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Automated Extension of Fixed Point PDE Solvers for Optimal Design with Bounded Retardation

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Constrained Optimization and Optimal Control for Partial Differential Equations

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 160))

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Abstract

We study PDE-constrained optimization problems where the state equation is solved by a pseudo-time stepping or fixed point iteration. We present a technique that improves primal, dual feasibility and optimality simultaneously in each iteration step, thus coupling state and adjoint iteration and control/design update. Our goal is to obtain bounded retardation of this coupled iteration compared to the original one for the state, since the latter in many cases has only a Q-factor close to one. For this purpose and based on a doubly augmented Lagrangian, which can be shown to be an exact penalty function, we discuss in detail the choice of an appropriate control or design space preconditioner, discuss implementation issues and present a convergence analysis. We show numerical examples, among them applications from shape design in fluid mechanics and parameter optimization in a climate model.

Mathematics Subject Classification (2000). Primary 90C30; Secondary 99Z99.

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References

  1. J.F. Bonnans, J. Charles Gilbert, C. Lemaréchal, C.A. Sagastizábal,Numerical Optimization Theoretical and Practical Aspects, Springer Berlin Heidelberg (2003).

    Book  Google Scholar 

  2. A. Brandt, Multi-Grid techniques: 1984 guide with applications to fluid dynamics, GMD-Studien. no. 85, St. Augustin, Germany (1984).

    MATH  Google Scholar 

  3. Christianson B., Reverse accumulation and attractive fixed points, Optimization Methods and Software, 3 (1994), 311–326.

    Article  Google Scholar 

  4. Christianson, B., Reverse Accumulation and Implicit Functions. Optimization Methods and Software9:4 (1998), 307–322.

    Article  MathSciNet  Google Scholar 

  5. J.E. Dennis Jr. and R.B. Schnabel, Numerical Methods for unconstrained optimization and Nonlinear Equations, Prentice-Hall (1983).

    Google Scholar 

  6. G. Di Pillo and L. Grippo. A Continuously Differentiable Exact Penalty Function for Nonlinear Programming Problems with Inequality Constraints. SIAM J. Control Optim. 23 (1986), 72–84.

    Article  MathSciNet  Google Scholar 

  7. G. Di Pillo, Exact penalty methods, in E. Spedicato (ed.), Algorithms for continuous optimization: the state of the art, Kluwer Academic Publishers (1994), 209–253.

    Google Scholar 

  8. L.C.W. Dixon, Exact Penalty Function Methods in Nonlinear Programming. Report NOC, The Hatfield Polytechnic, 103 (1979).

    Google Scholar 

  9. R. Fontecilla, T. Steihaug, R.A. Tapia, A Convergence Theory for a Class of Quasi-Newton Methods for Constrained Optimization. SIAM Num. An. 24 (1987), 1133–1151.

    Article  MathSciNet  Google Scholar 

  10. N.R. Gauger, A. Walther, C. Moldenhauer, M. Widhalm, Automatic Differentiation of an Entire Design Chain for Aerodynamic Shape Optimization, Notes on Numerical Fluid Mechanics and Multidisciplinary Design 96 (2007), 454–461.

    Article  Google Scholar 

  11. N.R. Gauger, A. Griewank, J. Riehme, Extension of fixed point PDE solvers for optimal design by one-shot method – with first applications to aerodynamic shape optimization, European J. Computational Mechanics (REMN), 17 (2008), 87–102.

    Article  Google Scholar 

  12. R. Giering, T. Kaminski, T. Slawig, Generating efficient derivative code with TAF: Adjoint and tangent linear Euler flow around an airfoil. Future Generation Computer Systems21:8 (2005), 1345–1355.

    Article  Google Scholar 

  13. I. Gherman, V. Schulz, Preconditioning of one-shot pseudo-timestepping methods for shape optimization, PAMM 5:1(2005), 741–759.

    Article  Google Scholar 

  14. J.C. Gilbert, Automatic Differentiation and iterative Processes, Optimization Methods and Software 1 (1992), 13–21.

    Article  Google Scholar 

  15. A. Griewank, D. Juedes, J. Utke, ADOL-C: A package for the automatic differentiation of algorithms written in C/C++. ACM Trans. Math. Softw. 22 (1996), 131–167.

    Article  Google Scholar 

  16. A. Griewank, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. No. 19 in Frontiers in Appl. Math. SIAM, Philadelphia, PA, (2000).

    MATH  Google Scholar 

  17. A. Griewank, Projected Hessians for Preconditioning in One-Step Design Optimization, Large Scale Nonlinear Optimization, Kluwer Academic Publ. (2006), 151–171.

    Google Scholar 

  18. A. Griewank, C. Faure, Reduced Functions, Gradients and Hessians from Fixed Point Iteration for state Equations, Numerical Algorithms 30,2 (2002), 113–139.

    Article  MathSciNet  Google Scholar 

  19. A. Griewank, D. Kressner, Time-lag in Derivative Convergence for Fixed Point Iterations, AR IMA Numéro spécial CARI’04 (2005), 87–102.

    Google Scholar 

  20. A. Hamdi, A. Griewank, Reduced quasi-Newton method for simultaneous design and optimization, Comput. Optim. Appl. online www.springerlink.com

  21. A. Hamdi, A. Griewank, Properties of an augmented Lagrangian for design optimization, Optimization Methods and Software 25:4 (2010), 645–664.

