A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP

Andreas Rauh; Michael Brill; Clemens Günther

International Journal of Applied Mathematics and Computer Science (2009)

  • Volume: 19, Issue: 3, page 381-397
  • ISSN: 1641-876X

Abstract

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The theoretical background and the implementation of a new interval arithmetic approach for solving sets of differentialalgebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on VALENCIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial value problems for ordinary differential equations. For the application to DAEs, VALENCIA-IVP has been extended by an interval Newton technique to solve nonlinear algebraic equations in a guaranteed way. In addition to verified simulation of initial value problems for DAE systems, the developed approach is applicable to the verified solution of the so-called inverse control problems. In this case, guaranteed enclosures for valid input signals of dynamical systems are determined such that their corresponding outputs are consistent with prescribed time-dependent functions. Simulation results demonstrating the potential of VALENCIA-IVP for solving DAEs in technical applications conclude this paper. The selected application scenarios point out relations to other existing verified simulation techniques for dynamical systems as well as directions for future research.

How to cite

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Andreas Rauh, Michael Brill, and Clemens Günther. "A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP." International Journal of Applied Mathematics and Computer Science 19.3 (2009): 381-397. <http://eudml.org/doc/207943>.

@article{AndreasRauh2009,
abstract = {The theoretical background and the implementation of a new interval arithmetic approach for solving sets of differentialalgebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on VALENCIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial value problems for ordinary differential equations. For the application to DAEs, VALENCIA-IVP has been extended by an interval Newton technique to solve nonlinear algebraic equations in a guaranteed way. In addition to verified simulation of initial value problems for DAE systems, the developed approach is applicable to the verified solution of the so-called inverse control problems. In this case, guaranteed enclosures for valid input signals of dynamical systems are determined such that their corresponding outputs are consistent with prescribed time-dependent functions. Simulation results demonstrating the potential of VALENCIA-IVP for solving DAEs in technical applications conclude this paper. The selected application scenarios point out relations to other existing verified simulation techniques for dynamical systems as well as directions for future research.},
author = {Andreas Rauh, Michael Brill, Clemens Günther},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {ordinary differential equations; differential-algebraic equations; VALENCIA-IVP; verified simulation; inverse control problems; ValEncIA-IVP},
language = {eng},
number = {3},
pages = {381-397},
title = {A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP},
url = {http://eudml.org/doc/207943},
volume = {19},
year = {2009},
}

TY - JOUR
AU - Andreas Rauh
AU - Michael Brill
AU - Clemens Günther
TI - A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP
JO - International Journal of Applied Mathematics and Computer Science
PY - 2009
VL - 19
IS - 3
SP - 381
EP - 397
AB - The theoretical background and the implementation of a new interval arithmetic approach for solving sets of differentialalgebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on VALENCIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial value problems for ordinary differential equations. For the application to DAEs, VALENCIA-IVP has been extended by an interval Newton technique to solve nonlinear algebraic equations in a guaranteed way. In addition to verified simulation of initial value problems for DAE systems, the developed approach is applicable to the verified solution of the so-called inverse control problems. In this case, guaranteed enclosures for valid input signals of dynamical systems are determined such that their corresponding outputs are consistent with prescribed time-dependent functions. Simulation results demonstrating the potential of VALENCIA-IVP for solving DAEs in technical applications conclude this paper. The selected application scenarios point out relations to other existing verified simulation techniques for dynamical systems as well as directions for future research.
LA - eng
KW - ordinary differential equations; differential-algebraic equations; VALENCIA-IVP; verified simulation; inverse control problems; ValEncIA-IVP
UR - http://eudml.org/doc/207943
ER -

