Simultaneous model spin-up and parameter identification with the one-shot method in a climate model example
We investigate the Oneshot Optimization strategy introduced by Hamdi and Griewank for the applicability and efficiency to identify parameters in models of the earth's climate system. Parameters of a box model of the North Atlantic Thermohaline Circulation are optimized with respect to the fit of model output to data given by another model of intermediate complexity. Since the model is run into a steady state by a pseudo time-stepping, efficient techniques are necessary to avoid extensive recomputations or storing when using gradient-based local optimization algorithms. The Oneshot approach simultaneously updates state, adjoint and parameter values. For the required partial derivatives, the algorithmic/automatic differentiation tool TAF was used. Numerical results are compared to results obtained by the BFGS-quasi-Newton method.
[1] Griewank, A., Evaluating Derivatives: Prin- ciples and Techniques of Algorithmic Differ- entiation. SIAM, Philadelphia, PA (2000).
[2] Christianson, B., Reverse accumulation and implicit functions. Optimization Methods and Software, 9(4), 307–322 (1998).
[3] Kaminski, T., Giering, R., and Voßbeck, M., Efficient sensitivities for the spin-up phase. Automatic Differentiation: Applica- tions, Theory, and Implementations, Lecture Notes in Computational Science and Engi- neering, Springer, New York, 50, 283–291 (2005).
[4] Hamdi, A. and Griewank, A., Reduced Quasi-Newton Method for Simultaneous Design and Optimization. Comput. Op- tim. Appl. online, Available at www . springerlink .com (2009).
[5] Hamdi, A. and Griewank, A., Properties of an Augmented Lagrangian for Design Op- timization. Optimization Methods and Soft- ware, 25(4), 645–664 (2010).
[6] O(¨)zkaya, E. and Gauger, N., Single-Step One-Shot Aerodynamic Shape Optimization. In- ternational Series of Numerical Mathemat- ics, 158, 191–204 (2009).
[7] Ta’asn, S., Pseudo-Time Methods for Con- strained Optimization Problems Governed by PDE. ICASE Report No. 95-32 (1995).
[8] Hazra, S. B. and Schulz, V., Simultane- ous Pseudo-Timestepping for PDE-Model Based Optimization Problems. BIT Numeri- cal Mathematics, 44, 457–472 (2004).
[9] Pham, D. and Karaboga, D., Intelligent Op- timisation Techniques: Genetic Algorithms, Tabu Search, Simulated Annealing and Neu- ral Networks. Springer London, Limited (2012).
[10] Ciric, L. B., A Generalization of Banach’s Contraction Principle. Proceedings of the American Mathematical Society, 45(2), 267– 273 (1974).
[11] Griewank, A. and Kressner, D., ”Time-lag in Derivative Convergence for Fixed Point It- erations. ARIMA Num´ero sp´ecial CARI’04, 87–102 (2005).
[12] Giering, R., Kaminski, T., and Slawig, T., Generating Efficient Derivative Code with TAF: Adjoint and Tangent Linear Euler Flow Around an Airfoil. Future Generation Com- puter Systems, 21(8), 1345–1355 (2005).
[13] Bischof, C. H., Lang, B., and Vehreschild, A., Automatic Differentiation for MATLAB Programs. Proceedings in Applied Mathemat- ics and Mechanics, 2(1), 50–53 (2003).
[14] Griewank, A., Juedes, D., and Utke, J., Algorithm 755: ADOL-C: A Package for the Automatic Differentiation of Algo- rithms Written in C/C++. ACM Transac- tions on Mathematical Software, 22(2), 131– 167 (1996).
[15] Zickfeld, K., Slawig, T., and Rahmstorf, S., A low-order model for the response of the Atlantic thermohaline circulation to climate change. Ocean Dynamics, 54, 8–26 (2004).
[16] Titz, S., Kuhlbrodt, T., Rahmstorf, S., and Feudel, U., On freshwater-dependent bifurca- tions in box models of the interhemispheric thermohaline circulation. Tellus A, 54, 89 – 98 (2002).
[17] Rahmstorf, S., Brovkin, V., Claussen, M., and Kubatzki, C., CLIMBER-2: A climate system model of intermediate complexity. Part II: Model sensitivity. Clim. Dyn., 17, 735–751 (2001).
[18] Zhu, C., Byrd, R. H., and Nocedal, J., L- BFGS-B: Algorithm 778: L-BFGS-B, FOR- TRAN routines for large scale bound con- strained optimization. ACM Transactions on Mathematical Software, 23(4), 550–560 (1997).