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POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model
 Journal of Computational Physics
"... In the present paper we consider a 2D shallowwater equations (SWE) model on a βplane solved using an alternating direction fully implicit (ADI) finitedifference scheme on a rectangular domain. The scheme was shown to be unconditionally stable for the linearized equations. The discretization yiel ..."
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In the present paper we consider a 2D shallowwater equations (SWE) model on a βplane solved using an alternating direction fully implicit (ADI) finitedifference scheme on a rectangular domain. The scheme was shown to be unconditionally stable for the linearized equations. The discretization yields a number of nonlinear systems of algebraic equations. We then use a proper orthogonal decomposition (POD) to reduce the dimension of the SWE model. Due to the model nonlinearities, the computational complexity of the reduced model still depends on the number of variables of the full shallow water equations model. By employing the discrete empirical interpolation method (DEIM) we reduce the computational complexity of the reduced order model due to its depending on the nonlinear full dimension model and regain the full model reduction expected from the POD model. To emphasize the CPU gain in performance due to use of POD/DEIM, we also propose testing an explicit Euler finite difference scheme (EE) as an alternative to the ADI implicit
A dualweighted trustregion adaptive POD 4DVar applied to a finiteelement shallowwater equations model
 International Journal for Numerical Methods in Fluids 2009; DOI: 10.1002/fld.2198
"... In this paper we study solutions of an inverse problem for a global shallow water model controlling its initial conditions specified from the 40yr ECMWF Reanalysis (ERA40) data sets, in the presence of full or incomplete observations being assimilated in a time interval (window of assimilation) w ..."
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In this paper we study solutions of an inverse problem for a global shallow water model controlling its initial conditions specified from the 40yr ECMWF Reanalysis (ERA40) data sets, in the presence of full or incomplete observations being assimilated in a time interval (window of assimilation) with or without background error covariance terms. As an extension of the work by Chen et al. (Int. J. Numer. Meth. Fluids 2009), we attempt to obtain a reduced order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4DVar for a finite volume global shallow water equation model based on the Lin–Rood fluxform semiLagrangian semiimplicit time integration scheme. Different approaches of POD implementation for the reduced inverse problem are compared, including a dualweighted method for snapshot selection coupled with a trustregion POD adaptivity approach. Numerical results with various observational densities and background error covariance operator are also presented. The POD 4D Var model results combined with the trustregion adaptivity exhibit similarity in terms of various error metrics to the full 4D Var results, but are obtained using a significantly lesser number of minimization iterations and require lesser CPU time. Based on our previous and current work, we conclude that POD 4D Var certainly warrants further studies, with promising potential of its extension
A POD reduced order unstructured mesh ocean modelling method for . . .
 OCEAN MODELLING 28 (2009) 127–136
, 2009
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Data Assimilation for Geophysical Fluids
"... The ultimate purpose of environmental studies is the forecast of its natural evolution. A prerequisite before a prediction is to retrieve at best the state of the environment. Data assimilation is the ensemble of techniques which, starting from heterogeneous information, permit to retrieve the initi ..."
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The ultimate purpose of environmental studies is the forecast of its natural evolution. A prerequisite before a prediction is to retrieve at best the state of the environment. Data assimilation is the ensemble of techniques which, starting from heterogeneous information, permit to retrieve the initial state of a flow. In the first part, the mathematical models governing geophysical flows are presented together with the networks of observations of the atmosphere and of the ocean. In variational methods, we seek for the minimum of a functional estimating the discrepancy between the solution of the model and the observation. The derivation of the optimality system, using the adjoint state, permits to compute a gradient which is used in the optimization. The definition of the cost function permits to take into account the available statistical information through the choice of metrics in the space of observation and in the space of the initial condition. Some examples are presented on simplified models, especially an application in oceanography. Among the tools of optimal control, the adjoint model permits to carry out sensitivity studies, but if we look for the sensitivity of the prediction with respect to the observations, then a secondorder analysis should be considered. One of the first methods used for assimilating data in oceanography is the nudging method, adding a forcing term in the equations. A variational variant of nudging method is described and also a socalled Computational Methods for the Atmosphere and the Oceans
Reducedorder modeling based on POD of a parabolized Navier–Stokes equation model I: forward model
 International Journal for Numerical Methods in Fluids 69
, 2012
"... A proper orthogonal decomposition (POD)based reducedorder model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A spacemarching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate ..."
