### 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 language | English |
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Title of host publication | Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Proceedings |

Pages | 578-589 |

Number of pages | 12 |

DOIs | |

Publication status | Published - Aug 20 2012 |

Externally published | Yes |

Event | 37th International Symposium on Mathematical Foundations of Computer Science 2012, MFCS 2012 - Bratislava, Slovakia Duration: Aug 27 2012 → Aug 31 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7464 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 37th International Symposium on Mathematical Foundations of Computer Science 2012, MFCS 2012 |
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Country | Slovakia |

City | Bratislava |

Period | 8/27/12 → 8/31/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

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 -