汚染物質の拡散の推定と予測のための逆問題の数学手法の開拓

• Project/Area Number: 21K18142
• Research Category: Grant-in-Aid for Challenging Research (Pioneering)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: 山本 昌宏 東京大学, 大学院数理科学研究科, 教授 (50182647)
• Project Period (FY): 2021-07-09 – 2027-03-31
• Project Status: Granted (Fiscal Year 2023)
• Budget Amount: ¥24,830,000 (Direct Cost: ¥19,100,000、Indirect Cost: ¥5,730,000); Fiscal Year 2026: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2025: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2024: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2023: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2022: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2021: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
• Keywords: 特異拡散 / 不均質媒質 / 非整数階偏微分方程式 / 非線形方程式 / 逆問題 / モデリング / 数学解析 / 拡散現象 / 放射線量等予測 / 汚染源推定 / 数値手法

Outline of Research at the Start

１．環境中の汚染予測という社会性が極めて高い課題に数学の立場から取組み、現場で利用可能な手法の構築を目指し、より合理的な予測手法を確率して、数学による社会貢献の「視える化」を実現する。
２．本課題はラプラス方程式をはじめとする定常場・非定常場の偏微分方程式に対する色々なタイプの逆問題として記述できるが、これまで等閑視されてきたこのような逆問題の数学的な研究を、現実からの強い要請に基づいて推進する。それにより従来の数学研究の体系において、現実を直視して柔軟な課題設定で数学に取り組む方向性を示し、数学自身の研究領域の拡大を促す。

Outline of Annual Research Achievements

土壌などの不均質媒質中の汚染物質の拡散などにおいては、通常の拡散と異なる現象を示すことがしばしば現場から報告されている。汚染などの将来予測を行うためには定量的な解析を可能とするモデル式が不可欠であり、古典的な拡散方程式にかわる有効なモデル式の確立とその数学解析が第一の課題である。汚染物質の拡散などの予測は、社会的にもインパクトが大きいので、説明責任を果たすことのできる健全な数理科学的な研究の素地を構築することが数学者の責務である。そのためのモデル式として、時間方向に履歴の効果を考慮した非整数階拡散方程式が受け入れられており、世界的にも夥しい研究がリアルタイムでなされている。しかしながら、既存の成果をなぞるか、理論的に脆弱なものが多く、実用の要請を満たすことが難しい。このような状況から、本研究の要点は以下の通りである。（１）非整数階線形偏微分方程式を伝統的な微分積分学の枠組みではなく、近代的な関数解析的な偏微分方程式論に見合う形で完成させる。そのような基礎付けは必ずしも一通りとは限らないが、近代的偏微分方程式論に適合し、そのうえで応用にも適した理論を独自に構築している。特に、より作用素論に基づいた非整数階偏微分方程式論をナンシー・ロレーヌ大学（フランス）の Mourad Choulli 教授と遂行した。（２）上記の線形理論に基づき、非整数階非線形偏微分方程式の理論を確立を目指している。非線形方程式は、緑化現象のモデル式である Klausmeister-Gray-Scott モデルなどとも関連しており、本研究の範囲を大きく広げることができた。（３）本課題の遂行のためには、モデル式の物理パラメータの定量的な推定のために、方程式の係数やソース項を解の限定された情報で決定するという逆問題についても数学解析を進めた。

Current Status of Research Progress

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

Reason

コロナ禍で前年度は研究遂行に遅れを認めざるを得なかったが、令和5年度は回復し所期の成果を収めたと判断している。理由は以下の通りである。（１）非整数階偏微分方程式の線形理論の主要部分を完成させたこと。（２）線形理論を活用して、非線形の非整数階偏微分方程式の解の時間局所的な存在や時間大局的な存在、漸近挙動などの性質を確立しつつあること。（３）対応する逆問題の数学解析について世界をリードする成果を挙げていること。
研究発表の項でも述べるように研究論文の出版実績が以上の判断の１つの根拠となる。

