Abstract
In the natural and enginiering sciences numerous sophisticated simulation models involving PDEs have been developed. In our research we focus on the transition from such simulation codes to optimization, where the design parameters are chosen in such a way that the underlying model is optimal with respect to some performance measure. In contrast to general non-linear programming we assume that the models are too large for the direct evaluation and factorization of the constraint Jacobian but that only a slowly convergent fixed-point iteration is available to compute a solution of the model for fixed parameters. Therefore, we pursue the so-called One-shot approach, where the forward simulation is complemented with an adjoint iteration, which can be obtained by handcoding, the use of Automatic Differentiation techniques, or a combination thereof. The resulting adjoint solver is then coupled with the primal fixed-point iteration and an optimization step for the design parameters to obtain an optimal solution of the problem. To guarantee the convergence of the method an appropriate sequencing of these three steps, which can be applied either in a parallel (Jacobi) or in a sequential (Seidel) way, and a suitable choice of the preconditioner for the design step are necessary. We present theoretical and experimental results for two choices, one based on the reduced Hessian and one on the Hessian of an augmented Lagrangian. Furthermore, we consider the extension of the One-shot approach to the infinite dimensional case and problems with unsteady PDE constraints.
The work was funded by the DFG (Deutsche Forschungsgesellschaft) as part of SPP1253.
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References
G. Biros, O. Ghattas, Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: the Krylov-Schur solver. SIAM J. Sci. Comput. 27(2), 687–713 (2005)
G. Biros, O. Ghattas, Parallel lagrange-Newton-Krylov-Schur methods for pde-constrained optimization. part i: the krylov-schur solver. SIAM J. Sci. Comput. 27(2), 687–713 (2005)
A. Borzì, V. Schulz, Multigrid methods for PDE optimization. SIAM Rev. 51(2), 361–39 (2009)
T. Bosse, A. Griewank, N.R. Gauger, S. Günther, V. Schulz, One-shot approaches to design optimzation, in Trends in PDE Constrained Optimization, ed. by P. Benner, G. Leugering, S. Engell, A. Griewank, H. Harbrecht, M. Hinze, R. Rannacher, S. Ulbrich. International Series of Numerical Mathematics (Springer, Basel, 2014). To appear
T. Bosse, L. Lehmann, A. Griewank, Adaptive sequencing of primal, dual, and design steps in simulation based optimization. Comput. Optim. Appl. (2013). doi:10.1007/s10589-013-9606-z
A. Carnarius, F. Thiele, E. Özkaya, A. Nemili, N.R. Gauger, Optimal control of unsteady flows using a discrete and a continuous adjoint approach, in System Modelling and Optimization. IFIP Advances in Information and Communication Technology, vol. 391, ed. by D. Hömberg, F. Tröltzsch (Springer, Berlin/Heidelberg, 2011), pp. 318–327
L.B. Ciric, A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45(2), 267–273 (1974)
N. Gauger, A. Griewank, A. Hamdi, C. Kratzenstein, E. Özkaya, T. Slawig, Automated extension of fixed point pde solvers for optimal design with bounded retardation, in Constrained Optimization and Optimal Control for Partial Differential Equations, ed. by G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, S. Ulbrich. International Series of Numerical Mathematics, vol. 160 (Springer, Basel, 2012), pp. 99–122
A. Griewank, E. Özkaya, Quantifying retardation in simulation based optimization, in Optimization, simulation, and control. Springer Optimization and its Application, vol. 76 (Springer, New York, 2013), pp. 79–96
S. Günther, N.R. Gauger, Q. Wang, Extension of the One-shot method for optimal control with unsteady PDEs, in Proceedings of the International Conference on Evolutionary and Deterministic Methods for Design, Optimization and Control with Applications to Industrial and Societal Problems (EUROGEN), Spain, 2013
A. Hamdi, A. Griewank, Reduced quasi-Newton method for simultaneous design and optimization. Comput. Optim. Appl. 49(3), 521–548 (2011)
M. Heinkenschloss, L.N. Vicente, Analysis of inexact trust-region sqp algorithms. SIAM J. Optim. 12(2), 283–302 (2002)
L. Kaland, J.C. De Los Reyes, N.R. Gauger, One-shot methods in function space for PDE-constrained optimal control problems. Optim. Methods Softw. 1–30 (2013). doi:10.1080/10556788.2013.774397
C. Kratzenstein, T. Slawig, Simultaneous model spin-up and parameter identification with the One-shot method in a climate model example. Int. J. Optim. Control 3(2), 99–110 (2013)
U. Naumann, The art of differentiating computer programs: an introduction to algorithmic differentiation. Software, Environments and Tools (Society for Industrial and Applied Mathematics, Philadelphia, 2011)
J. Nocedal, S.J. Wright, Numerical Optimization. Springer Series in Operations Research and Financial Engineering, 2nd edn. (Springer, New York, 2006)
P. Parekh, M.J. Follows, E.A. Boyle, Decoupling of iron and phosphate in the global ocean. Glob. Biogeochem. Cycles 19(2), GB2020 (2005)
T. Slawig, K. Zickfeld, Parameter optimization using algorithmic differentiation in a reduced-forms model of the atlantic thermohaline circulation. Nonlinear Anal. Real World Appl. 5/3, 501–518 (2004)
C. Zhu, R.H. Byrd, J. Nocedal, L-bfgs-b: algorithm 778: L-bfgs-b, fortran routines for large scale bound constrained optimization. ACM Trans. Math. Softw. 23(4), 550–560 (1997)
K. Zickfeld, T. Slawig, S. Rahmstorf, A low-order model for the response of the atlantic thermohaline circulation to climate change. Ocean. Dyn. 54, 8–26 (2004)
J.C. Ziems, S. Ulbrich, Adaptive multilevel inexact sqp methods for pde-constrained optimization. SIAM J. Optim. 21(1), 1–40 (2011)
Acknowledgements
The work was funded by the DFG (Deutsche Forschungsgesellschaft) as part of the DFG Schwerpunktprogramm 1253 – Optimization with partial differential equations. Our sincere thanks are due to several other groups of the SPP 1253 (Bock et al., Schulz et al.) for their stimulating comments and discussions. We are especially grateful for the many helpful suggestions and for the encouraging interest shown by other research groups outside of the SPP 1253: Alonso et al., Farrell et al., Koziel et al., Oschlies et al., De los Reyes et al., Thiele et al., and Wang et al.
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Bosse, T. et al. (2014). Optimal Design with Bounded Retardation for Problems with Non-separable Adjoints. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_6
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DOI: https://doi.org/10.1007/978-3-319-05083-6_6
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