Computational complexity of smooth differential equations

Akitoshi Kawamura, Hiroyuki Ota, Carsten Rösnick, Martin Ziegler

Research output: Chapter in Book/Report/Conference proceedingConference contribution

11 Citations (Scopus)

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 .

Original languageEnglish
Title of host publicationMathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings
Pages578-589
Number of pages12
DOIs
Publication statusPublished - Aug 20 2012
Externally publishedYes
Event37th International Symposium on Mathematical Foundations of Computer Science 2012, MFCS 2012 - Bratislava, Slovakia
Duration: Aug 27 2012Aug 31 2012

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7464 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other37th International Symposium on Mathematical Foundations of Computer Science 2012, MFCS 2012
CountrySlovakia
CityBratislava
Period8/27/128/31/12

Fingerprint

Computational complexity
Computational Complexity
Differential equations
Differential equation
G-function
Hardness
Lipschitz
Continuous Time
Smoothness
Counting
Polynomial time
Ordinary differential equation
Requirements
Ordinary differential equations
Polynomials
Class
Hierarchy

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Kawamura, A., Ota, H., Rösnick, C., & Ziegler, M. (2012). Computational complexity of smooth differential equations. In Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings (pp. 578-589). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7464 LNCS). https://doi.org/10.1007/978-3-642-32589-2_51

Computational complexity of smooth differential equations. / Kawamura, Akitoshi; Ota, Hiroyuki; Rösnick, Carsten; Ziegler, Martin.

Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings. 2012. p. 578-589 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7464 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kawamura, A, Ota, H, Rösnick, C & Ziegler, M 2012, Computational complexity of smooth differential equations. in Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7464 LNCS, pp. 578-589, 37th International Symposium on Mathematical Foundations of Computer Science 2012, MFCS 2012, Bratislava, Slovakia, 8/27/12. https://doi.org/10.1007/978-3-642-32589-2_51
Kawamura A, Ota H, Rösnick C, Ziegler M. Computational complexity of smooth differential equations. In Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings. 2012. p. 578-589. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-32589-2_51
Kawamura, Akitoshi ; Ota, Hiroyuki ; Rösnick, Carsten ; Ziegler, Martin. / Computational complexity of smooth differential equations. Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings. 2012. pp. 578-589 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{d9d6d74f77df468f99367056202f5f28,
title = "Computational complexity of smooth differential equations",
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 .",
author = "Akitoshi Kawamura and Hiroyuki Ota and Carsten R{\"o}snick and Martin Ziegler",
year = "2012",
month = "8",
day = "20",
doi = "10.1007/978-3-642-32589-2_51",
language = "English",
isbn = "9783642325885",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
pages = "578--589",
booktitle = "Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings",

}

TY - GEN

T1 - Computational complexity of smooth differential equations

AU - Kawamura, Akitoshi

AU - Ota, Hiroyuki

AU - Rösnick, Carsten

AU - Ziegler, Martin

PY - 2012/8/20

Y1 - 2012/8/20

N2 - 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 .

AB - 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 .

UR - http://www.scopus.com/inward/record.url?scp=84865013518&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865013518&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-32589-2_51

DO - 10.1007/978-3-642-32589-2_51

M3 - Conference contribution

AN - SCOPUS:84865013518

SN - 9783642325885

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 578

EP - 589

BT - Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings

ER -