A dual-weighted trust-region adaptive POD 4D-VAR applied to a finite-element shallow-water equations model

We consider a limited-area finite-element discretization of the shallow-water equations model. Our purpose in this paper is to solve an inverse problem for the above model controlling its initial conditions in presence of observations being assimilated in a time interval (window of assimilation). We then attempt to obtain a reduced-order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4-D VAR. Different approaches of POD implementation of the reduced inverse problem are compared, including a dual-weighed method for snapshot selection coupled with a trust-region POD approach. Numerical results obtained point to an improved accuracy in all metrics tested when dual-weighing choice of snapshots is combined with POD adaptivity of the trust-region type. Results of ad-hoc adaptivity of the POD 4-D VAR turn out to yield less accurate results than trust-region POD when compared with high-fidelity model. Directions of future research are finally outlined. Copyright 2009 John Wiley & Sons, Ltd.