New development of numerical analysis based on the space-time variational method

• Project/Area Number: 21H04431
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: 齊藤 宣一 東京大学, 大学院数理科学研究科, 教授 (00334706)
• Co-Investigator(Kenkyū-buntansha): 野津 裕史 金沢大学, 数物科学系, 教授 (00588783) ; 及川 一誠 筑波大学, 数理物質系, 准教授 (10637466) ; 金森 正史 国立研究開発法人宇宙航空研究開発機構, 航空技術部門, 主任研究開発員 (50770872) ; 三竹 大寿 東京大学, 大学院数理科学研究科, 准教授 (90631979)
• Project Period (FY): 2021-04-05 – 2026-03-31
• Project Status: Granted (Fiscal Year 2024)
• Budget Amount: ¥41,340,000 (Direct Cost: ¥31,800,000、Indirect Cost: ¥9,540,000); Fiscal Year 2025: ¥8,580,000 (Direct Cost: ¥6,600,000、Indirect Cost: ¥1,980,000); Fiscal Year 2024: ¥8,580,000 (Direct Cost: ¥6,600,000、Indirect Cost: ¥1,980,000); Fiscal Year 2023: ¥8,320,000 (Direct Cost: ¥6,400,000、Indirect Cost: ¥1,920,000); Fiscal Year 2022: ¥8,710,000 (Direct Cost: ¥6,700,000、Indirect Cost: ¥2,010,000); Fiscal Year 2021: ¥7,150,000 (Direct Cost: ¥5,500,000、Indirect Cost: ¥1,650,000)
• Keywords: 時空間変分法 / 平均場ゲーム方程式 / 離散外微分形式 / 有限要素法 / 数値シミュレーション / Discontinuous / 有限要素外積解析

Outline of Research at the Start

時空間変分法とは、時空間領域で定義された関数空間上で汎関数を最小化したり、停留点を求めることで時間発展偏微分方程式の解を求める方法である。社会現象の制御やデータ科学などの今日的な動機により、数学的にも数値計算上も、従来の初期値問題の枠組みでは十分に扱えない問題が認識され、その解決が要請されている。本研究では、時空間変分法をキーワードに数値解析学が柱となり、純粋数学の立場から、解析理論・数値解析理論の深化と解析学における新たな問題の発掘を遂行する一方で、応用数学の立場からは、新しいシミュレーション技術の提供と、数学理論による体系化・言語化を通じた技術の大衆化を実現する。

Outline of Annual Research Achievements

本研究は、A. 平均場ゲーム方程式の時空間変分法による数値解析、B. Maxwell方程式の時空間変分法による数値解析、C. 時空間変分法に基づくデータ同化の3つのユニットに別れて研究が遂行された。Aについては次の成果を得た。前年度に提案したfictitious playに基づく反復とCole-Hopf変換を応用した実用的な差分解法について、その収束解析を精密化し、より数学的に自然な仮定の下での誤差評価を得た。さらに、最適制御問題に現れるHamilton-Jacobi-Bellman方程式に対する差分法の収束解析を行った。特に、HJB方程式のコスト関数が解析的に表現できない場合を厳密に扱った。時空間の微分を含む偏微分方程式のための高精度な数値解法の開発・応用の研究を行った。特に、質量保存型ラグランジュ・ガレルキン法の時間２次精度化に成功した。カプトー型時間分数冪拡散方程式の弱解の同値性、ハミルトン・ヤコビ・ベルマン方程式の解の長時間挙動、外力付等高面平均曲率流方程式のコーシー・ノイマン問題のリプシッツ評価について、解析的な結果を得た。Bについては、Maxwell方程式の時間周期問題に対する時空間での有限要素外積解析に基づく有限要素スキームの数学解析を精密化した。Cについては、Hybridizable Discontinuous Galerkin (HDG) 法の非重複領域分割法について研究を行った。モデル問題として2次元Poisson方程式を考え、よく知られたDirichlet-Neumann （交代型）反復アルゴリズム、およびNeumann-Neumann法、FETI (Dirichlet-Dirichlet) 法をHDG法に適用して数値計算を行ったところ、通常の有限要素法と同じように収束することが確認された。機械学習的な手法による偏微分方程式の解法を調査した。

