2022 Fiscal Year Final Research Report
Study of dynamical systems of random relaxed Newton methods
| Project/Area Number |
21K20323
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2021-08-30 – 2023-03-31
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| Keywords | ランダム力学系 / ニュートン法 / ランダムアルゴリズム / 複素解析 |
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| Outline of Final Research Achievements |
The PI implemented the random relaxed Newton methods and made several mathematical predictions through numerical experiments. For example, it was known that the root-finding algorithm worked well when the noise was sufficiently large, but for some examples, the random algorithm worked well even with very small noise. The PI also found numerically that the size of noise required for the algorithm is closely related to the parameter at which the family of the deterministic relaxed Newton maps bifurcates. The PI expects that mathematical proof of this conjecture will lead to the development of better algorithms.
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| Free Research Field |
ランダム力学系理論
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| Academic Significance and Societal Importance of the Research Achievements |
工学を含むあらゆる数理的な課題の中で，与えられた関数の零点（根）を求めることはとても基本的で重要な問題です．本研究は，有名な求根アルゴリズムであるニュートン法にあえてノイズを入れることでアルゴリズムを改善できるか，という着想に基づいています．得られた成果として，ランダム力学系の確率分岐は決定論的な分岐よりも早く起こるだろう，という数学的にも実用上も重要な予想を発見するに至りました．これは，力学系理論の研究を新しい観点から開拓するという学術的な意義があります．また，実社会に対しても，（ランダム）求根アルゴリズムの改善を通して大きな影響を与えられる可能性を秘めています．
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