2018 Fiscal Year Annual Research Report
Towards efficient solvers for ordinary differential equations in exact real arithmetic
| Project/Area Number |
18J10407
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| Research Institution | Kyushu University |
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Principal Investigator |
THIES HOLGER 九州大学, システム情報科学研究院, 特別研究員(PD)
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| Project Period (FY) |
2018-04-25 – 2020-03-31
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| 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 |
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| 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.
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| 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.
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| 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.
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Research Products
(8 results)
