Abstract
Optimum experimental design (OED) for parameter identification has become a key technique in the model validation process for dynamical systems. This paper deals with optimum experimental design for systems modelled by differential-algebraic equations. We show how to formulate OED as a nonstandard nonlinear optimal control problem. The direct multiple shooting method is a state of the art method for the solution of standard optimal control problems that leads to structured nonlinear programs. We present two possibilities how to adapt direct multiple shooting to OED by introducing additional variables and constraints. We highlight special structures in the constraint and objective derivatives whose evaluation is usually the bottleneck when solving dynamic optimization problems by multiple shooting. We have implemented a structure exploiting algorithm that takes all these structures into account. Two benchmark examples show the efficiency of the new algorithm.
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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)
Atkinson, A.C., Donev, A.: Optimum Experimental Designs. Oxford Statistical Sciences Series, vol. 8. Oxford University Press, Oxford (1992)
Bauer, I., Bock, H.G., Schlöder J.P.: DAESOL – a BDF-code for the numerical solution of differential algebraic equations. Internal Report, IWR, SFB 359, Universität Heidelberg (1999)
Bauer, I., Bock, H.G., Körkel, S., Schlöder, J.P.: Numerical methods for optimum experimental design in DAE systems. J. Comput. Appl. Math. 120(1–2), 1–15 (2000)
Bock, H.G.: Numerical treatment of inverse problems in chemical reaction kinetics. In: Ebert, K.H., Deuflhard, P., Jäger, W. (eds.) Modelling of Chemical Reaction Systems. Springer Series in Chemical Physics, vol. 18, pp. 102–125. Springer, Heidelberg (1981)
Bock, H.G.: Randwertproblemmethoden zur parameteridentifizierung in systemen nichtlinearer differentialgleichungen. Bonner Mathematische Schriften, vol. 183. Universität Bonn, Bonn (1987)
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. Pergamon Press, Budapest (1984). Available at http://www.iwr.uni-heidelberg.de/groups/agbock/FILES/Bock1984.pdf
Bock, H.G. Eich, E., Schlöder. J.P.: Numerical solution of constrained least squares boundary value problems in differential-algebraic equations. In: Strehmel, K. (ed.) Numerical Treatment of Differential Equations. Proceedings of the NUMDIFF-4 Conference, Halle-Wittenberg, 1987. Texte zur Mathematik, vol. 104, pp. 269–280. Teubner, Leipzig (1988)
Contreras, M., Tapia, R.A.: Sizing the BFGS and DFP updates: numerical study. J. Optim. Theory Appl. 78(1), 93–108 (1993)
Fedorov, V.V.: Theory of Optimal Experiments. Elsevier, Amsterdam (1972)
Ferreau, H.J., Kirches, C., Potschka, A., Bock, H.G., Diehl, M.: qpOASES: A parametric active-set algorithm for quadratic programming. Math. Program. Comput. 6(4), 327–363 (2014)
Franceschini, G., Macchietto, S.: Model-based design of experiments for parameter precision: state of the art. Chem. Eng. Sci. 63, 4846–4872 (2008)
Hoang, M.D., Barz, T., Merchan, V.A., Biegler, L.T., Arellano-Garcia, H.: Simultaneous solution approach to model-based experimental design. AIChE J. 59(11), 4169–4183 (2013)
Janka, D.: Optimum experimental design and multiple shooting. Master’s thesis, Universität Heidelberg, Heidelberg (2010)
Körkel, S.: Numerische methoden für optimale versuchsplanungsprobleme bei nichtlinearen DAE-modellen. Ph.D. thesis, Universität Heidelberg, Heidelberg (2002)
Körkel, S., Potschka, A., Bock, H.G., Sager, S.: A multiple shooting formulation for optimum experimental design. Math. Program. (2012, submitted revisions)
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)
Pukelsheim, F.: Optimal design of experiments. In: Classics in Applied Mathematics, vol. 50. SIAM, Philadelphia (2006). ISBN:978-0-898716-04-7.
Sager, S.: MIOCP benchmark site. (2014) http://mintoc.de
Sager, S.: Sampling decisions in optimum experimental design in the light of Pontryagin’s maximum principle. SIAM J. Control. Optim. 51(4), 3181–3207 (2013)
Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim. 16(1), 1–31 (2005)
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Janka, D., Körkel, S., Bock, H.G. (2015). Direct Multiple Shooting for Nonlinear Optimum Experimental Design. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds) Multiple Shooting and Time Domain Decomposition Methods. Contributions in Mathematical and Computational Sciences, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-23321-5_4
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DOI: https://doi.org/10.1007/978-3-319-23321-5_4
Publisher Name: Springer, Cham
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