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

年度はじめに定めた計画が概ね達成されている。

Strategy for Future Research Activity

対面での（研究チーム全体での）研究打ち合わせの機会を増やす。海外から、関係研究者の招聘を再検討する。

Research Products

• [Journal Article] A Two-Step Lagrange?Galerkin Scheme for the Shallow Water Equations with a Transmission Boundary Condition and Its Application to the Bay of Bengal Region?Part I: Flat Bottom Topography (2023)
  • Author(s): Rasid Md Mamunur、Kimura Masato、Murshed Md Masum、Wijayanti Erny Rahayu、Notsu Hirofumi
  • Journal Title: Mathematics Volume: 11 Issue: 7 Pages: 1633-1633
  • DOI: 10.3390/math11071633
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A mass-lumping finite element method for radially symmetric solution of a multidimensional semilinear heat equation with blow-up (2022)
  • Author(s): T. Nakanishi and N. Saito
  • Journal Title: Int. J. Comput. Math. Volume: 99 Issue: 9 Pages: 1890-1917
  • DOI: 10.1080/00207160.2021.2019716
  • Peer Reviewed / Open Access
• [Journal Article] A Mass-Preserving Two-Step Lagrange?Galerkin Scheme for Convection-Diffusion Problems (2022)
  • Author(s): Futai Kouta、Kolbe Niklas、Notsu Hirofumi、Suzuki Tasuku
  • Journal Title: Journal of Scientific Computing Volume: 92 Issue: 2 Pages: 37-37
  • DOI: 10.1007/s10915-022-01885-w
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A shape design problem for the Navier?Stokes flow with a convective boundary condition (2022)
  • Author(s): Simon J. S. H、Notsu Hirofumi
  • Journal Title: Computational and Applied Mathematics Volume: 41 Issue: 4 Pages: 167-167
  • DOI: 10.1007/s40314-022-01876-5
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A shape optimization problem constrained with the Stokes equations to address maximization of vortices (2022)
  • Author(s): Simon John Sebastian、Notsu Hirofumi
  • Journal Title: Evolution Equations & Control Theory Volume: - Issue: 6 Pages: 1873-1902
  • DOI: 10.3934/eect.2022003
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A convective boundary condition for the Navier-Stokes equations (2022)
  • Author(s): Simon John Sebastian H.、Notsu Hirofumi
  • Journal Title: Applied Mathematics Letters Volume: 128 Pages: 107876-107876
  • DOI: 10.1016/j.aml.2021.107876
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation (2022)
  • Author(s): Giga Yoshikazu、Mitake Hiroyoshi、Sato Shoichi
  • Journal Title: Journal of Differential Equations Volume: 316 Pages: 364-386
  • DOI: 10.1016/j.jde.2022.01.057
  • NAID: 120007145351
  • Peer Reviewed / Open Access
• [Journal Article] The large time profile for Hamilton-Jacobi-Bellman equations (2022)
  • Author(s): D. A. Gomes, H. Mitake, H. V. Tran
  • Journal Title: Math. Ann. Volume: 384 Issue: 3-4 Pages: 1409-1459
  • DOI: 10.1007/s00208-021-02320-5
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Level-set forced mean curvature flow with the Neumann boundary condition (2022)
  • Author(s): J. Jang, D. Kwon, H. Mitake, H. V. Tran
  • Journal Title: J. Math. Pures Appl. Volume: 168 Pages: 143-167
  • DOI: 10.1016/j.matpur.2022.11.002
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations (2021)
  • Author(s): N. Saito
  • Journal Title: IMA J. Numer. Anal. Volume: 41 Issue: 2 Pages: 1253-1278
  • DOI: 10.1093/imanum/draa017
  • Peer Reviewed / Open Access
• [Journal Article] Second-Order Finite Difference Approximations of the Upper-Convected Time Derivative (2021)
  • Author(s): Medeiros Debora D.、Notsu Hirofumi、Oishi Cassio M.
  • Journal Title: SIAM Journal on Numerical Analysis Volume: 59 Issue: 6 Pages: 2955-2988
  • DOI: 10.1137/20m1364990
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Twin vortex computer in fluid flow (2021)
  • Author(s): Goto Ken、Nakajima Kohei、Notsu Hirofumi
  • Journal Title: New Journal of Physics Volume: 23 Issue: 6 Pages: 063051-063051
  • DOI: 10.1088/1367-2630/ac024d
  • Peer Reviewed / Open Access
• [Journal Article] Existence of asymptotic speed for birth and spread model equations (2021)
  • Author(s): Y. Giga, H. Mitake, T. Ohtsuka and H. V. Tran
  • Journal Title: Indiana Univ. Math. J. Volume: 70 Issue: 1 Pages: 121-156
  • DOI: 10.1512/iumj.2021.70.8305
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Large time behavior for a Hamilton-Jacobi equation in a critical Coagulation-Fragmentation model (2021)
  • Author(s): H. Mitake, H. V. Tran, T. S. Van
  • Journal Title: Commun. Math. Sci. Volume: 19 Issue: 2 Pages: 495-512
