4-D Var Data Assimilation and POD Model Reduction Applied to Geophysical Dynamics Models

Standard spatial discretization schemes for dynamical system (DS), usually lead to large-scale, high-dimensional, and in general, nonlinear systems of ordinary differential equations.Due to limited computational and storage capabilities, Reduced Order Modeling (ROM) techniques from system and control theory provide an attractive approach to approximate the large-scale discretized state equations using low-dimensional models. The objective of 4-D variational data assimilation (4-D Var) is to obtain the minimum of a cost functional estimating the discrepancy between the model solutions and distributed observations in time and space. A control reduction methodology based on Proper Orthogonal Decomposition (POD), referred to as POD 4-D Var, has been widely used for nonlinear systems with tractable computations. However, the appropriate criteria for updating a POD ROM are not yet known in the application to optimal control. This is due to the limited validity of the POD ROM for inverse problems. Therefore, the classical Trust-Region (TR) approach combined with POD (TRPOD) was recently proposed as a way to alleviate the above difficulties. There is a global convergence result for TR, and benefiting from the trust-region philosophy, rigorous convergence results guarantee that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem. In order to reduce the POD basis size and still achieve the global convergence, a method was proposed to incorporate information from the 4-D Var system into the ROM procedure by implementing a dual weighted POD (DWPOD) method. The first new contribution in my dissertation consists in studying a new methodology combining the dual weighted snapshots selection and trust region POD adaptivity (DWTRPOD). Another new contribution is to combine the incremental POD 4-D Var, balanced truncation techniques and method of snapshots methodology. In the linear DS, this is done by integrating the linear forward model many times using different initial conditions in order to construct an ensemble of snapshots so as to generate the forward POD modes. Then those forward POD modes will serve as the initial conditions for its corresponding adjoint system. We then integrate the adjoint system a large number of times based on different initial conditions generated by the forward POD modes to construct an ensemble of adjoint snapshots. From this ensemble of adjoint snapshots, we can generate an ensemble of so-called adjoint POD modes. Thus we can approximate the controllability Grammian of the adjoint system instead of solving the computationally expensive coupled Lyapunov equations. To sum up, in the incremental POD 4-D Var, we can approximate the controllability Grammian by integrating the TLM a number of times and approximate observability Grammian by integrating its adjoint also a number of times. A new idea contributed in this dissertation is to extend the snapshots based POD methodology to the nonlinear system. Furthermore, we modify the classical algorithms in order to save the computations even more significantly. We proposed a novel idea to construct an ensemble of snapshots by integrating the tangent linear model (TLM) only once, based on which we can obtain its TLM POD modes. Then each TLM POD mode will be used as an initial condition to generate a small ensemble of adjoint snapshots and their adjoint POD modes. Finally, we can construct a large ensemble of adjoint POD modes by putting together each small ensemble of adjoint POD modes. To sum up, our idea in a forthcoming study is to test approximations of the controllability Grammian by integrating TLM once and observability Grammian by integrating adjoint model a reduced number of times. Optimal control of a finite element limited-area shallow water equations model is explored with a view to apply variational data assimilation(VDA) by obtaining the minimum of a functional estimating the discrepancy between the model solutions and distributed observations. In our application, some simplified hypotheses are used, namely the error of the model is neglected, only the initial conditions are considered as the control variables, lateral boundary conditions are periodic and finally the observations are assumed to be distributed in space and time. Derivation of the optimality system including the adjoint state, permits computing the gradient of the cost functional with respect to the initial conditions which are used as control variables in the optimization. Different numerical aspects related to the construction of the adjoint model and verification of its correctness are addressed. The data assimilation set-up is tested for various mesh resolutions scenarios and different time steps using a modular computer code. Finally, impact of large-scale unconstrained minimization solvers L-BFGS is assessed for various lengths of the time windows. We then attempt to obtain a reduced-order model (ROM) of 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. Finally, we study solutions of an inverse problem for a global shallow water model controlling its initial conditions specified from the 40-yr ECMWF Re-Analysis (ERA-40) datasets, in presence of full or incomplete observations being assimilated in a time interval (window of assimilation) presence of background error covariance terms. As an extension of this research, we attempt to obtain a reduced-order model of above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4-D Var for a finite volume global shallow water equations model based on the Lin-Rood flux-form semi-Lagrangian semi-implicit time integration scheme. Different approaches of POD implementation for the reduced inverse problem are compared, including a dual-weighted method for snapshot selection coupled with a trust-region POD adaptivity approach. Numerical results with various observational densities and background error covariance operator are also presented. The POD 4-D Var model results combined with the trust region adaptivity exhibit similarity in terms of various error metrics to the full 4-D Var results, but are obtained using a significantly lesser number of minimization iterations and require lesser CPU time. Based on our previous and current research work, we conclude that POD 4-D Var certainly warrants further studies, with promising potential for its extension to operational 3-D numerical weather prediction models.