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A proper orthogonal decomposition (POD)based reducedorder model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A spacemarching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reducedorder model. In order to improve the accuracy and the stability of the reducedorder model in the presence of a high Reynolds number, we applied a Sobolev H1 norm calibration to the POD construction process. Finally, some numerical tests with a highfidelity model as well as the POD reducedorder model were carried out to demonstrate the efficiency and the accuracy of the reducedorder model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H1 norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reducedorder PNS models. Copyright © 2011 John Wiley & Sons, Ltd.
Numerical Simulations with Data Assimilation Using an Adaptive
 POD Procedure, Lecture Notes in Computer Science
"... Abstract. In this study the proper orthogonal decomposition (POD) methodology to model reduction is applied to construct a reducedorder control space for simple advectiondiffusion equations. Several 4DVar data assimilation experiments associated with these models are carried out in the reduced co ..."
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Abstract. In this study the proper orthogonal decomposition (POD) methodology to model reduction is applied to construct a reducedorder control space for simple advectiondiffusion equations. Several 4DVar data assimilation experiments associated with these models are carried out in the reduced control space. Emphasis is laid on the performance evaluation of an adaptive POD procedure, with respect to the solution obtained with the classical 4DVar (full model), and POD 4DVar data assimilation. Despite some perturbation factors characterizing the model dynamics, the adaptive POD scheme presents better numerical robustness compared to the other methods, and provides accurate results. 1
Comparative Study with Data Assimilation Experiments Using Proper Orthogonal Decomposition
 Method, Lecture Notes in Computer Science
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
MIXED FINITE ELEMENT FORMULATION AND ERROR ESTIMATES BASED ON PROPER ORTHOGONAL DECOMPOSITION FOR THE NONSTATIONARY NAVIER–STOKES EQUATIONS ∗
"... 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 ..."
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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 PODreduced 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.
A reducedorder kalman filter for data assimilation in physical oceanography
 SIAM Rev
"... Abstract. A central task of physical oceanography is the prediction of ocean circulation at various time scales. Mathematical techniques are used in this domain not only for the modeling of ocean circulation but also for the enhancement of simulation through data assimilation. The ocean circulation ..."
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Abstract. A central task of physical oceanography is the prediction of ocean circulation at various time scales. Mathematical techniques are used in this domain not only for the modeling of ocean circulation but also for the enhancement of simulation through data assimilation. The ocean circulation model of concern here, namely, HYCOM, is briefly presented through its variables, equations, and specific vertical coordinate system. The main part of this paper focuses on the Kalman filter as a data assimilation method, and especially on how this mathematical technique, usually associated with a prohibitively high computing cost for operational sciences, is simplified in order to make it applicable to the simulation of realistic ocean circulation models. Some practical issues are presented, such as a brief explanation about ocean observation systems, together with examples of data assimilation results.
2007: Technical summary
 In Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change
"... A postcompletion error occurs when the final step of a task is omitted because the main goal of the task is thought to be completed (Byrne & Bovair, 1997). Postcompletion errors are more likely to occur after interruptions (Ratwani, McCurry & Trafton, 2008). Global placekeeping cues (Gray, 2 ..."
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A postcompletion error occurs when the final step of a task is omitted because the main goal of the task is thought to be completed (Byrne & Bovair, 1997). Postcompletion errors are more likely to occur after interruptions (Ratwani, McCurry & Trafton, 2008). Global placekeeping cues (Gray, 2000) allow a user to track their progress in a task and may be a method for reducing the rate of postcompletion errors. A computerbased procedural task with a postcompletion step was used in this experiment to determine how the interaction of global placekeeping cues with interruptions would affect postcompletion errors. These results suggest that global placekeeping cues reduce the postcompletion error rate after interruptions, but that global placekeeping does not completely eliminate postcompletion errors.