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Multisymplectic Geometry and Geometric Numerical Integrator for Variational Problems

Research Project

Project/Area Number 20K14365
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 12040:Applied mathematics and statistics-related
Research InstitutionKeio University

Principal Investigator

Peng Linyu  慶應義塾大学, 理工学部(矢上), 准教授 (90725780)

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2023: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
KeywordsMultisymplectic geometry / Geometric integration / Variational calculus / Symmetry / Noether's theorem / Conservation law / Lagrangian / Moving frame / Symmetry-preserving / Burgers' equation / Kepler problem / Symplectic structure / Euclidean group / Formal Lagrangian / Variational integrator / KdV equation / Symmetry reduction / Group-invariant solution / Similarity solution / Soliton / Variational principle / DPD / Hamiltonian system / 修正形式ラグランジアン / ハミルトン偏微分方程式 / Geometric integrator / Variational bicomplex / Variational problems / PDEs
Outline of Research at the Start

Geometric integrator is among one of the most efficient numerical methods for differential equations. In this project, we establish a unified and systematical analogue for understanding both continuous and discrete multisymplectic structures of arbitrary order variational differential equations.

Outline of Final Research Achievements

In the current project, we have delved into the variational analysis of differential and discrete equations from various perspectives. For variational differential equations, we have developed a discrete counterpart of the variational bicomplex structure. This framework aids in understanding fundamental geometric and algebraic features, such as multisymplectic structure, symmetries, and conservation laws. For nonvariational problems, we have defined a modified formal Lagrangian formulation (MFLF), enabling the treatment of any differential equations. Specifically, this approach facilitates the systematic construction of conservation laws using Noether’s Theorem and variational integration even for nonvariational equations.

Academic Significance and Societal Importance of the Research Achievements

離散variational bicomplexは、差分方程式や数値方法の幾何及び代数的な研究するための基本的なツールになっている。対称性、保存則、マルティシンプレクティック構造、逆問題などはvariational bicomplexのコホモロジー群に関係しています。また、非変分問題のためのmodified formal Lagrangian formulationが導入されており、Noetherの定理から保存則を導出し、変分積分法の構築を容易にしています。これらの革新的な構造と理論は、物理現象の理解と効率的な数値積分法の開発に期待されます。

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (55 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (12 results) Journal Article (10 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 10 results) Presentation (32 results) (of which Int'l Joint Research: 16 results,  Invited: 13 results) Funded Workshop (1 results)

  • [Int'l Joint Research] Central Queensland University(オーストラリア)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of Leeds/University of Kent(英国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Max Planck Institute(ドイツ)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] 北京信息科技大学/上海大学(中国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Universita Politecnica delle Marche(イタリア)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] UNC-Chapel Hill(米国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] University of Leeds/University of Kent(英国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Shanghai University(中国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Marche Polytechnic University(イタリア)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] University of Kent/University of Leeds(英国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Beijing Institute of Technology/Shanghai University(中国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Max Planck Institute(ドイツ)

    • Related Report
      2020 Research-status Report
  • [Journal Article] The Comparison of Riemannian Geometric Matrix-CFAR Signal Detectors2024

    • Author(s)
      Yusuke Ono, Linyu Peng
    • Journal Title

      IEEE Transactions on Aerospace and Electronic Systems

      Volume: 60 Issue: 2 Pages: 1679-1691

    • DOI

      10.1109/taes.2023.3339403

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The effect of accuracy of initial velocity discretisations on discrete energy in variational integration2024

    • Author(s)
      Mamoru Gunji, Yusuke Ono, Linyu Peng
    • Journal Title

      Proceedings of the IUTAM Symposium on Nonlinear Dynamics for Design of Mechanical Systems Across Different Length/Time Scales

      Volume: -

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Developing a cloud evidence method for dynamic early warning of tunnel construction safety risk in undersea environment2023

    • Author(s)
      Hong Zhou, Binwei Gao, Xianbo Zhao, Linyu Peng, Shichao Bai
    • Journal Title

      Developments in the Built Environment

      Volume: 16 Pages: 100225-100225

    • DOI

      10.1016/j.dibe.2023.100225

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Some novel physical structures of a (2+1)-dimensional variable-coefficient Korteweg-de Vries system2023

    • Author(s)
      Yaqing Liu, Linyu Peng
    • Journal Title

      Chaos, Solitons & Fractals

      Volume: 171 Pages: 113430-113430

    • DOI

      10.1016/j.chaos.2023.113430

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Towards a simple mathematical model for the legal concept of balancing of interests2022

    • Author(s)
      Frederike Zufall, Rampei Kimura, Linyu Peng
    • Journal Title

      Artificial Intelligence and Law

      Volume: - Issue: 4 Pages: 807-827

    • DOI

      10.1007/s10506-022-09338-3

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Towards a median signal detector through the total Bregman divergence and its robustness analysis2022

