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Adaptive sequencing of primal, dual, and design steps in simulation based optimization

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Abstract

Many researchers have used Oneshot optimization methods based on user-specified primal state iterations, the corresponding adjoint iterations, and appropriately preconditioned design steps. Our goal here is to develop heuristics for sequencing these three subtasks, in order to optimize the convergence rate of the resulting coupled iteration cycle. A key ingredient is the preconditioning in the design step by a BFGS approximation of the projected Hessian. We provide a hard bound on the spectral radius of the coupled iteration cycle at local minima satisfying second order sufficiency conditions. Finally, we show how certain problem specific parameters can be estimated by local samples and be used to steer the whole process adaptively. We present limited numerical results that confirm the theoretical analysis.

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References

  1. Averick, B.M., Carter, R.G., Moré, J.J.: The minpack-2 test problem collection (1991). doi:10.1.1.45.259

  2. Byrd, R.H., Nocedal, J., Schnabel, R.B.: Representations of quasi-Newton matrices and their use in limited memory methods. Math. Program., Ser. A 63(2), 129–156 (1994). doi:10.1007/BF01582063

    Article  MATH  MathSciNet  Google Scholar 

  3. Evans, L.C.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, vol. 19. Am. Math. Soc., Providence (2010)

    MATH  Google Scholar 

  4. Gauger, N., Griewank, A., Hamdi, A., Kratzenstein, C., Oezkaya, E., Slawig, T.: Automated extension of fixed point PDE solvers for optimal design with bounded retardation. Int. Ser. Numer. Math. 106, 99–122 (2012)

    Google Scholar 

  5. Griewank, A.: Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation. Frontiers in Applied Mathematics, vol. 19. SIAM, Philadelphia (2000)

    MATH  Google Scholar 

  6. Griewank, A.: Projected Hessians for preconditioning in one-step one-shot design optimization. In: Large-Scale Nonlinear Optimization. Nonconvex Optim. Appl., vol. 83, pp. 151–171. Springer, New York (2006). doi:10.1007/0-387-30065-1_10

    Chapter  Google Scholar 

  7. Griewank, A., Faure, C.: Reduced functions, gradients and Hessians from fixed-point iterations for state equations. Numer. Algorithms 30(2), 113–139 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Griewank, A., Faure, C.: Piggyback differentiation and optimization. In: Large-Scale PDE-Constrained Optimization (Santa Fe, NM, 2001). Lect. Notes Comput. Sci. Eng., vol. 30, pp. 148–164. Springer, Berlin (2003)

    Chapter  Google Scholar 

  9. Griewank, A., Oezkaya, E.: Quantifying retardation in simulation based optimization. In: Optimization, Simulation, and Control. Springer Optimization and Its Application, vol. 76, pp. 79–96. Springer, New York (2013)

    Chapter  Google Scholar 

  10. Hamdi, A., Griewank, A.: Reduced quasi-Newton method for simultaneous design and optimization. Comput. Optim. Appl. 49(3), 521–548 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hazra, S.B., Schulz, V., Brezillon, J., Gauger, N.R.: Aerodynamic shape optimization using simultaneous pseudo-timestepping. J. Comput. Phys. 204(1), 46–64 (2005). doi:10.1016/j.jcp.2004.10.007

    Article  MATH  MathSciNet  Google Scholar 

  12. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, New York (2009)

    Book  MATH  Google Scholar 

  13. Jameson, A.: Aerodynamic design via control theory. In: Recent Advances in Computational Fluid Dynamics, Princeton, NJ, 1988. Lect. Notes in Eng., vol. 43, pp. 377–401. Springer, Berlin (1989). doi:10.1007/978-3-642-83733-3_14

    Chapter  Google Scholar 

  14. Kaland, L., Reyes, J.C.D.L., Gauger, N.R.: One shot methods in function space for PDE-constrained optimal control problems. Optim. Methods Softw. (2013). doi:10.1080/10556788.2013.774397

    Google Scholar 

  15. Kanzow, C.: Numerik Linearer Gleichungssysteme—Direkte und Iterative Verfahren, 1 edn. Gabler, Wiesbaden (2004)

    Google Scholar 

  16. Kelley, C.T.: Iterative Methods for Optimization. Frontiers in Applied Mathematics, vol. 18. SIAM, Philadelphia (1999). doi:10.1137/1.9781611970920

    Book  MATH  Google Scholar 

  17. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Die Grundlehren der mathematischen Wissenschaften, vol. 170. Springer, New York (1971). Translated from the French by S.K. Mitter

    Book  MATH  Google Scholar 

  18. Naumann, U.: The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation. Software, Environments and Tools. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  19. Nemili, A., Özkaya, E., Gauger, N., Carnarius, A., Thiele, F.: Optimal control of unsteady flows using discrete adjoints. AIAA 2011-3720 (2011). doi:10.1007/978-3-642-35680-3_53

  20. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)

    MATH  Google Scholar 

  21. Ostrowski, A.M.: Solution of Equations and Systems of Equations, 2nd edn. Pure and Applied Mathematics, vol. 9. Academic Press, New York (1966)

    MATH  Google Scholar 

  22. Özkaya, E., Gauger, N.R.: Single-step one-shot aerodynamic shape optimization. In: Optimal Control of Coupled Systems of Partial Differential Equations. Int. Ser. Numer. Math., vol. 158, pp. 191–204. Birkhäuser, Basel (2009). doi:10.1007/978-3-7643-8923-9_11

    Chapter  Google Scholar 

  23. Özkaya, E., Gauger, N.R.: Automatic transition from simulation to one-shot shape optimization with Navier-Stokes equations. GAMM-Mitt. 33(2), 133–147 (2010). doi:10.1002/gamm.201010011

    Article  MATH  MathSciNet  Google Scholar 

  24. Potschka, A., Bock, H.G.: On the connection of forward and optimization problem in simultaneous inexact SQP methods. In: Recent Trends in Mathematics Related to PDE Constrained Optimization (2010)

    Google Scholar 

  25. Quarteroni, A.: Numerical Models for Differential Problems. MS&A. Modeling, Simulation and Applications vol. 2. Springer, Milan (2009). Translated from the 4th (2008) Italian edition by Silvia Quarteroni. doi:10.1007/978-88-470-1071-0

    Book  MATH  Google Scholar 

  26. Tröltzsch, F., Sprekels, J.: Optimal Control of Partial Differential Equations—Theory, Methods, and Applications. Am. Math. Soc., Heidelberg (2010)

    MATH  Google Scholar 

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Acknowledgements

The research for this paper was funded by the Schwerpunktprogramm 1253 Optimization with partial Differential Equations of the Deutsche Forschungsgesellschaft DFG in the project Automated Extension of Fixed Point PDE Solvers for Optimal Design with Bounded Retardation. The authors are indebted to the other members of this project, namely N. Gauger, E. Ozkaya, C. Kratzenstein and Th. Slawig.

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  1. Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Torsten Bosse, Lutz Lehmann & Andreas Griewank

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  1. Torsten Bosse
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  2. Lutz Lehmann
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  3. Andreas Griewank
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Correspondence to Torsten Bosse.

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Bosse, T., Lehmann, L. & Griewank, A. Adaptive sequencing of primal, dual, and design steps in simulation based optimization. Comput Optim Appl 57, 731–760 (2014). https://doi.org/10.1007/s10589-013-9606-z

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