SUMMARYFour-dimensional variational data assimilation (4DVAR) is a powerful tool for data assimilation in meteorology and oceanography. However, a major hurdle in use of 4DVAR for realistic general circulation models is the dimension of the control space (generally equal to the size of the model state variable and typically of order 10 7 -10 8 ) and the high computational cost in computing the cost function and its gradient that require integration model and its adjoint model.In this paper, we propose a 4DVAR approach based on proper orthogonal decomposition (POD). POD is an efficient way to carry out reduced order modelling by identifying the few most energetic modes in a sequence of snapshots from a time-dependent system, and providing a means of obtaining a low-dimensional description of the system's dynamics. The POD-based 4DVAR not only reduces the dimension of control space, but also reduces the size of dynamical model, both in dramatic ways. The novelty of our approach also consists in the inclusion of adaptability, applied when in the process of iterative control the new control variables depart significantly from the ones on which the POD model was based upon. In addition, these approaches also allow to conveniently constructing the adjoint model.The proposed POD-based 4DVAR methods are tested and demonstrated using a reduced gravity wave ocean model in Pacific domain in the context of identical twin data assimilation experiments. A comparison with data assimilation experiments in the full model space shows that with an appropriate selection of the basis functions the optimization in the POD space is able to provide accurate results at a reduced computational cost. The POD-based 4DVAR methods have the potential to approximate the performance of full order 4DVAR with less than 1/100 computer time of the full order 4DVAR. The HFTN (Hessian-free truncated-Newton)algorithm benefits most from the order reduction (see (Int. J. Numer. Meth. Fluids, in press)) since computational savings are achieved both in the outer and inner iterations of this method.

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The proper orthogonal decomposition (POD) is shown to be an efficient model reduction technique for simulating physical processes governed by partial differential equations. In this paper, we make an initial effort to investigate problems related to POD reduced modeling of a large-scale upper ocean circulation in the tropic Pacific domain. We construct different POD models with different choices of snapshots and different number of POD basis functions. The results from these different POD models are compared with that of the original model. The main findings are: (1) the large-scale seasonal variability of the tropic Pacific obtained by the original model is well captured by a low dimensional system of order of 22, which is constructed using 20 snapshots and 7 leading POD basis functions. (2) the RMS errors for the upper ocean layer thickness of the POD model of order of 22 are less than 1m that is less than 1% of the average thickness and the correlations between the upper ocean layer thickness with that from the POD model is around 0.99. (3) Retaining modes that capture 99% energy is necessary in order to construct POD models yielding a high accuracy.

Abstract. In this paper, proper orthogonal decomposition (POD) is used for model reduction of mixed finite element (MFE) for the nonstationary Navier-Stokes equations and error estimates between a reference solution and the POD solution of reduced MFE formulation are derived. The basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual MFE method for the nonstationary Navier-Stokes equations or from physics system trajectories by drawing samples of experiments and interpolation (or data assimilation), and then the basis functions of the usual MFE method are substituted with the POD basis functions reconstructed by the elements of the ensemble to derive the POD-reduced MFE formulation for the nonstationary Navier-Stokes equations. It is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between POD solution of reduced MFE formulation and the reference solution is consistent with theoretical results. Moreover, it is also shown that this result validates the feasibility and efficiency of the POD method.Key words. mixed finite element method, proper orthogonal decomposition, the nonstationary Navier-Stokes equations, error estimate AMS subject classifications. 65N30, 35Q10DOI. 10.1137/070689498 1. Introduction. The mixed finite element (MFE) method is one of the important approaches for solving systems of partial differential equations, for example, the nonstationary Navier-Stokes equations (see [1], [2], or [3]). However, the computational model for the fully discrete system of MFE solutions of the nonstationary Navier-Stokes equations yields very large systems that are computationally intensive. Thus, an important problem is how to simplify the computational load and save time-consuming calculations and resource demands in the actual computational process in a way that guarantees a sufficiently accurate and efficient numerical solution. Proper orthogonal decomposition (POD), also known as Karhunen-Loève expansions in signal analysis and pattern recognition (see [4]), or principal component analysis in statistics (see [5]), or the method of empirical orthogonal functions in geophysical fluid dynamics (see [6], [7]) or meteorology (see [8]), is a technique offering adequate approximation for representing fluid flow with reduced number of degrees of freedom, i.e., with lower dimensional models (see [9]), so as to alleviate the computational load