Wednesday , June 20 2018

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

Dedicated to Professor Neculai Andrei on the occasion of his 60th birthday.

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.

>>Full text
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.

1. Introduction

This paper explores the feasibility of carrying out a modular structured variational data assimilation (VDA) using a finite-element method of the nonlinear shallow water equations model on a limited area domain, in which we improve the methodology(Courtier and Talagrand 1987; Zhu et al. 1994) and addresses issues in the development of the adjoint of a basic finite-element model. Specific numerical difficulties in the adjoint derivation, for example, the treatment of the adjoint of the iterative process required for solving the systems of linear algebraic equations resulting from the finite-element discretizations using Crank-Nicholson time differencing scheme (see Wang et al. 1972; Douglas and Dupont 1970) are explicitly addressed. The systems of algebraic linear equations resulting from the finite-element discretizations of the shallow-water equations model were solved by a Gauss-Seidel iterative method. To save computer memory, a compact storage scheme for the banded and sparse global matrices was used (see Hinsman, 1975). We emphasize the development of the tangent linear (TLM) and the adjoint models of the finite-element shallow-water equations model and illustrate its use on various retrieval cases when the initial conditions are served as control variables.

The plan of this paper is as follows. The finite-element Galerkin method for the shallow-water equations model on an f plane, the derivation of its tangent linear model and its adjoint are briefly described in section 2. The full finite element discretizations of the model of the nonlinear shallow-water equations model is described in section 3. Section 4 introduces the optimal control methodology including the development of the tangent linear model and its adjoint as well as formulation of the cost functional aimed at allowing the derivation of optimal initial conditions reconciling model forecast and observations in a window of data assimilation by minimizing the cost functional measuring lack of fit between model forecast and observations. Particular attention is paid to the development of adjoint of iterative Gauss-Seidel solver. Verification of the correctness of the adjoint is carried out in a detailed manner for all stages of the calculations (i.e. TLM, adjoint and gradient test).

Set-up of numerical experiments and the experimental design are detailed in Section 5. Basic assimilation experiments using a random perturbation of the initial conditions as observations and their results are presented. Particular attention is paid to the effectiveness of limited memory Quasi-Newton method L-BFGS for minimizing the cost functional in retrieving optimal initial conditions.

Various scenarios involving mesh resolution, different time steps as well as various lengths of the assimilation windows are tested and numerical conclusions are drawn. Finally Section 6 presents Summary and Conclusions. A detailed description of the entire optimal control set-up code organization is provided and illustrated in Appendix A.

REFERENCES:

  1. Axellson, O., V. A. Barker, Finite element solution of boundary value problems: Theory and computation, Computer Science and Applied Mathematics Series, Academic Press, 1984, p. 437.
  2. Blum, J., F. Le Dimet, I. Michael Navon, Data Assimilation for Geophysical Fluids, in press with Computational Methods for the Atmosphere and the Oceans, Volume 14: Special Volume (Handbook of Numerical Analysis). R. Temam and J. Tribbia, eds. Elsevier Science Ltd, New York ( Philippe G. Ciarlet, Editor), 2008.
  3. Courtier, P., O. Talagrand, Variational assimilation of meteorological observations with the the adjoint vorticity equation, Part II, Numerical results, Quart. J. Roy. Meteor. Soc, 113, 1987, pp. 1329 – 1347.
  4. Douglas, J., T. Dupont, Galerkin method for parabolic problems, S.I.A.M., J. Numer. Anal., 7, 1970, pp. 575 – 626.
  5. Galewsky, J., R.K. Scott and L.M. Polvani, An initial-value problem for testing numerical models of the global shallow water equations, Tellus, A 56 (5), 2004, pp. 429-440.
  6. Grammeltvedt, A., A survey of finite-difference schemes for the primitive equations for a barotropic fluid, Mon. Wea. Rev. 97, 1969, pp. 384-404.
  7. Hinsman, D.E., Application of a finite-element method to the barotropic primitive equations, M. Sc. Thesis, Naval Postgraduate School, Department of Meteorology, Monterrey, CA, 1975.
  8. Huebner, K. H., The finite-element method for engineers, John Wiley & Sons, Chichester, 1975.
  9. Liu, D. C., J. Nocedal, On the limited memory BFGS method for large scale optimization, Mathematical Programming, 45, 1989, pp. 503-528.
  10. Martins, J. R. R. A., P. Sturdza, and J. J. Alonso, The complex-step derivative approximation, ACM Transactions on Mathematical Software, Vol. 29 No. 3, 2003, pp. 245-262.
  11. Navon, I. M., Finite-element simulation of the shallow-water equations model on a limited area domain, Appl. Math. Model, 3, 1979, pp. 337-348.
  12. Navon, I. M., FEUDX: a two-stage, high-accuracy, finite-element FORTRAN program for solving shallow-water equations, Computers and Geosciences, 13, 1987, pp. 255-285.
  13. Navon, I. M., X. Zou, J. Derber and J. Sela, Variational data assimilation with an adiabatic version of the NMC spectral mode, Mon. Wea. Rev., Vol. 120, No.7, 1992, pp. 1435-143.
  14. Nocedal, J., Updating quasi-Newton matrices with limited storage, Math. Comp., 24, 1980, pp. 773 – 782.
  15. Payne, N. A., B. M. Irons, Private communication to O. Zienkiewicz, 1963.
  16. Tan, W. Y., Shallow water hydrodynamics: mathematical theory and numerical solution for a two dimensional system of shallow water equations, Beijing, China, Water & Power Press, Elsevier oceanography series, Elsevier Science Ltd, August 1992.
  17. Vreugdenhil, C.B., Numerical methods for shallow-water flow, Dordrecht, Boston, Kluwer Academic Publishers, 1994.
  18. Wang, H. H., P. Halpern, J. Douglas, and I. Dupont, Numerical solutions of the one-dimensional primitive equations using Galerkin approximation with localized basic functions, Mon. Wea. Rev., 100, 1972, pp. 738 -746.
  19. Zhu, K., I. M. Navon and X. Zou, Variational data assimilation with a variable resolution finite-element shallow-water equations model, Mon. Wea. Rev., Vol. 122, No. 5, 1994, pp. 946-965.
  20. Zhu, J., Z. R. L. Taylor, O. C. Zienkiewicz, The Finite Element Method: Its Basis And Fundamentals, Butterworth – Heinemann, 2005, pp. 54-102.
  21. Zienkiewicz, O.C., R.L. Taylor, J. Z. Zhu, P. Nithiarasu, The Finite Element Method, Hardcover, Butterworth- Heinemann, 2005.
  22. Zou, X., I. M. Navon, and J. Sela: Control of gravitational Oscillations in Variational Data Assimilation, Mon. Wea. Rev., 121, 1993, pp. 272 – 289,1993