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Optimal Control of a Finite-Element Limited-Area Shallow-Water Equations Model

Xiao CHEN
Department of Mathematics
Florida State University, Tallahassee, Florida

I. M. NAVON
Department of Scientific Computing
Florida State University, Tallahassee, Florida

Abstract: Optimal control of a finite element limited-area shallow water equations model is explored with a view to applying 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 minimization solvers L-BFGS is assessed for various lengths of the time windows.

Keywords: Variational data assimilation; Shallow-Water equations model; Galerkin Finite-Element; Adjoint model; Limited-area boundary condition.

Xiao Chen holds a M.Sc. degree in Applied Mathematics(2006) from Zhejiang University, Hangzhou, China. Currently, he is enrolled as a doctoral student in the department of mathematics at Florida State University, USA. His research interests include optimal control in Fluid Dynamics, four dimensional variational data assimilation method, probability and stochastic process, scientific computing and automatic differentiation.

Ionel Michael Navon graduated in Mathematics and Physics from Hebrew University,Jerusalem and holds a M.Sc. in Atmospheric Sciences from the same University and a Ph.D. degree in Applied Mathematics from University of Witwatersrand, Johannesburg (1979). He is presently Program Director and Professor in the Department of Scientific Computing at Florida State University , Tallahassee, FL where he joined in 1985. He served previously as Chief Research Officer at National Research institute for Mathematical Sciences in Pretoria, South-Africa.( 1975-1984) He is the author of more than 160 peer reviewed high impact journal papers in areas of optimal control, data assimilation, parameter estimation, finite element modeling, large-scale numerical optimization and model reduction applied to the geosciences that are highly cited on ISI Web of Science, along with more than 100 scientific and technical reports.. He is also co-author of one book on adjoint sensitivity as well as contributor of chapters in a dozen books including recently the Handbook for Numerical Analysis Series ( Elsevier) and a chapter on Data Assimilation in a recent Springer book on data assimilation.(2009). He is a Fellow of the American Meteorological Society, was senior NRC fellow, former Editor of major journals in applied mathematics and atmospheric sciences and is presently Editor of the International Journal for Numerical Methods in Fluids (Wiley) and is included since 1992 in Who’s Who in America. Prof. Navon educated 7 doctoral students in Applied Mathematics serving as their major Professor. His current research interests focus on all aspects of POD model reduction, data assimilation in the geosciences, large-scale optimization, optimal control and ensemble filters. His research is funded by the National Science Foundation and NASA. He collaborates since 2002 with a major research group at Imperial College as well as with a research group on ocean data assimilation at IAP, Academia Sinica, Beijing, China.

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CITE THIS PAPER AS:
Xiao CHEN, I. M. NAVON, Optimal Control of a Finite-Element Limited-Area Shallow-Water Equations Model, Studies in Informatics and Control, ISSN 1220-1766, vol. 18 (1), pp. 41-62, 2009.