References

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  1. Auer, E., Rauh, A., Hofer, E. P. and Luther, W. (2008). Validated modeling of mechanical systems with S MART MOBILE: Improvement of Performance by VALENCIA-IVP, Proceedings of the Dagstuhl Seminar 06021-Reliable Implementation of Real Number Algorithms: Theory and Practice, Dagstuhl, Germany, Lecture Notes in Computer Science, Vol. 5045, Springer-Verlag, Berlin/Heidelberg, pp. 1-27. Zbl1165.65307
  2. Bendsten, C. and Stauning, O. (2007). FADBAD++, Version 2.1, Available at: http://www.fadbad.com. 
  3. Berz, M. and Makino, K. (1998). Verified integration of ODEs and flows using differential algebraic methods on highorder Taylor models, Reliable Computing 4(4): 361-369. Zbl0976.65061
  4. Cash, J. R. and Considine, S. (1992). An MEBDF code for stiff initial value problems, ACM Transactions on Mathematical Software (TOMS) 18(2): 142-155. Zbl0893.65049
  5. Chua, L., Desoer, C. A. and Kuh, E. S. (1990). Linear and Nonlinear Circuits, McGraw-Hill, New York, NY. Zbl0631.94017
  6. Czechowski, P. P., Giovannini, L. and Ordys, A. W. (2006). Testing algorithms for inverse simulation, Proceedings of the 2006 IEEE International Conference on Control Applications, Munich, Germany, pp. 2607-2612. 
  7. de Swart, J. J. B., Lioen, W. M. and van der Veen, W. A. (1998). Specification of PSIDE, Technical Report MAS-R9833, CWI, Amsterdam, Available at: http://walter.lioen.com/papers/SLV98.pdf. 
  8. Deville, Y., Janssen, M. and van Hentenryck, P. (2002). Consistency techniques for ordinary differential equations, Constraint 7(3-4): 289-315. Zbl1020.65035
  9. Eijgenraam, P. (1981). The solution of initial value problems using interval arithmetic, Mathematical Centre Tracts No. 144, Stichting Mathematisch Centrum, Amsterdam. Zbl0471.65043
  10. Galassi, M. (2006). GNU Scientific Library Reference Manual. Revised Second Edition (v1.8), Available at: http://www.gnu.org/software/gsl/. 
  11. Hairer, E., Lubich, C. and Roche, M. (1989). The Numerical Solution of Differential-Algebraic Systems by RungeKutta Methods, Lecture Notes in Mathematics, Vol. 1409, Springer-Verlag, Berlin. Zbl0683.65050
  12. Hairer, E. and Wanner, G. (1991). Solving Ordinary Differential Equations II-Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin/Heidelberg. Zbl0729.65051
  13. Hammersley, J. M. and Handscomb, D. C. (1964). Monte-Carlo Methods, John Wiley & Sons, New York, NY. Zbl0121.35503
  14. Hoefkens, J. (2001). Rigorous Numerical Analysis with High-Order Taylor Models, Ph.D. thesis, Michigan State University, East Lansing, MI, Available at: http://www.bt.pa.msu.edu/cgi-bin/display.pl?name=hoefkensphd. 
  15. Iavernaro, F. and Mazzia, F. (1998). Solving ordinary differential equations by generalized Adams methods: Properties and implementation techniques, Applied Numerical Mathematics 28(2): 107-126. Zbl0926.65076
  16. Jaulin, L., Kieffer, M., Didrit, O. and Walter, É. (2001). Applied Interval Analysis, Springer-Verlag, London. Zbl1023.65037
  17. Keil, C. (2007). Profil/BIAS, Version 2.0.4, Available at: www.ti3.tu-harburg.de/keil/profil/. 
  18. Krawczyk, R. (1969). Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing 4(3): 189-201, (in German). Zbl0187.10001
  19. Kunkel, P., Mehrmann, V., Rath, W. and Weickert, J. (1997). GELDA: A Software Package for the Solution of General Linear Differential Algebraic Equations, pp. 115-138, Available at: http://www.math.tu-berlin.de/numerik/mt/NumMat/Software/GELDA/. Zbl0868.65041
  20. Lin, Y. and Stadtherr, M. A. (2007). Deterministic global optimization for dynamic systems using interval analysis, CD-Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, IEEE Computer Society, Los Alamitos, CA. 
  21. Moore, R. E. (1966). Interval Arithmetic, Prentice-Hall, Englewood Cliffs, New Jersey, NY. 
  22. Nedialkov, N. S. (2007). Interval tools for ODEs and DAEs, CD-Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, IEEE Computer Society, Los Alamitos, CA. 
  23. Nedialkov, N. S. and Pryce, J. D. (2008). DAETS-DifferentialAlgebraic Equations by Taylor Series, Available at: http://www.cas.mcmaster.ca/~nedialk/daets/. Zbl1188.65111
  24. Petzold, L. (1982). A description of DASSL: A differential/algebraic systems solver, IMACS Transactions on Scientific Computation 1: 65-68. 
  25. Rauh, A. (2008). Theorie und Anwendung von Intervallmethoden für Analyse und Entwurf robuster und optimaler Regelungen dynamischer Systeme, FortschrittBerichte VDI, Reihe 8, Nr. 1148, Ph.D. thesis, University of Ulm, Ulm, (in German). 
  26. Rauh, A. and Auer, E. (2008). Verified simulation of ODEs and DAEs in VALENCIA-IVP, Proceedings of the 13th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2008, El Paso, TX, USA, (under review). 
  27. Rauh, A., Auer, E., Freihold, M., Hofer, E. P. and Aschemann, H. (2008). Detection and reduction of overestimation in guaranteed simulations of hamiltonian systems, Proceedings of the 13th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2008, El Paso, TX, USA, (under review). 
  28. Rauh, A., Auer, E. and Hofer, E. P. (2007a). VALENCIA-IVP: A comparison with other initial value problem solvers, CD-Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, IEEE Computer Society, Los Alamitos, CA. 
  29. Rauh, A., Auer, E., Minisini, J. and Hofer, E. P. (2007b). Extensions of VALENCIA-IVP for reduction of overestimation, for simulation of differential algebraic systems, and for dynamical optimization, Proceedings of the 6th International Congress on Industrial and Applied Mathematics, Minisymposium on Taylor Model Methods and Interval Methods-Applications, PAMM, Zurich, Switzerland, Vol. 7(1), pp. 1023001-1023002. 
  30. Rauh, A. and Hofer, E. P. (2009). Interval methods for optimal control, in A. Frediani and G. Buttazzo (Eds.), Proceedings of the 47th Workshop on Variational Analysis and Aerospace Engineering 2007, Erice, Italy, Springer-Verlag, New York, NY, pp. 397-418. Zbl1182.49024
  31. Rauh, A., Minisini, J. and Hofer, E. P. (2009). Towards the development of an interval arithmetic environment for validated computer-aided design and verification of systems in control engineering, Proceedings of the Dagstuhl Seminar 08021: Numerical Validation in Current Hardware Architectures, Dagstuhl, Germany, Lecture Notes in Computer Science, Vol. 5492, Springer-Verlag, Berlin/Heidelberg, pp. 175-188. 
  32. Röbenack, K. (2002). On the efficient computation of higher order maps using Taylor arithmetic and the Campbell-Baker-Hausdorff formula, in A. Zinober and D. Owens (Eds.), Nonlinear and Adaptive Control, Lecture Notes in Control and Information Science, Vol. 281, Springer, London, pp. 327-336. Zbl1157.93378

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