Abstract
We discuss PDE-constrained optimization problems with iterative state solvers. As typical and challenging example, we present an application in climate research, namely a parameter optimization problem for a marine ecosystem model. Therein, a periodic state is obtained via a slowly convergent fixed-point type iteration. We recall the algorithm that results from a direct or black-box optimization of such kind of problems, and discuss ways to obtain derivative information to use in gradient-based methods. Then we describe two optimization approaches, the One-shot and the Surrogate-based Optimization method. Both methods aim to reduce the high computational effort caused by the slow state iteration. The idea of the One-shot approach is to construct a combined iteration for state, adjoint and parameters, thus avoiding expensive forward and reverse computations of a standard adjoint method. In the Surrogate-based Optimization method, the original model is replaced by a surrogate which is here based on a truncated iteration with fewer steps. We compare both approaches, provide implementation details for the presented application, and give some numerical results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bandler, J.W., Cheng, Q.S., Dakroury, S.A., Mohamed, A.S., Bakr, M.H., Madsen, K., Søndergaard, J.: Space mapping: the state of the art. IEEE Trans. Microwave Theory Tech. 52(1), 337–361 (2004)
Christianson, B.: Reverse accumulation and implicit functions. Optim. Methods Softw. 9(4), 307–322 (1998)
Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-region methods. In: MPS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2000)
Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Prog. Aerosp. Sci. 45(1–3), 50–79 (2009)
Gauger, N., Griewank, A., Hamdi, A., Kratzenstein, C., Özkaya, E., Slawig, T.: Automated extension of fixed point PDE solvers for optimal design with bounded retardation. In: Leugering, G., Engel, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., Ulbrich, S. (eds.) Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol. 160. Birkhäuser, Basel (2011)
Giering, R., Kaminski, T.: Applying TAF to generate efficient derivative code of Fortran 77–95 programs. Proc. Appl. Math. Mech. 2(1), 54–57 (2003)
Griewank, A.: Evaluating Derivatives Principles and Techniques of Algorithmic Differentiation. Frontiers in Applied Mathematics, vol. 19. SIAM, Philadelphia (2000)
Griewank, A., Hamdi, A.: Properties of an augmented Lagrangian for design optimization. Optim. Methods Softw. 25(4), 645–664 (2010)
Griewank, A., Hamdi, A.: Reduced quasi-Newton method for simultaneous design and optimization. Comput. Optim. Appl. Online 49, 521–548 (2011)
Griewank, A., Kressner, D.: Time-lag in derivative convergence for fixed point iterations. ARIMA Numéro spécial CARI’04, pp. 87–102 (2005)
Griewank, A., Walther, A.: Evaluating derivatives: principles and techniques of algorithmic differentiation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)
Kaland, L., De Los Reyes, J.C., Gauger, N.: One shot methods in function space for pde-constrained optimal control problems. Optim. Methods Softw. 29(2), 376–405 (2013)
Khatiwala, S., Visbeck, M., Cane, M.A.: Accelerated simulation of passive tracers in ocean circulation models. Ocean Model. 9(1), 51–69 (2005)
Koziel, S., Bandler, J.W., Cheng, Q.S.: Robust trust-region space-mapping algorithms for microwave design optimization. IEEE Trans. Microw. Theory Tech. 58(8), 2166–2174 (2010)
Kratzenstein, C., Slawig, T.: Simultaneous model spin-up and parameter identification with the one-shot method in a climate model example. Int. J. Optim. Control Theory Appl. 3(2), 99–110 (2013)
Kriest, I., Khatiwala, S., Oschlies, A.: Towards an assessment of simple global marine biogeochemical models of different complexity. Prog. Oceanogr. 86(3–4), 337–360 (2010)
Leifsson, L., Koziel, S.: Multi-fidelity design optimization of transonic airfoils using physics-based surrogate modeling and shape-preserving response prediction. J. Comput. Sci. 1(2), 98–106, 6 (2010)
Özkaya, E., Gauger, N.: Single-step one-shot aerodynamic shape optimization. Technical report (2008)
Parekh, P., Follows, M.J., Boyle, E.A.: Decoupling of iron and phosphate in the global ocean. Glob. Biogeochem. Cycles 19(2), GB2020 (2005)
Prieß, M., Koziel, S., Slawig, T.: Surrogate-based optimization of climate model parameters using response correction. J. Comput. Sci. 2(4), 335–344, 12 (2011)
Prieß, M., Piwonski, J., Koziel, S., Oschlies, A., Slawig, T.: Accelerated parameter identification in a 3D marine biogeochemical model using surrogate-based optimization. Ocean Model. 68, 22–36 (2013)
Queipo, N.V., Haftka, R.T., Shyy, W., Goel, T., Vaidyanathan, R., Tucker, P.K.: Surrogate-based analysis and optimization. Prog. Aerosp. Sci. 41(1), 1–28 (2005)
Sarmiento, J.L., Gruber, N.: Ocean Biogeochemical Dynamics. Princeton University Press, Princeton (2006)
Simpson, T.W., Poplinski, J.D., Koch, P.N., Allen, J.K.: Metamodels for computer-based engineering design: Survey and recommendations. Eng. Comput. 17, 129–150 (2001)
Smola, A.J., Schölkopf, B.: A tutorial on support vector regression. Stat. Comput. 14, 199–222 (2004)
Søndergaard, J.: Optimization using surrogate models - by the space mapping technique. PhD thesis, Informatics and Mathematical Modelling, Technical University of Denmark, DTU (2003)
Acknowledgements
The work was supported by DFG in the Cluster “Future Ocean” and the priority program 1253 “Optimization with Partial Differential Equations”, and by the EU in the FP7 project “CarboChange”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Slawig, T., Prieß, M., Kratzenstein, C. (2014). Surrogate-Based and One-Shot Optimization Methods for PDE-Constrained Problems with an Application in Climate Models. In: Koziel, S., Leifsson, L., Yang, XS. (eds) Solving Computationally Expensive Engineering Problems. Springer Proceedings in Mathematics & Statistics, vol 97. Springer, Cham. https://doi.org/10.1007/978-3-319-08985-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-08985-0_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08984-3
Online ISBN: 978-3-319-08985-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)