    Article  MathSciNet  Google Scholar 

  22. A. Jameson, Optimum aerodynamic design using CFD and control theory, In 12th AIAA Computational Fluid Dynamics Conference, AIAA paper 95-1729, San Diego, CA, 1995. American Institute of Aeronautics and Astronautics (1995).

    Google Scholar 

  23. T. Kaminski, R. Giering, M. Vossbeck, Efficient sensitivities for the spin-up phase. In Automatic Differentiation: Applications, Theory, and Implementations, H.M. Bücker, G. Corliss, P. Hovland, U. Naumann, and B. Norris, eds., Springer, New York. Lecture Notes in Computational Science and Engineering 50 (2005), 283–291.

    Google Scholar 

  24. C.T. Kelley, Iterative Methods for optimizations, Society for Industrial and Applied Mathematics (1999).

    Google Scholar 

  25. C. Kratzenstein, T. Slawig, One-shot parameter optimization in a box-model of the North Atlantic thermohaline circulation. DFG Preprint SPP 1253-11-03 (2008).

    Google Scholar 

  26. C. Kratzenstein, T. Slawig, One-shot Parameter Identification – Simultaneous Model Spin-up and Parameter Optimization in a Box Model of the North Atlantic Thermohaline Circulation. DFG Preprint SPP1253-082 (2009).

    Google Scholar 

  27. B. Mohammadi, O. Pironneau. Applied Shape Optimization for Fluids. Numerical Mathematics and scientific Computation, Cladenen Press, Oxford (2001).

    MATH  Google Scholar 

  28. P.A. Newman, G.J.-W. Hou, H.E. Jones, A.C. Taylor. V.M. Korivi. Observations on computational methodologies for use in large-scale, gradient-based, multidisciplinary design incorporating advanced CFD codes. Technical Memorandum 104206, NASA Langley Research Center. AVSCOM Technical Report 92-B-007 (1992).

    Google Scholar 

  29. E. Özkaya, N. Gsauger, Single-step one-shot aerodynamic shape optimization. DFG Preprint SPP1253-10-04 (2008).

    Google Scholar 

  30. S. Rahmstorf, V. Brovkin, M. Claussen, C. Kubatzki, Climber-2: A climate system model of intermediate complexity. Part ii. Clim. Dyn.17 (2001), 735–751.

    Article  Google Scholar 

  31. C.C. Rossow, A flux splitting scheme for compressible and incompressible flows, Journal of Computational Physics, 164 (2000), 104–122.

    Article  Google Scholar 

  32. S. Schlenkrich, A. Walther, N.R. Gauger, R. Heinrich, Differentiating fixed point iterations with ADOL-C: Gradient calculation for fluid dynamics, in H.G. Bock, E. Kostina, H.X. Phu, R. Rannacher, editors, Modeling, Simulation and Optimization of Complex Processes – Proceedings of the Third International Conference on High Performance Scientific Computing 2006 (2008), 499–508.

    Google Scholar 

  33. S. Titz, T. Kuhlbrodt, S. Rahmstorf, U. Feudel, On freshwater-dependent bifurcations in box models of the interhemispheric thermohaline circulation. Tellus A54 (2002), 89–98.

    Article  Google Scholar 

  34. K. Zickfeld, T. Slawig, T.S. Rahmstorf, A low-order model for the response of the Atlantic thermohaline circulation to climate change. Ocean Dyn.54 (2004), 8–26.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and CCES, RWTH Aachen University, Schinkelstr. 2, D-52062, Aachen, Germany

    Nicolas Gauger & Emre Özkaya

  2. Institut für Mathematik Fakultät Math.-Nat. II, Humboldt-Universität, Unter den Linden 6, D-10099, Berlin, Germany

    Andreas Griewank & Adel Hamdi

  3. Institut für Informatik and DFG Cluster of Excellence The Future Ocean, Christian-Albrechts-Universität zu Kiel, D-24098, Kiel, Germany

    Claudia Kratzenstein & Thomas Slawig

Authors
  1. Nicolas Gauger
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  2. Andreas Griewank
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  3. Adel Hamdi
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  4. Claudia Kratzenstein
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  5. Emre Özkaya
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  6. Thomas Slawig
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Corresponding author

Correspondence to Nicolas Gauger .

Editor information

Editors and Affiliations

  1. , Department Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany

    Günter Leugering

  2. , Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Geschossbau 2, Dortmund, 44221, Germany

    Sebastian Engell

  3. , Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Andreas Griewank

  4. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  5. , Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 293, Heidelberg, 69120, Germany

    Rolf Rannacher

  6. , FB 4 - Department Mathematik, Universität Trier, Trier, 54286, Germany

    Volker Schulz

  7. Fakultät für Mathematik, M1, Lehrstuhl für Mathematische Optimierung, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85748, Germany

    Michael Ulbrich

  8. , Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, 64293, Germany

    Stefan Ulbrich

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Gauger, N., Griewank, A., Hamdi, A., Kratzenstein, C., Özkaya, E., Slawig, T. (2012). Automated Extension of Fixed Point PDE Solvers for Optimal Design with Bounded Retardation. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_6

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