  • DOI: 10.4310/cms.2021.v19.n2.a8
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Remarks on the generalized Cauchy-Dirichlet problem for graph mean curvature flow with driving force (2021)
  • Author(s): H. Mitake, L. Zhang
  • Journal Title: Partial Differ. Equ. Appl. Volume: 2 Issue: 3 Pages: 1-24
  • DOI: 10.1007/s42985-020-00066-4
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] An analysis on the penalty and Nitsche's methods for the Stokes?Darcy system with a curved interface (2021)
  • Author(s): Zhou Guanyu、Kashiwabara Takahito、Oikawa Issei、Chung Eric、Shiue Ming-Cheng
  • Journal Title: Applied Numerical Mathematics Volume: 165 Pages: 83-118
  • DOI: 10.1016/j.apnum.2021.02.006
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Well-posedness of one-dimensional models of blood flow in arteries (2022)
  • Author(s): N. Saito
  • Organizer: WCCM-APCOM 2022
  • Int'l Joint Research
• [Presentation] 平均場ゲーム方程式のCole-Hopf変換とFictitious Play反復による数値計算 (2022)
  • Author(s): 井上大輔，伊藤優司，柏原崇人，齊藤宣一，吉田広顕
  • Organizer: 日本応用数理学会2022年度年会
• [Presentation] Lax--Milgramとその後の70年 (2022)
  • Author(s): 齊藤宣一
  • Organizer: 西浦廉政先生の古希記念研究集会
  • Invited
• [Presentation] The Keller--Segel system of chemotaxis, finite element method, and finite volume method (2022)
  • Author(s): N. Saito
  • Organizer: EASIAM Conference 2022
  • Int'l Joint Research / Invited
• [Presentation] A mass-preserving two-step Lagrange-Galerkin scheme for convection-diffusion problems (2022)
  • Author(s): H. Notsu, K. Futai, N. Kolbe, T. Suzuki
  • Organizer: RIMS Workshop
  • Invited
• [Presentation] Second-order finite difference approximations of the upper-convected time derivative (2022)
  • Author(s): H. Notsu, D.O. Medeiros, C.M. Oishi
  • Organizer: WCCM-APCOM 2022
  • Int'l Joint Research / Invited
• [Presentation] Twin vortex computer in fluid flow and a related topic (2022)
  • Author(s): H. Notsu
  • Organizer: RIMS Workshop
  • Int'l Joint Research / Invited
• [Presentation] On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation (2022)
  • Author(s): H. Mitake
  • Organizer: Inverse Problems and Control for PDEs and the Hamilton-Jacobi Equation
  • Int'l Joint Research / Invited
• [Presentation] On Lipschitz regularity for level-set forced mean curvature flow under the Neumann boundary condition (2022)
  • Author(s): H. Mitake
  • Organizer: RIMS共同研究（公開型）「発展方程式論の革新：異分野との融合がもたらす理論の深化」
  • Int'l Joint Research / Invited
• [Presentation] On Lipschitz regularity for level-set forced mean curvature flow under the Neumann boundary condition (2022)
  • Author(s): H. Mitake
  • Organizer: NTU-Tokyo joint conference
  • Int'l Joint Research / Invited
• [Presentation] The Keller--Segel system of chemotaxis, finite element method, and finite volume method (2021)
  • Author(s): N. Saito
  • Organizer: The Hong Kong Polytechnic University, Colloquium
  • Int'l Joint Research / Invited
• [Presentation] 数値解析における様々な反例 (2021)
  • Author(s): 齊藤宣一
  • Organizer: ものづくり企業に役立つ応用数理手法の研究会
  • Invited
• [Presentation] Analysis of the finite element method for the Stokes problems with singular source terms (2021)
  • Author(s): N. Saito
  • Organizer: 2021 EIMS International Conference on Computational Mathematics
  • Int'l Joint Research / Invited
• [Presentation] Hamilton-Jacobi-Bellman方程式に対する風上差分法の収束証明 (2021)
  • Author(s): 井上大輔，伊藤優司，吉田広顕，柏原崇人，齊藤宣一
  • Organizer: 日本応用数理学会2021年度年会
• [Presentation] 計算モデルの正当性を保証する数理 (2021)
  • Author(s): 齊藤宣一
  • Organizer: CREST「現代の数理科学と連携するモデリング手法の構築」シンポジウム
• [Presentation] Twin vortex computer in fluid flow (2021)
  • Author(s): H. Notsu, K. Goto, K. Nakajima
  • Organizer: International Conference: Differential Equations for Data Science 2021 (DEDS2021)
  • Int'l Joint Research / Invited
• [Presentation] On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation (2021)
  • Author(s): H. Mitake
  • Organizer: The Hong Kong University of Science and Technology, PDE seminar
  • Int'l Joint Research / Invited
• [Presentation] On weak solutions to first-order discount mean field games (2021)
  • Author(s): H. Mitake
  • Organizer: OIST Analysis Seminar
  • Int'l Joint Research / Invited
• [Presentation] 時間分数冪拡散方程式に対する粘性解と超関数解の同値性 (2021)
  • Author(s): H. Mitake
  • Organizer: 京都大学NLPDEセミナー
  • Invited
• [Presentation] On weak solutions to first-order discount mean field games (2021)
  • Author(s): H. Mitake, K. Terai
  • Organizer: 日本数学会年会