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Rigorous integration of non-linear ordinary differential equations in Chebyshev basis. (English) Zbl 1325.65101
This paper shows how to compute enclosures for the solution of initial value problems using Chebyshev expansions rather than a Taylor-like approach. The method relies on the development of an algorithm that computes the multiplication of two function enclosures in truncated Chebyshev form. A Picard-type iteration scheme is used to obtain the enclosures and in order to integrate over larger time windows, then several steps of integration are needed in which the so-called wrapping effect can be suppressed. The approach seems to allow for higher precision with lower-order approximations than in the Taylor model. However, there are still issues with stiffness and even modest sized systems that this work does not address.

65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
Full Text: DOI
[1] Auer, E; Rauh, A, Vericomp: a system to compare and assess verified ivp solvers, Computing, 94, 163-172, (2012) · Zbl 1238.65064
[2] Battles, Z; Trefethen, LN, An extension of MATLAB to continuous functions and operators, SIAM J. Sci. Comput., 25, 1743-1770, (2004) · Zbl 1057.65003
[3] Berz, M; Makino, K, Suppression of the wrapping effect by Taylor model-based verified integrators: long-term stabilization by shrink wrapping, Int. J. Differ. Equ. Appl., 10, 385-403, (2005) · Zbl 1133.65045
[4] Boyd, J.P.: Chebyshev and Fourier Spectral Methods, p. 688. Courier Dover Publications, New York (2001) · Zbl 0994.65128
[5] Brisebarre, N., Joldeş, M.: Chebyshev interpolation polynomial-based tools for rigorous computing. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC ’10, pp. 147-154. ACM, New York (2010) · Zbl 1321.65018
[6] Camacho, C; De Figueiredo, LH, The dynamics of the jouanolou foliation on the complex projective 2-space, Ergodic Theory Dyn. Syst., 5, 757-766, (2001) · Zbl 0995.37031
[7] Clenshaw, CW, A note on the summation of Chebyshev series, MTAC, 9, 118-120, (1955) · Zbl 0065.05403
[8] Corliss, G. F.: Guaranteed error bounds for ordinary differential equations. In: Theory of Numerics in Ordinary and Partial Differential Equations, pp. 1-75. Oxford University Press (1994) · Zbl 0843.65060
[9] De Figueiredo, LH; Stolfi, J, Affine arithmetic: concepts and applications, Numer. Algorithm., 37, 147-158, (2004) · Zbl 1074.65050
[10] Dekker, T, A floating-point technique for extending the available precision, Numer. Math., 18, 224-242, (1971/72) · Zbl 0226.65034
[11] Dzetkulič, T.: ODEIntegrator. http://odeintegrator.sourceforge.net, 2012. Software package · Zbl 0226.65034
[12] Eble, I.: ber Taylor-Modelle. PhD thesis, Institut fr Angewandte und Numerische Mathematik (2007)
[13] Henzinger, T. A., Horowitz, B., Majumdar, R., Wong-Toi, H.: Beyond HyTech: hybrid systems analysis using interval numerical methods. In: Lynch, N., Krogh, B. (eds.) Proceedings of the HSCC’00, vol. 1790 of LNCS, pp. 130-144, Springer (2000) · Zbl 0938.93552
[14] IEEE Standards Board: IEEE standard for binary floating-point arithmetic. Technical report, The Institute of Electrical and Electronics Engineers. Technical Report IEEE Std 754-1985 (1985)
[15] Kühn, W, Rigorously computed orbits of dynamical systems without the wrapping effect, Computing, 61, 47-67, (1998) · Zbl 0910.65052
[16] Lohner, R. J.: Enclosing the solutions of ordinary initial and boundary value problems. In: Computer Arithmetic: Scientific Computation and Programming Languages, pp. 255-286. Teubner, Stuttgart (1987)
[17] Lohner, R.J.: Computations of guaranteed enclosures for the solutions of ordinary initial and boundary value problems. In: Cash, J., Gladwell, I. (eds.) Computational Ordinary Differential Equationas, pp. 425-435. Clarendon Press, Oxford (1992) · Zbl 0767.65069
[18] Makino, K; Berz, M, Taylor models and other validated functional inclusion methods, Int. J. Pur. Appl. Math., 4, 379-456, (2003) · Zbl 1022.65051
[19] Makino, K; Berz, M, Suppression of the wrapping effect by Taylor model-based verified integrators: long-term stabilization by preconditioning, Int. J. Differ. Equ. Appl., 10, 353-384, (2005) · Zbl 1133.65045
[20] Makino, K; Berz, M, Suppression of the wrapping effect by Taylor model-based verified integrators: the single step, Int. J. Pur. Appl. Math., 36, 175-197, (2006) · Zbl 1131.65060
[21] Makino, K., Berz, M.: Rigorous integration of flows and ODEs using Taylor models. Symb. Numer. Comput., 79-84 (2009) · Zbl 1356.65168
[22] Moore, R. E.: Interval analysis. Prentice hall, Englewood cliffs (1966) · Zbl 0176.13301
[23] Moore, R. E., Kearfott, R. B., Cloud, M. J.: Introduction to interval analysis. SIAM (2009) · Zbl 1168.65002
[24] Nedialkov, NS; Jackson, KR; Corliss, GF, Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput., 105, 21-68, (1999) · Zbl 0934.65073
[25] Ratschan, S., She, Z.: Safety verification of hybrid systems by constraint propagation-based abstraction refinement. ACM Trans. Embed. Comput. Syst. 6 (1) (2007) · Zbl 1078.93508
[26] Rauh, A., Hofer, E. P., Valencia-ivp, E. Auer.: A comparison with other initial value problem solvers. In: Proceedings of the 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, SCAN ’06, pp. 36-. IEEE Computer Society, Washington (2006) · Zbl 0910.65052
[27] Tucker, W, A rigorous ODE solver and smale’s 14th problem, Found. Comput. Math., 2, 53-117, (2002) · Zbl 1047.37012
[28] Wittig, A., Berz, M.: Rigorous high precision interval arithmetic in COSY INFINITY. In: Proceedings of the Fields Institute (2009)
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