Skip to main content
SpringerLink
Log in

An Efficient Optimization Approach for Computationally Expensive Timesteppers Using Tabulation

  • Chapter
Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena

Summary

A methodology is outlined for the efficient solution of dynamic optimization problems when the system evolution is described by computationally expensive timestepper-based models. The computational requirements issue is circumvented by extending the notion of in situ adaptive tabulation to stochastic systems. Conditions are outlined that allow unbiased estimation of the mapping gradient matrix and, subsequently, expressions to compute the ellipsoid of attraction are derived. The proposed approach is applied towards the solution of two representative dynamic optimization problems, (a) a bistable reacting system describing catalytic oxidation of CO and, (b) a homogeneous chemically reacting system describing dimerization of a monomer. In both cases, tabulation resulted in significant reduction in the solution time of the optimization problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Armaou, I.G. Kevrekidis: Equation-free optimal switching policies using coarse time-steppers. International Journal of Robust and Nonlinear Control, in press (2005)

    Google Scholar 

  2. A. Armaou, C.I. Siettos, I.G. Kevrekidis: Time-steppers and ‘coarse’ control of distributed microscopic processes. Int. J. Robust & Nonlin. Contr. 14, 89–111 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. A.J. Booker, J.E. Dennis, P.D. Frank, D.B. Serafini, V. Torczon, M.W. Trosset: A rigorous framework for optimization of expensive functions by surrogates. NASA/CR-1998-208735, pages 1–19 (1998)

    Google Scholar 

  4. L. Dai: Rate of convergence for derivative estimation of discrete-time Markov chains via finite-difference approximation with common randon numbers. SIAM J. Appl. Math. 57, 731–751 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. W.F. Feehery, P.I. Barton: Dynamic optimization with equality path constraints. Ind. Eng. Chem. Res. 38, 2350–2363 (1999)

    Article  Google Scholar 

  6. C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg, K. Theodoropoulos, I.G. Kevrekidis: Equation-free multiscale computation: enabling microscopic simulators to perform system-level tasks. Comm. Math. Sciences 1, 715–762 (2003)

    MATH  MathSciNet  Google Scholar 

  7. D.T. Gillespie: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phys. 22, 403–34 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  8. D.T. Gillespie: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115, 1716–1733 (2001)

    Article  ADS  Google Scholar 

  9. P. Glasserman, D.D. Yao: Some guidelines and guarantees for common random numbers. Management Science 38, 884–908 (1992)

    Article  MATH  Google Scholar 

  10. P.W. Glynn: Likelihood ration gradient estimation for stochastic systems. Communications of the ACM 33, 76–84 (1990)

    Article  Google Scholar 

  11. P.W. Glynn, P. L’ecuyer: Likehood ratio gradient estimation for stochastic recursions. Adv. Appl. Prob. 27, 1019–1053 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Heidelberger, X.R. Cao, M.A. Zazanis, R. Suri: Convergence properties of infinitesimal perturbation analysis estimates. Management Science 34, 1281–1302 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Y.C. Ho: Performance evaluation and perturbation analysis of discrete event dynamic systems. IEEE Transactions on Automatic Control AC-32, 563–572 (1987)

    Google Scholar 

  14. D.R. Jones, M. Schonlau, W.J. Welch: Efficient global optimization of expensive black-box functions. J. Global Opt. 13, 455–492 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. C.T. Kelley, E.W. Sachs: Solution of optimal control problems by a pointwise projected newton method. SIAM Journal on Control and Optimization 33(6), 1731 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. T.G. Kolda, R.M. Lewis, V. Torczon: Optimization by direct search: New perspectives on some classical and modern methods. SIAM Review 45, 385–482 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. P. L’Ecuyer: A unified view of the IPA, SF and LR gradient estimation techniques. Management Science 36, 1364–1383 (1990)

    Article  MATH  Google Scholar 

  18. P. L’Ecuyer, G. Perron: On the convergence rates of IPA and FDC derivative estimators. Operations Research 42, 643–656 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. R.M. Lewis, V. Torczon: Pattern search algorithms for bound constrained minimization. SIAM J. Optimization 9, 1082–1099 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. R.M. Lewis, V. Torczon: Pattern search algorithms for linearly constrained minimization. SIAM J. Optimization 10, 917–947 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Li, D. Liao, S. Yip: Nearly exact solution for coupled continuum/MD fluid simulation. J. Comp. Mat. Des. 6, 95–102 (1999)

