An ensemble particle filter was recently developed as a fully nonlinear filter of Bayesian condit... more An ensemble particle filter was recently developed as a fully nonlinear filter of Bayesian conditional probability estimation, along with the well known ensemble Kalman filter. A Gaussian resampling method is proposed here to generate the posterior analysis ensemble in an effective and efficient way. As a result the ensemble particle filter has good stability and potential applicability to large-scale problems. The Lorenz model is used here to test the proposed method. Multi-modal probability distributions can appear either with state dependent stochastic model errors or nonlinear observations. Ensemble Kalman filter (EnKF)is known to have a difficulty in tracking state transitions accurately. Current implementations of EnKF have not taken non-Gaussian contributions into account. With the posterior Gaussian resampling method the ensemble particle filter can track state transitions more accurately. Moreover, it is applicable to systems with typical multi-modal behavior, provided that certain prior knowledge becomes available about the general structure of posterior probability distribution. A simple scenario is considered to illustrate this point based on Lorenz model attractors. The present work demonstrates that the proposed ensemble particle filter can provide an accurate estimation of multi-modal distribution and is potentially applicable to large-scale data assimilation problems.
This work studies reduced order modeling (ROM) approaches to speed up the solution of variational... more 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 requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based 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 framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.
ABSTRACT The adjoint model of a finite-element shallow-water equations model was obtained with a ... more ABSTRACT The adjoint model of a finite-element shallow-water equations model was obtained with a view to calculate the gradient of a cost functional in the framework of using this model to carry out variational data assimilation (VDA) experiments using optimal control of partial differential equations. The finite-element model employs a triangular finite-element Galerkin scheme and serves as a prototype of 2D shallow-water equation models with a view of tackling problems related to VDA with finite-element numerical weather prediction models. The derivation of the adjoint of this finite-element model involves overcoming specific computational problems related to obtaining the adjoint of iterative procedures for solving systems of nonsymmetric linear equations arising from the finite-element discretization and dealing with irregularly ordered discrete variables at each time step. The correctness of the adjoint model was verified at the subroutine level and was followed by a gradient check conducted once the full adjoint model was assembled. VDA experiments were performed using model-generated observations. In our experiments, assimilation was carried out assuming that observations consisting of a full-model-state vector are available at every time step in the window of assimilation. Successful retrieval was obtained using the initial conditions as control variables, involving the minimization of a cost function consisting of the weighted sum of difference between model solution and model-generated observations. An additional set of experiments was carried out aiming at evaluating the impact of carrying out VDA involving variable mesh resolution in the finite-element model over the entire assimilation period. Several conclusions are drawn related to the efficiency of VDA with variable horizontal mesh resolution finite-element discretization and the transfer of information between coarse and fine meshes. 51 refs., 17 figs., 5 tabs.
A Sasaki variational approach is for the first time applied to enforce conservation of potential ... more A Sasaki variational approach is for the first time applied to enforce conservation of potential enstrophy and total mass in long term integrations of two ADI finite-difference approximations of the nonlinear shallow-water equations on a beta plane. The performance of the variational approach is compared with that of a modified Bayliss-Isaacson technique also designed to enforce conservation of potential enstrophy and total mass at each time step of the numerical integration. Both techniques yielded very satisfactory results after 20 days of numerical integration. It appears, however, that the Bayliss-Isaacson technique is more robust and less demanding of CPU time, while the modified Sasaki variational technique is highly dependent on the method used to update the Lagrange multiplier.
An ensemble particle filter was recently developed as a fully nonlinear filter of Bayesian condit... more An ensemble particle filter was recently developed as a fully nonlinear filter of Bayesian conditional probability estimation, along with the well known ensemble Kalman filter. A Gaussian resampling method is proposed here to generate the posterior analysis ensemble in an effective and efficient way. As a result the ensemble particle filter has good stability and potential applicability to large-scale problems. The Lorenz model is used here to test the proposed method. Multi-modal probability distributions can appear either with state dependent stochastic model errors or nonlinear observations. Ensemble Kalman filter (EnKF)is known to have a difficulty in tracking state transitions accurately. Current implementations of EnKF have not taken non-Gaussian contributions into account. With the posterior Gaussian resampling method the ensemble particle filter can track state transitions more accurately. Moreover, it is applicable to systems with typical multi-modal behavior, provided that certain prior knowledge becomes available about the general structure of posterior probability distribution. A simple scenario is considered to illustrate this point based on Lorenz model attractors. The present work demonstrates that the proposed ensemble particle filter can provide an accurate estimation of multi-modal distribution and is potentially applicable to large-scale data assimilation problems.
This work studies reduced order modeling (ROM) approaches to speed up the solution of variational... more 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 requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based 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 framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.
ABSTRACT The adjoint model of a finite-element shallow-water equations model was obtained with a ... more ABSTRACT The adjoint model of a finite-element shallow-water equations model was obtained with a view to calculate the gradient of a cost functional in the framework of using this model to carry out variational data assimilation (VDA) experiments using optimal control of partial differential equations. The finite-element model employs a triangular finite-element Galerkin scheme and serves as a prototype of 2D shallow-water equation models with a view of tackling problems related to VDA with finite-element numerical weather prediction models. The derivation of the adjoint of this finite-element model involves overcoming specific computational problems related to obtaining the adjoint of iterative procedures for solving systems of nonsymmetric linear equations arising from the finite-element discretization and dealing with irregularly ordered discrete variables at each time step. The correctness of the adjoint model was verified at the subroutine level and was followed by a gradient check conducted once the full adjoint model was assembled. VDA experiments were performed using model-generated observations. In our experiments, assimilation was carried out assuming that observations consisting of a full-model-state vector are available at every time step in the window of assimilation. Successful retrieval was obtained using the initial conditions as control variables, involving the minimization of a cost function consisting of the weighted sum of difference between model solution and model-generated observations. An additional set of experiments was carried out aiming at evaluating the impact of carrying out VDA involving variable mesh resolution in the finite-element model over the entire assimilation period. Several conclusions are drawn related to the efficiency of VDA with variable horizontal mesh resolution finite-element discretization and the transfer of information between coarse and fine meshes. 51 refs., 17 figs., 5 tabs.
A Sasaki variational approach is for the first time applied to enforce conservation of potential ... more A Sasaki variational approach is for the first time applied to enforce conservation of potential enstrophy and total mass in long term integrations of two ADI finite-difference approximations of the nonlinear shallow-water equations on a beta plane. The performance of the variational approach is compared with that of a modified Bayliss-Isaacson technique also designed to enforce conservation of potential enstrophy and total mass at each time step of the numerical integration. Both techniques yielded very satisfactory results after 20 days of numerical integration. It appears, however, that the Bayliss-Isaacson technique is more robust and less demanding of CPU time, while the modified Sasaki variational technique is highly dependent on the method used to update the Lagrange multiplier.
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Papers by Ionel M Navon