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Towards efficient solvers for ordinary differential equations in exact real arithmetic

Research Project

Project/Area Number 18J10407
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Theory of informatics
Research InstitutionKyushu University
Research Fellow THIES HOLGER  九州大学, システム情報科学研究院, 特別研究員(PD)
Project Period (FY) 2018-04-25 – 2020-03-31
Project Status Granted (Fiscal Year 2019)
Budget Amount *help
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2018: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywords計算理論 / 計算量理論 / 計算可能解析 / 実数の計算理論 / 常微分方程式 / 力学系 / 平均計算量
Outline of Annual Research Achievements

A main goal of the research was to give efficient algorithms for problems involving continuous-time dynamical systems that are given by ordinary differential equations. Building on previous research, results for many operators on one-dimensional analytic functions could be generalized to the multidimensional case. Part of this work was published in the proceedings of the International Workshop on Logic, Language, Information, and Computation.
As an approach to better understand the typical behaviour regarding the computational complexity of problems involving dynamical systems, average-case complexity was applied. As stated in the research plan, previous results on the N-body problem could be generalized to Hamiltonian dynamical systems and properties that such a system is average-case polynomial-time computable have been defined. The results were published in the Proceedings of Mathematical Foundations of Computer Science (MFCS 2018) in Liverpool.
Progress was also made on more general aspects of the theory of second order computation. Mainly, a reasonable definition for a complexity class of type-two linear-time has been found. The new theory has several applications to computable analysis. The main results of this research were published in the proceedings of the International Conference on Theory and Applications of Models of Computation.
Recent work deals with formalizing some results from computable analysis using the coq proof assistant. Some first results were presented at the Third Workshop on Mathematical Logic and its Application in Nancy, France in March 2019.

Current Status of Research Progress
Current Status of Research Progress

1: Research has progressed more than it was originally planned.

Reason

Many results that were stated in the research plan could be achieved and were published at proceedings and presented at international conferences.
Further, additional results related to the research topic that have not been stated in the research plan were attained.

Strategy for Future Research Activity

For future work, a main goal is to formalize and verify properties of exact real arithmetic and dynamical systems.
In particular it is planned to formally proof the correctness of the work in the previous fiscal year on efficient ODE solving in exact real arithmetic.
With regards of computational complexity, it is planned to deeper study different complexity models like average-case complexity and parameterized complexity and their relation to problems in computable analysis, with an emphasis on dynamical systems.
It is planned to extend the results from recent publications as well as publish new results and formal verification.

Report

(1 results)
  • 2018 Annual Research Report

Research Products

(8 results)

All 2019 2018 Other

All Journal Article Presentation Remarks

  • [Journal Article] Second-Order Linear-Time Computability with Applications to Computable Analysis2019

    • Author(s)
      Kawamura Akitoshi、Steinberg Florian、Thies Holger
    • Journal Title

      Proc. of the 15th Annual Conference on Theory and Applications of Models of Computation (TAMC 2019), April 13-16, 2019, Kitakyushu, Japan

      Volume: 11436 Pages: 337-358

    • DOI

      10.1007/978-3-030-14812-6_21

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Parameterized Complexity for Uniform Operators on Multidimensional Analytic Functions and ODE Solving2018

    • Author(s)
      A. Kawamura, F. Steinberg and H. Thies
    • Journal Title

      Proc. of the 25th International Workshop on Logic, Language, Information, and Computation (WoLLIC 2018), July 24-27, 2018, Bogota, Colombia

      Volume: 10944 Pages: 223-236

    • DOI

      10.1007/978-3-662-57669-4_13

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Average-Case Polynomial-Time Computability of Hamiltonian Dynamics2018

    • Author(s)
      A. Kawamura, H. Thies and M. Ziegler
    • Journal Title

      Proc. of the 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), August 27-31, 2018, Liverpool, UK

      Volume: 117 Pages: 30-30

    • DOI

      10.4230/LIPICS.MFCS.2018.30

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Some formal proofs of isomorphy and discontinuity2019

    • Author(s)
      Steinberg Florian, Thies Holger
    • Organizer
      Third Workshop on Mathematical Logic and its Applications, Nancy, France
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] A class for second-order linear-time computability2018

    • Author(s)
      Kawamura Akitoshi, Steinberg Florian, Thies Holger
    • Organizer
      Continuity, Computability, Constructivity (CCC) 2018, Faro, Portugal
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Computable analysis and computability in linear time2018

    • Author(s)
      Kawamura Akitoshi, Steinberg Florian, Thies Holger
    • Organizer
      Workshop on Computability Theory and Foundations of Mathematics 2018, Tokyo, Japan
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Applications of average-case complexity to problems in analysis2018

    • Author(s)
      Kawamura Akitoshi, Thies Holger, Ziegler Martin
    • Organizer
      LA Symposium, Chiba, Japan
    • Related Report
      2018 Annual Research Report
  • [Remarks] Holger Thies (personal homepage)

    • URL

      http://www.holgerthies.com

    • Related Report
      2018 Annual Research Report

URL: 

Published: 2018-05-01   Modified: 2019-12-27  

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