    Google Scholar 

  22. Y. Lou, P.D. Christofides: Estimation and control of surface roughness in thin film growth using kinetic Monte-Carlo methods. Chem Eng. Sci. 58, 3115–3129 (2003)

    Article  Google Scholar 

  23. H.H. McAdams, A. Arkin: Stochastic mechanisms in gene expression. Proc. Natl. Acad. Sci. 94, 814–819 (1997)

    Article  ADS  Google Scholar 

  24. C.A. Meyer, C.A. Floudas, A. Neumaier: Global optimization with nonfactorable constraints. Ind. Eng. Chem. Res. 41, 6413–6424 (2002)

    Article  Google Scholar 

  25. S.B. Pope: Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation. Combust. Theory & Modelling 1, 41–63 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. C.V. Rao, A.P. Arkin: Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. J. Chem. Phys. 118, 4999–5010 (2003)

    Article  ADS  Google Scholar 

  27. M.I. Reiman, A. Weiss: Sensitivity analysis of simulations via likelihood ratios. Operations Research 37, 830–844 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  28. C.I. Siettos, A. Armaou, A.G. Makeev, I.G. Kevrekidis: Microscopic/stochastic timesteppers and coarse control: a kinetic Monte Carlo example. AIChE J. 49, 1922–1926 (2003)

    Article  Google Scholar 

  29. R. Srivastava, L. You, J. Yin: Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theor. Biol. 218, 309–321 (2002)

    Article  MathSciNet  Google Scholar 

  30. R. Suri: Perturbation analysis: The state of the art and research issues explained via the GI/G/l queue. Proc. IEEE 77, 114–137 (1989)

    Article  Google Scholar 

  31. E.B. Tadmor, G.S. Smith, N. Bernstein, E. Kaxiras: Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B 59, 235–245 (1999)

    Article  ADS  Google Scholar 

  32. V. Torczon: On the convergence of pattern search algorithms. SIAM J. Optimization 7, 1–25 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. A. Varshney, A. Armaou: Multiscale optimization of thin-film growth using hybrid PDE/kMC process systems. Chem. Eng. Sci. 60 6780–6794 (2005)

    Article  Google Scholar 

  34. S. Vasantharajan, J. Viswanathan, L.T. Biegler: Reduced successive quadratic programming implementation for large-scale optimization problems with smaller degrees of freedom. Comp. Chem. Eng. 14, 907–915 (1990)

    Article  Google Scholar 

  35. V.S. Vassiliadis, R.W.H. Sargent, C.C. Pantelides: Solution of a class of multistage dynamic optimization problems, parts I & II. Ind. & Eng. Chem. Res. 33, 2111–2133 (1994)

    Article  Google Scholar 

  36. M.A. Zazanis, R. Suri: Convergence rates of finite-difference sensitivity estimates for stochastic systems. Operations Research 41, 694–703 (1993)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Pennsylvania State University, University Park, PA, 16802, USA

    A. Varshney & A. Armaou

Authors
  1. A. Varshney
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Armaou
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Alexander N. Gorban

  2. Institute of Computational Modeling, Russian Academy of Sciences, Russia

    Alexander N. Gorban

  3. Department of Chemical Engineering, Princeton University, Engineering Quadrangle A-217, Princeton, NJ, 08544-5263, USA

    Ioannis G. Kevrekidis

  4. School of Chemical Engineering and Analytical Science, University of Manchester, PO Box 88, Sackville St., Manchester, M60 1QD, UK

    Constantinos Theodoropoulos

  5. Department of Chemical Engineering, Worcester Polytechnic Institute, Institute Road 100, Worcester, MA, 01609-2280, USA

    Nikolaos K. Kazantzis

  6. Institut für Polymere, ETH Zürich, Wolfgang Pauli-Straße 10, CH-8093, Zürich, Switzerland

    Hans Christian Öttinger

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Varshney, A., Armaou, A. (2006). An Efficient Optimization Approach for Computationally Expensive Timesteppers Using Tabulation. In: Gorban, A.N., Kevrekidis, I.G., Theodoropoulos, C., Kazantzis, N.K., Öttinger, H.C. (eds) Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35888-9_23

Download citation

Publish with us

Policies and ethics

Search

Navigation