Structure of Optimal Samples in Continuous Nonlinear Experimental Design for Parameter Estimation

Conference paper

Abstract

In the continuous case, Optimal Experimental Design (OED) deals with designs that are described by probability distributions or samples over the experimental domain. An optimal design may correspond to a distribution having finite or infinite support or being continuous. In this paper, the structure of optimal samples for experimental designs is elucidated. It is shown that any design is in fact equivalent to a design with a finite number of support points. The lower bound and upper bound of this number, especially for optimal designs, are given and examples indicate their sharpness. Moreover, we propose an algorithm to construct optimal designs which have finite support. Several applications to OED for dynamic systems with inputs are also discussed.

Notes

Acknowledgements

The authors would like to thank anonymous reviewers for their constructive comments. The first author’s research is funded by the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences.

References

  1. 1.
    Bock, H.G., Körkel, S., Kostina, E., Schlöder, J.P.: Robustness aspects in parameter estimation, optimal design of experiments and optimal control. In: Jäger, W., Rannacher, R., Warnatz, J. (eds.) Reactive Flows, Diffusion and Transport, pp. 117–146. Springer, Berlin (2007)CrossRefGoogle Scholar
  2. 2.
    Bock, H.G., Körkel, S., Schlöder, J. P.: Parameter estimation and optimal experimental design for nonlinear differential equation models. In Bock, H.G., Carraro, T., Jäger, W., Körkel, S., Rannacher, R., Schlöder, J.P. (eds.) Model Based Parameter Estimation: Theory and Application, vol. 4, pp. 1–30. Springer, Berlin (2013)CrossRefGoogle Scholar
  3. 3.
    Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47, 99–131 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Körkel, S.: Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen. PhD thesis, Universität Heidelberg (2002)Google Scholar
  5. 5.
    Pázman, A.: Foundations of Optimum Experimental Design. D. Reidel Publishing Company, Dordrecht (1986)MATHGoogle Scholar
  6. 6.
    Pukelsheim, F.: Optimal Designs of Experiments. SIAM, Philadelphia (2006)CrossRefMATHGoogle Scholar
  7. 7.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  8. 8.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)MATHGoogle Scholar
  9. 9.
    Sager, S.: Sampling decisions in optimal experimental design in the light of Pontryagin’s maximum principle. SIAM J. Control Optim. 51(4), 3181–3207 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Interdisciplinary Center for Scientific Computing (IWR)HeidelbergGermany

Personalised recommendations