Computational Complexity of Smooth Differential Equations

  • Akitoshi Kawamura
  • Hiroyuki Ota
  • Carsten Rösnick
  • Martin Ziegler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

Abstract

The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h′(t) = g(t, h(t)) under various assumptions on the function g has been investigated in hope of understanding the intrinsic hardness of solving the equation numerically. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C 1; for each k ≥ 2, the solution h can be hard for the counting hierarchy if g is of class C k .

Keywords

Computational Complexity Real Function Limited Feedback String Function Uniform Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ko, K.: Complexity Theory of Real Functions. Birkhäuser Boston (1991)Google Scholar
  2. 2.
    Weihrauch, K.: Computable Analysis: An Introduction. Springer (2000)Google Scholar
  3. 3.
    Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Computational Complexity 19(2), 305–332 (2010)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Pour-El, M., Richards, I.: A computable ordinary differential equation which possesses no computable solution. Annals of Mathematical Logic 17(1-2), 61–90 (1979)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill (1955)Google Scholar
  6. 6.
    Ko, K.: On the computational complexity of ordinary differential equations. Information and Control 58(1-3), 157–194 (1983)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Miller, W.: Recursive function theory and numerical analysis. Journal of Computer and System Sciences 4(5), 465–472 (1970)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Müller, N.: Uniform Computational Complexity of Taylor Series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  9. 9.
    Ko, K., Friedman, H.: Computing power series in polynomial time. Advances in Applied Mathematics 9(1), 40–50 (1988)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kawamura, A.: Complexity of initial value problems. Fields Institute Communications (to appear)Google Scholar
  11. 11.
    Kawamura, A., Cook, S.: Complexity theory for operators in analysis. ACM Transactions on Computation Theory 4(2), Article 5 (2012)Google Scholar
  12. 12.
    Bournez, O., Graça, D.S., Pouly, A.: Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 170–181. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Wagner, K.: The complexity of combinatorial problems with succinct input representation. Acta Informatica 23(3), 325–356 (1986)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Torán, J.: Complexity classes defined by counting quantifiers. Journal of the ACM 38(3), 752–773 (1991)CrossRefGoogle Scholar
  15. 15.
    Kawamura, A.: On function spaces and polynomial-time computability. Dagstuhl Seminar 11411: Computing with Infinite Data (2011)Google Scholar
  16. 16.
    Kawamura, A., Müller, N., Rösnick, C., Ziegler, M.: Uniform polytime computable operators on univariate real analytic functions. In: Proceedings of the Ninth International Conference on Computability and Complexity in Analysis (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Akitoshi Kawamura
    • 1
  • Hiroyuki Ota
    • 1
  • Carsten Rösnick
    • 2
  • Martin Ziegler
    • 2
  1. 1.University of TokyoTokyoJapan
  2. 2.Technische Universität DarmstadtDarmstadtGermany

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