    • Author(s)
      Yusuke Ono, Linyu Peng
    • Journal Title

      Signal Processing

      Volume: 201 Pages: 108728-108728

    • DOI

      10.1016/j.sigpro.2022.108728

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] A modified formal Lagrangian formulation for general differential equations2022

    • Author(s)
      Linyu Peng
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: - Issue: 2 Pages: 573-598

    • DOI

      10.1007/s13160-022-00500-7

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Transformations, symmetries and Noether theorems for differential-difference equations2022

    • Author(s)
      Linyu Peng, Peter E. Hydon
    • Journal Title

      Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

      Volume: 478 Issue: 2259 Pages: 20210944-20210944

    • DOI

      10.1098/rspa.2021.0944

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A stochastic Hamiltonian formulation applied to dissipative particle dynamics2022

    • Author(s)
      Linyu Peng, Noriyoshi Arai, Kenji Yasuoka
    • Journal Title

      Applied Mathematics and Computation

      Volume: - Pages: 127126-127126

    • DOI

      10.1016/j.amc.2022.127126

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Fisher-Rao geometry and Jeffreys prior for Pareto distribution2020

    • Author(s)
      Mingming Li, Huafei Sun, Linyu Peng
    • Journal Title

      Communications in Statistics - Theory and Methods

      Volume: - Issue: 6 Pages: 1895-1910

    • DOI

      10.1080/03610926.2020.1771593

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Toda type latticeの対称性と群不変解2024

    • Author(s)
      富田繁, 彭林玉
    • Organizer
      日本応用数理学会第20回研究部会連合発表会
    • Related Report
      2023 Annual Research Report
  • [Presentation] A finite difference algorithm for mean field games2024

    • Author(s)
      Jiongyi Wang, Linyu Peng
    • Organizer
      日本応用数理学会若手の会 第9回学生研究発表会
    • Related Report
      2023 Annual Research Report
  • [Presentation] 行列空間上のLq平均を活用した目標信号検出2024

    • Author(s)
      佐藤大輝, , 彭林玉
    • Organizer
      日本応用数理学会若手の会 第9回学生研究発表会
    • Related Report
      2023 Annual Research Report
  • [Presentation] Symmetry analysis of semi-discrete equations and their conservation laws2023

    • Author(s)
      Linyu Peng
    • Organizer
      6th International Conference on Dynamics, Vibration and Control (ICDVC2022)
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The influence of accuracy of initial values on the discrete energy in variational integrator'2023

    • Author(s)
      Mamoru Gunji, Yusuke Ono, Linyu Peng
    • Organizer
      IUTAM Symposium on Nonlinear Dynamics for Design of Mechanical Systems across Different Length/Time Scales
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research
  • [Presentation] A discretization of Dirac structures and Lagrange-Dirac dynamical systems2023

    • Author(s)
      Hiroaki Yoshimura, Linyu Peng
    • Organizer
      10th International Congress on Industrial and Applied Mathematics (ICIAM2023)
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Applications of Bures-Wasserstein geometry of HPD matrices to signal detection2023

    • Author(s)
      Yusuke Ono, Linyu Peng
    • Organizer
      10th International Congress on Industrial and Applied Mathematics (ICIAM2023)
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Noether’s conservation laws based on the modified formal variational method2023

    • Author(s)
      Linyu Peng
    • Organizer
      10th International Congress on Industrial and Applied Mathematics (ICIAM2023)
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The modified formal variational formulation for general differential equations and applications2023

    • Author(s)
      彭林玉
    • Organizer
      早稲田大学 持続的環境エネルギー社会共創研究機構 研究所間交流会
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Discrete Lagrangian multiforms on the difference variational bicomplex2023

    • Author(s)
      Linyu Peng
    • Organizer
      BIRS Workshop Lagrangian Multiform Theory and Pluri-Lagrangian Systems
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research
  • [Presentation] 自己共分散行列空間における信号検出手法とそのロバスト性解析2023

    • Author(s)
      小野 悠介, 彭 林玉
    • Organizer
      信号処理研究会
    • Related Report
      2022 Research-status Report
  • [Presentation] 半離散方程式の対称性解析とその応用2023

    • Author(s)
      彭 林玉
    • Organizer
      ミニワークショップ「非線形現象の応用数理-モデリング,解析と計算」
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Symmetry analysis of semi-discrete equations and their conservation laws2023

    • Author(s)
      Linyu Peng
    • Organizer
      6th International Conference on Dynamics, Vibration and Control (ICDVC2022+1)
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Nonlinear Motions of a Self-adaptive Resonator2022

