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Direct multiple shooting for parabolic PDE constrained optimization. (English) Zbl 1337.65054

Carraro, Thomas (ed.) et al., Multiple shooting and time domain decomposition methods. MuS-TDD, Heidelberg, Germany, May 6–8, 2013. Cham: Springer (ISBN 978-3-319-23320-8/hbk; 978-3-319-23321-5/ebook). Contributions in Mathematical and Computational Sciences 9, 159-181 (2015).
Summary: Direct multiple shooting is a flexible and efficient method to solve difficult optimal control problems constrained by ordinary differential equations or differential-algebraic equations. The aim of this article is to concisely summarize the main conceptual and methodological approaches to solve also optimal control problems with parabolic partial differential equations constraints via a direct multiple shooting method. The main obstacle is the sheer size of the discretized optimization problems. We explain a typical direct discretization approach and discuss an inexact sequential quadratic programming method based on two-grid Newton-Picard preconditioning. Special attention is given to a-posteriori \(\kappa\)-estimators that monitor contraction and to the structure-exploiting treatment of the resulting large-scale quadratic programming subproblems, including an extended condensing technique that exploits multiple shooting and two-grid Newton-Picard structures. Finally, we present numerical results for an advection-diffusion and a bacterial chemotaxis example.
For the entire collection see [Zbl 1333.65003].

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M37 Numerical methods based on nonlinear programming
92B10 Taxonomy, cladistics, statistics in mathematical biology
65F08 Preconditioners for iterative methods
49M15 Newton-type methods
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[1] Albersmeyer, J.: Adjoint based algorithms and numerical methods for sensitivity generation and optimization of large scale dynamic systems. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (2010) · Zbl 1210.65003
[2] Albersmeyer, J.; Bock, H. G.; Bock, H. G.; Kostina, E.; Phu, X. H.; Rannacher, R., Sensitivity generation in an adaptive BDF-method, Modeling, Simulation and Optimization of Complex Processes. Proceedings of the International Conference on High Performance Scientific Computing, pp. 15-24, Hanoi, 6-10 March 2006 (2008), Berlin: Springer, Berlin
[3] Bellman, R. E., Dynamic Programming (1957), Princeton: University Press, Princeton · Zbl 0077.13605
[4] Biegler, L. T., Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation, Comput. Chem. Eng., 8, 243-248 (1984) · doi:10.1016/0098-1354(84)87012-X
[5] Bock, H. G.; Deuflhard, P.; Hairer, E., Recent advances in parameter identification techniques for ODE, Numerical Treatment of Inverse Problems in Differential and Integral Equations, 95-121 (1983), Boston: Birkhäuser, Boston · Zbl 0516.65067 · doi:10.1007/978-1-4684-7324-7_7
[6] Bock, H.G., Plitt, K.J.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC World Congress, pp. 242-247, Budapest. Pergamon Press (1984)
[7] Dautray, R.; Lions, J.-L.; Craig, A., Evolution problems I, Mathematical Analysis and Numerical Methods for Science and Technology (1992), Berlin: Springer, Berlin · Zbl 0755.35001
[8] Deuflhard, P.: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics, vol. 35. Springer, Berlin (2006) · Zbl 1056.65051
[9] Griewank, A.; Walther, A., Evaluating Derivatives (2008), Philadelphia: SIAM, Philadelphia · Zbl 1159.65026 · doi:10.1137/1.9780898717761
[10] Hesse, H.K.: Multiple shooting and mesh adaptation for PDE constrained optimization problems. Ph.D. thesis, University of Heidelberg (2008) · Zbl 1149.65308
[11] Hinze, M.; Pinnau, R.; Ulbrich, M.; Ulbrich, S., Optimization with PDE Constraints (2009), New York: Springer, New York · Zbl 1167.49001
[12] Lebiedz, D.; Brandt-Pollmann, U., Manipulation of self-aggregation patterns and waves in a reaction-diffusion system by optimal boundary control strategies, Phys. Rev. Lett., 91, 20, 208301 (2003) · doi:10.1103/PhysRevLett.91.208301
[13] Lust, K.; Roose, D.; Spence, A.; Champneys, A. R., An adaptive Newton-Picard algorithm with subspace iteration for computing periodic solutions, SIAM J. Sci. Comput., 19, 4, 1188-1209 (1998) · Zbl 0915.65088 · doi:10.1137/S1064827594277673
[14] Nocedal, J.; Wright, S. J., Numerical Optimization (2006), Berlin: Springer, Berlin · Zbl 1104.65059
[15] Osborne, M. R., On shooting methods for boundary value problems, J. Math. Anal. Appl., 27, 417-433 (1969) · Zbl 0177.20402 · doi:10.1016/0022-247X(69)90059-6
[16] Pontryagin, L. S.; Boltyanski, V. G.; Gamkrelidze, R. V.; Miscenko, E. F., The Mathematical Theory of Optimal Processes (1962), Chichester: Wiley, Chichester · Zbl 0102.32001
[17] Potschka, A.: A direct method for the numerical solution of optimization problems with time-periodic PDE constraints. Ph.D. thesis, Universität Heidelberg (2011) · Zbl 1237.65062
[18] Potschka, A., A Direct Method for Parabolic PDE Constrained Optimization Problems (2013), Springer, Berlin: Advances in Numerical Mathematics, Springer, Berlin
[19] Potschka, A.; Bock, H. G.; Schlöder, J. P., A minima tracking variant of semi-infinite programming for the treatment of path constraints within direct solution of optimal control problems, Optim. Methods Softw., 24, 2, 237-252 (2009) · Zbl 1286.49005 · doi:10.1080/10556780902753098
[20] Potschka, A.; Mommer, M. S.; Schlöder, J. P.; Bock, H. G., Newton-Picard-based preconditioning for linear-quadratic optimization problems with time-periodic parabolic PDE constraints, SIAM J. Sci. Comput., 34, 2, 1214-1239 (2012) · Zbl 1266.65104 · doi:10.1137/100807776
[21] Russell, R. D.; Shampine, L. F., A collocation method for boundary value problems, Numer. Math., 19, 1-28 (1972) · Zbl 0221.65129 · doi:10.1007/BF01395926
[22] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), Philadelpha: SIAM, Philadelpha · Zbl 1031.65046 · doi:10.1137/1.9780898718003
[23] Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25, 2nd edn. Springer, Berlin (2006) · Zbl 1105.65102
[24] Tröltzsch, F., Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen (2009), Wiesbaden: Vieweg+Teubner Verlag, Wiesbaden · Zbl 1320.49001 · doi:10.1007/978-3-8348-9357-4
[25] Tsang, T. H.; Himmelblau, D. M.; Edgar, T. F., Optimal control via collocation and non-linear programming, Int. J. Control., 21, 763-768 (1975) · Zbl 0318.49028 · doi:10.1080/00207177508922030
[26] Tyson, R.; Lubkin, S. R.; Murray, J. D., Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Biol., 38, 359-375 (1999) · Zbl 0921.92005
[27] Tyson, R.; Lubkin, S. R.; Murray, J. D., A minimal mechanism for bacterial pattern formation, Proc. R. Soc. B Biol. Sci., 266, 299-304 (1999) · doi:10.1098/rspb.1999.0637
[28] Walther, A.; Kowarz, A.; Griewank, A., ADOL-C: a package for the automatic differentiation of algorithms written in C/C++ (2005), Technical report: Institute of Scientific Computing, Technical University Dresden, Technical report
[29] Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987). Translated from the German by C.B. Thomas and M.J. Thomas · Zbl 0623.35006
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