Sensitivity analysis of ODEs/DAEs using the Taylor series method

SIAM Journal on Scientific Computing

Volumn 27, Issue 6, 2006, Pages 1929-1947

  Barrio, Roberto ^a  

^a UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

Automatic differentiation; Partial derivatives; Sensitivity analysis; Taylor method

Indexed keywords

COMPUTATIONAL METHODS; DIFFERENTIATION (CALCULUS); NUMERICAL METHODS; SENSITIVITY ANALYSIS;

AUTOMATIC DIFFERENTIATION; PARTIAL DERIVATIVES; TAYLOR METHODS; TAYLOR SERIES METHODS;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 33751249935     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/030601892     Document Type: Article

References (43)

1. R. BARRIO, Performance of the Taylor series method for ODEs/DAEs, Appl. Math. Comput., 163 (2005), pp. 525-545.
2. R. BARRIO, Sensitivity tools vs. Poincaré sections, Chaos Solitons Fractals, 25 (2005), pp. 711-726.
3. R. BARRIO, F. BLESA, AND M. LARA, VSVO formulation of the Taylor method for the numerical solution of ODEs, Comput. Math. Appl., 50 (2005), pp. 93-111.
4. M. BERZ, The method of power series trucking for the mathematical description of beam dynamics, Nuclear Inst. Methods, A258 (1987), pp. 431-436.
5. M. BERZ, Differential algebraic description of beam dynamics to very high orders, Particle Accelerators, 24 (1989), pp. 109-124.
6. D. BARTON, On Taylor series and stiff equations, ACM Trans. Math. Software, 6 (1980), pp. 280-294.
7. C. BISCHOF, A. CARLE, P. KHADEMI, AND A. MAUER, ADIFOR 2.0: Automatic differentiation of Fortran 77 programs, IEEE Comput. Sci. Engrg., 3 (1996), pp. 18-32.
8. Y. CAO, S. LI, AND L. PETZOLD, Adjoint sensitivity analysis for differential-algebraic equations: Algorithms and software, J. Comput. Appl. Math., 125 (2002), pp. 131-145.
9. Y. CAO, S. LI, L. PETZOLD, AND R. SERBAN, Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution, SIAM J. Sci. Comput., 24 (2003), pp. 1076-1089.
10. R. CARRETERO-GONZÁLEZ, H. N. NÚÑEZ-YÉPEZ, AND A. L. SALAS-BRITO, Regular and chaotic behaviour in extensible pendulum, European J. Phys., 15 (1994), pp. 139-148.
11. Y. F. CHANG AND G. F. CORLISS, ATOMFT: Solving ODEs and DAEs using Taylor series, Comput. Math. Appl., 28 (1994), pp. 209-233.
12. W. G. CHOE AND J. GUCKENHEIMER, Computing periodic orbits with high accuracy, Comput. Methods Appl. Mech. Engrg., 170 (1999), pp. 331-341.
13. G. F. CORLISS AND Y. F. CHANG, Solving ordinary differential equations using Taylor series, ACM Trans. Math. Software, 8 (1982), pp. 114-144.
14. G. F. CORLISS AND Y. F. CHANG, G-stop facility in ATOMFT, a Taylor series ordinary differential equation solver, in Computational Ordinary Differential Equations, S. O. Fatunla, ed., University Press, Champaign, IL, 1992, pp. 37-77.
15. G. F. CORLISS AND G. KIRLINGER, On implicit Taylor series methods for stiff ODEs, in Computer Arithmetic and Enclosure Methods (Oldenburg, 1991), North-Holland, Amsterdam, 1992, pp. 371-379.
16. P. EBERHARD AND C. BISCHOF, Automatic differentiation of numerical integration algorithms, Math. Comp., 68 (1999), pp. 717-731.
17. W. F. FEEHERY, J. E. TOLSMA, AND P. I. BARTON, Efficient sensitivity analysis of large-scale differential-algebraic systems, Appl. Numer. Math., 25 (1997), pp. 41-54.
18. M. FOUCHARD, E. LEGA, C. FROESCHLÉ, AND C. FROESCHLÉ, On the relationship between fast Lyapunov indicator and periodic orbits for continuous flows, Celestial Mech. Dynam. Astronom., 83 (2002), pp. 205-222.
19. A. GRIEWANK, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Frontiers Appl. Math. 19, SIAM, Philadelphia, PA, 2000.