    • Author(s)
      Toshihiko Sugiura, Koki Ochiai, Xuefeng Liu, Yuki Oikawa, Linyu Peng
    • Organizer
      10th European Nonlinear Dynamics Conference (ENOC2022)
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research
  • [Presentation] 変分的積分法を用いたベイジアンニューラルネットワークの応用2022

    • Author(s)
      大竹 桃子, 彭 林玉
    • Organizer
      第35回計算力学講演会 (CMD2022)
    • Related Report
      2022 Research-status Report
  • [Presentation] ディラック力学系のモデリングと離散的変分法による定式化2022

    • Author(s)
      吉村 浩明, 彭 林玉
    • Organizer
      第35回計算力学講演会 (CMD2022)
    • Related Report
      2022 Research-status Report
  • [Presentation] 修正形式ラグランジアンのバーガース方程式への応用2022

    • Author(s)
      宮前 圭佑, 大竹 桃子, 小幡 研治, 彭 林玉
    • Organizer
      第35回計算力学講演会 (CMD2022)
    • Related Report
      2022 Research-status Report
  • [Presentation] The difference variational bicomplex and discrete integrable systems2022

    • Author(s)
      Linyu Peng
    • Organizer
      New Trends in Lagrangian and Hamiltonian Aspects of Integrable Systems
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Application of the modified formal variational formulation to the Burgers’ equation2022

    • Author(s)
      Momoko Otake, Kenji Obata, Linyu Peng
    • Organizer
      15th World Congress on Computational Mechanics (WCCM-XV) & 8th Asian Pacific Congress on Computational Mechanics (APCOM-VIII)
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research
  • [Presentation] 散逸粒子動力学のStormer-Verlet法2022

    • Author(s)
      彭 林玉, 荒井 規允, 泰岡 顕治
    • Organizer
      日本応用数理学会2022年度年会
    • Related Report
      2022 Research-status Report
  • [Presentation] The total Bregman divergenceを用いた外れ値検知とそのロバスト性解析2022

    • Author(s)
      小野 悠介, 彭 林玉
    • Organizer
      日本応用数理学会2022年度年会
    • Related Report
      2022 Research-status Report
  • [Presentation] 半離散方程式の対称性解析及びネーターの定理2022

    • Author(s)
      彭 林玉
    • Organizer
      研究集会「非線形波動から可積分系へ 2022」
    • Related Report
      2022 Research-status Report
  • [Presentation] Numerical methods based on the modified formal Lagrangian formulation of differential equations2022

    • Author(s)
      Linyu Peng
    • Organizer
      KSIAM 2022 Annual Meeting
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The matrix-CFAR via total Bregman divergence medians in signal detection2021

    • Author(s)
      Yusuke Ono, Linyu Peng, Hiroki Sato
    • Organizer
      3rd International Conference on Advances in Signal Processing and Artificial Intelligence
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] The modified formal variational structure and variational integrator2021

    • Author(s)
      Linyu Peng
    • Organizer
      26th International Conference on Difference Equations and Applications
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 半離散力学系の連続対称性と保存則2021

    • Author(s)
      Linyu Peng, Peter E. Hydon
    • Organizer
      日本応用数理学会2021年度年会
    • Related Report
      2021 Research-status Report
  • [Presentation] Continuous symmetries of semi-discrete equations and conserved quantities2021

    • Author(s)
      Linyu Peng
    • Organizer
      OCAMI Workshop 'ヘリシティと時空対称性,古典場から量子場まで'
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Symmetries and Noether's conservation laws of semi-discrete equations2021

    • Author(s)
      Linyu Peng
    • Organizer
      Moving Frames and their Modern Applications
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 半離散方程式の対称性から2021

    • Author(s)
      Linyu Peng
    • Organizer
      ワークショップ 可積分系研究の最近の進展 --理論,シミュレーション,応用--
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Modified formal variational structures for general differential equations2020

    • Author(s)
      Linyu Peng, Kenji Obata
    • Organizer
      日本応用数理学会2020年度年会
    • Related Report
      2020 Research-status Report
  • [Presentation] The infinitesimal invariance criterion for statistical transformation models2020

    • Author(s)
      Linyu Peng
    • Organizer
      International Conference on Mathematical Characterization, Analysis and Applications of Complex Information, Beijing Institute of Technology, China
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Conservation laws of general DEs: A modified formal Lagrangian formulation2020

    • Author(s)
      Linyu Peng
    • Organizer
      Seminar at the Department of Mathematics, Nanjing University of Aeronautics and Astronautics, China
    • Related Report
      2020 Research-status Report
    • Invited
  • [Funded Workshop] BIRS Workshop Lagrangian Multiform Theory and Pluri-Lagrangian Systems2023

    • Related Report
      2023 Annual Research Report

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Published: 2020-04-28   Modified: 2025-01-30  

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