin projections. For the first time POD, tensorial POD,and discrete empirical interpolation method (DEIM) are employed to develop reduceddata assimilation systems for a geophysical flow model, namely, the twodimensional shallow water equations. Numerical experiments confirm the theoreticalfindings. In case of Petrov-Galerkin reduced data assimilation stabilizationstrategies must be considered for the reduced order models. The new hybrid tensorialPOD/DEIM shallow water ROM data assimilation system provides analysessimilar to those produced by the full resolution data assimilation system in onetenth of the computational time.