20. A. GRIEWANK, J. UTKE, AND A. WALTHER, Evaluating higher derivative tensors by forward propagation of univariate Taylor series, Math. Comp., 69 (2000), pp. 1117-1130.
21. J. GUCKENHEIMER AND B. MELOON, Computing periodic orbits and their bifurcations with automatic differentiation, SIAM J. Sci. Comput, 22 (2000), pp. 951-985.
22. E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving Ordinary Differential Equations I, Nonstiff Problems, 2nd ed, Springer Ser. Comput. Math. 8, Springer-Verlag, New York, 1993.
23. E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II. Stiff and Differential Algebraic Problems, Springer Ser. Comput. Math. 14, Springer-Verlag, New York, 1991.
24. E. HAIRER AND G. WANNER, Stiff differential equations solved by Radau methods, J. Comput. Appl. Math., 111 (1999), pp. 93-111.
25. M. HÉNON AND C. HEILES, The applicability of the third integral of motion: Some numerical experiments, Astron. J., 1 (1964), pp. 73-79.
26. J. HOEFKENS. M. BERZ, AND K. MAKING, Computing validated solutions of implicit differential equations, Adv. Comput. Math., 19 (2003), pp. 231-253.
27. P. HOVLAND, C. BISCHOF, D. SPIEGELMAN, AND M. CASELLA, Efficient derivative codes through automatic differentiation and interface contraction: An application in biostatistics, SIAM J. Sci. Comput., 18 (1997), pp. 1056-1066.
28. A. JORBA AND M. ZOU, A software package for the numerical integration of ODE by means of high-order Taylor methods, Experiment. Math., 14 (2005), pp. 99-117.
29. M. LARA, A. DEPRIT, AND A. ELIPE, Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory, Celestial Mech. Dynam. Astronom., 62 (1995), pp. 167-181.
30. M. LARA, A. ELIPE, AND M. PALACIOS, Automatic programming of recurrent power series, Math. Comput. Simulation, 49 (1999), pp. 351-362.
31. S. LI, L. PETZOLD, AND W. ZHU, Sensitivity analysis of differential-algebraic equations: A comparison of methods on a special problem, Appl. Numer. Math., 32 (2000), pp. 161-174.
32. W. M. LIOEN AND J. J. B. DE SWART, Test Set for Initial Value Problem Solvers, Release 2.1, 1999; available online from http://ftp.cwi.nl/IVPtestset/ report/testset.pdf.
33. K. MAKING AND M. BERZ, Remainder differential algebras and their applications, in Computational Differentiation (Santa Fe, NM, 1996), SIAM, Philadelphia, PA, 1996, pp. 63-74.
34. R. E. MOORE, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.
35. N. S. NEDIALKOV, K. R. JACKSON, AND G. F. CORLISS, Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput., 105 (1999), pp. 21-68.
36. N. S. NEDIALKOV AND J. D. PRYCE, Solving differential-algebraic equations by Taylor series (I): Computing Taylor coefficients, BIT, 45 (2005), pp. 561-591.
37. R. NEIDINGER, An efficient method for the numerical evaluation of partial derivatives of arbitrary order, ACM Trans. Math. Software, 18 (1992), pp. 159-173.
38. P. J. PRINCE AND J. R. DORMAND, High order embedded Runge-Kutta formulae, J. Comput. Appl. Math., 7 (1981), pp. 67-75.
39. J. D. PRYCE, Solving high-index DAEs by Taylor series, Numer. Algorithms, 19 (1998), pp. 195-211.
40. J. D. PRYCE, A simple structural analysis method for DAEs, BIT, 41 (2001), pp. 364-394.
41. L. B. RALL, Automatic Differentiation: Techniques and Applications, Lecture Notes in Comput. Sci. 120, Springer-Verlag, Berlin, 1981.
42. A. SANDU, G. R. CARMICHAEL, AND F. A. POTRA, Sensitivity analysis for atmospheric chemistry models via automatic differentiation, Atmos. Environ., 31 (1997), pp. 475-489.
43. 1-Lohner algorithm, Found. Comput. Math., 2 (2002), pp. 429-465.