Numerical Algorithms

, Volume 58, Issue 2, pp 261–292 | Cite as

A new algorithm for index determination in DAEs using algorithmic differentiation

Original Paper

Abstract

We present an approach for determining the tractability index using truncated polynomial arithmetic. In particular, computing the index this way generates a sequence of matrices that contains itself derivatives. We implement the time differentiations using algorithmic differentiation techniques, specially using the standard ADOL-C package, with which calculating the derivatives becomes a simple shift and scaling of coefficients. We present the theory supporting the procedure we propose, as well as the implementation issues behind it to provide a convenient interface to the standard ADOL-C functionality. We give also examples of academic and practical problems and report several experimental results we have obtained with them.

Keywords

DAE Index determination Tractability index AD Algorithmic differentiation Automatic differentiation 

AMS 2000 Subject Classifications

65L80 65D25 68W30 

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References

  1. 1.
    Abram, E.: Netzwerkbasierte Analyse von elektromechanischen DAE-Systemen. Diplomarbeit, Institut für Mathematik, Technische Universität Berlin (2008)Google Scholar
  2. 2.
    Bendtsen, C., Stauning, O.: FADBAD, a flexible C+ + package for Automatic Differentiation. Tech. Rep. IMM–REP–1996–17, IMM, Dept. of Mathematical Modelling, Technical University of Denmark (1996)Google Scholar
  3. 3.
    Campbell, S.L.: A general form for solvable linear time varying singular systems of differential equations. SIAM J. Math. Anal. 18(4), 1101–1115 (1987)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Campbell, S.L., Gear, C.: The index of general nonlinear DAEs. Numer. Math. 72(2), 173–196 (1995)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    De Luca, A., Isidori, A.: Feedback linearization of invertible systems. In: 2nd Colloq. Aut. & Robots, Duisburg (1987)Google Scholar
  6. 6.
    Duff, I., Gear, C.: Computing the structural index. SIAM J. Algebr. Discrete Methods 7, 594–603 (1986)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore, MD, USA (1996)MATHGoogle Scholar
  8. 8.
    Griewank, A., Walter, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd edn. SIAM (2008)Google Scholar
  9. 9.
    Günther, M., Feldmann, U.: CAD based electric circuit modeling in industry I: mathematical structure and index of network equations. Surv. Math. Ind. 8, 97–129 (1999)MATHGoogle Scholar
  10. 10.
    Higueras, I., März, R.: Differential algebraic equations with properly stated leading terms. Comput. Math. Appl. 48(1–2), 215–235 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    König, D.: Indexcharakterisierung bei nichtlinearen Algebro-Differentialgleichungen. Master’s thesis, Institut für Mathematik, Humboldt-Universität zu Berlin (2006)Google Scholar
  12. 12.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations - Analysis and Numerical Solution. EMS Publishing House, Zürich, Switzerland (2006)MATHCrossRefGoogle Scholar
  13. 13.
    Kunkel, P., Mehrmann, V., Seufer, I.: GENDA – Homepage, http://www.math.tu-berlin.de/numerik/mt/NumMat/Software/GENDA/
  14. 14.
    Lamour, R.: Index determination and calculation of consistent initial values for DAEs. Comput. Math. Appl 50, 1125–1140 (2005)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Lamour, R., März, R., Tischendorf, C.: Projector-Based DAE Analysis. (2011, in preparation)Google Scholar
  16. 16.
    Lamour, R., Mattheij, R., März, R.: On the Stability Behaviour of Systems obtained by Index Reduction. J. Comput. Appl. Math. 56, 305–319 (1994)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Lamour, R., Monett, D.: Index determination of DAEs – a wide field for automatic differentiation. In: Simos, T., Psihoyios, G., Tsitouras, C. (eds.) International Conference on Numerical Analysis and Applied Mathematics. AIP Conference Proceedings 1168, vol. 2, pp. 727–730. Rethymno, Crete, Greece (2009)Google Scholar
  18. 18.
    Lamour, R., Mazzia, F.: Computation of consistent initial values for properly stated index 3 DAEs. BIT 49(1), 161–175 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    März, R.: The index of linear differential algebraic equations with properly stated leading terms. In: Result. Math., vol. 42, pp. 308–338. Birkhäuser Verlag, Basel (2002)Google Scholar
  20. 20.
    März, R.: Differential algebraic systems with properly stated leading term and MNA equations. In: Antreich, K., Bulirsch, R., Gilg, A., Rentrop, P. (eds.) Modeling, Simulation and Optimization of Integrated Circuits, International Series of Numerical Mathematics, vol. 146, pp. 135–151. Birkhäuser Verlag, Basel (2003)CrossRefGoogle Scholar
  21. 21.
    Monett, D., Lamour, R., Griewank, A.: Index determination in DAEs using the library indexdet and the ADOL–C package for algorithmic differentiation. In: Bischof, C.H., Bücker, H.M., Hovland, P., Naumann, U., Utke, J. (eds.) Advances in Automatic Differentiation. Lecture Notes in Computational Science and Engineering, vol. 64, pp. 247–257. Springer, Berlin (2008)CrossRefGoogle Scholar
  22. 22.
    Nedialkov, N., Pryce, J.: Solving Differential-Algebraic Equations by Taylor Series (I): Computing Taylor coefficients. BIT Numerical Mathematics 45, Springer (2005)Google Scholar
  23. 23.
    Nedialkov, N., Pryce, J.: Solving Differential-Algebraic Equations by Taylor Series (II): Computing the System Jacobian. BIT Numerical Mathematics 47, Springer (2007)Google Scholar
  24. 24.
    Nedialkov, N., Pryce, J.: Solving Differential-Algebraic Equations by Taylor Series (III): the DAETS Code. J. Numer. Anal. Indust. Appl. Math. 1(1), Springer (2007)Google Scholar
  25. 25.
    Pantelides, C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Statist. Comput. 9(2), 213–231 (1988)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Pryce, J.D.: A simple structural analysis method for DAEs. BIT 41(2), 364–294 (2001)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Reissig, G., Martinson, W., Barton, P.I.: Differential-algebraic equations of index 1 may have an arbitrarily high structural index. SIAM J. Sci. Comput. 21(6), 1987–1990 (2000)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Tischendorf, C.: Coupled systems of differential algebraic and partial differential equations in circuit and device simulation. Modeling and numerical analysis. Habilitationsschrift, Institut für Mathematik, Humboldt-Universität zu Berlin (2004)Google Scholar
  29. 29.
    Walther, A., Griewank, A.: ADOL-C: A Package for the Automatic Differentiation of Algorithms Written in C/C+ +, Version 2.0.0 (2008)Google Scholar
  30. 30.
    Wente, C.: Tools for automatic differentiation, community portal for automatic differentiation. http://www.autodiff.org/?module=Tools

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of MathematicsHumboldt-University of BerlinBerlinGermany
  2. 2.Computer Science Department and Department of Cooperative StudiesBerlin School of Economics and LawBerlinGermany

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