Strategy for Future Research Activity

（１）現場には本研究計画の数理的な解析を必要としている問題と解決への要請がいろいろある。そのような問題の探索、数学解析、工学者など現場に近い研究者との連携を当初の計画通りに遂行する。（２）理論面での研究を引き続き発展させ、現場の研究者と連携し、厳密で整合性があるだけではなく現場の課題に役立つ数学理論を構築していく。（３）非整数階偏微分方程式については、世界中でリアルタイムで数多い研究がなされている。そのような状況から、適切な研究者と連携し、本研究計画をコアにした国際共同研究体制を維持、発展させる。そのために指導的な立場を担うことができる海外の中堅の研究者の招へい・共同研究の推進とともに研究代表者、連携研究者の出張を計画している。国内・国外の研究集会の参加、成果発表、研究連絡なども実行していくことはいうまでもない。

Research Products

• [Int'l Joint Research] Fudan University(中国)
• [Int'l Joint Research] Texas A&M University(米国)
• [Int'l Joint Research] Sapienza University of Rome(イタリア)
• [Int'l Joint Research] Aix-Marseille University/Nancy-Lorraine University(フランス)
• [Int'l Joint Research] Beuth Tech. Hochschule(ドイツ)
• [Int'l Joint Research]
• [Int'l Joint Research] Fudan University/Chinese University of Hong Kong/City University of Hong Kong(中国)
• [Int'l Joint Research] Sapienza University of Rome/University of Parma/University of Bari(イタリア)
• [Int'l Joint Research] Aix Marseille University/University of Nancy-Lorraine(フランス)
• [Int'l Joint Research] Beuth Tech. Hochschule(ドイツ)
• [Int'l Joint Research] University of Sevilla(スペイン)
• [Int'l Joint Research]
• [Journal Article] Homogenization and inverse problems for fractional diffusion equations (2023)
  • Author(s): A. Kawamoto, M. Machida and M. Yamamoto
  • Journal Title: Ｆract.Calc. Appl. Anal. Volume: 26 Pages: 2118-2165
  • Peer Reviewed
• [Journal Article] Blowup in L^1(Omega)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms (2023)
  • Author(s): G. Floridia, Y. Liu and M. Yamamoto
  • Journal Title: Adv. Nonlinear Anal. Volume: 12
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Determination of source or initial values for acoustic equations with a time-fractional attenuation (2023)
  • Author(s): X. Huang, Y. Kian, E. Soccorsi and M. Yamamoto
  • Journal Title: Anal. Appl. Volume: 21 Pages: 1105-1130
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part I: Linear equations (2023)
  • Author(s): Y. Luchko and M. Yaammoto
  • Journal Title: Fract. Calc. Appl. Anal. Volume: 26 Pages: 1504-1544
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Uniqueness of inverse source problems for general evolution equations (2023)
  • Author(s): Y. Kian, Y. Liu and M. Yamamoto
  • Journal Title: Commun. Contemp. Math. Volume: 25
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Uniqueness for an inverse coefficient problem for a one-dimensional time-fractional diffusion equation with non-zero boundary conditions (2023)
  • Author(s): W. Rundell and M. Yamamoto
  • Journal Title: App,. Anal. Volume: 102 Pages: 815-829
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations (2023)
  • Author(s): M. Yamamoto
  • Journal Title: Math. Control Relat. Fields Volume: 13 Pages: 833-851
  • Peer Reviewed
• [Journal Article] Uniqueness for inverse source problems for fractional diffusion-wave equations by data during not acting time (2023)
  • Author(s): M. Yamamoto
  • Journal Title: Inverse Prob. Volume: 39
  • Peer Reviewed
• [Journal Article] Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem (2023)
  • Author(s): D. Jiang, Z. Li, M. Pauron and M. Yamamoto
  • Journal Title: Math. Methods Appl. Sci. Volume: 46 Pages: 2275-2287
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior (2023)
  • Author(s): Y. Liu and M. Yamamoto
  • Journal Title: Inverse Prob. Volume: 39
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions (2023)
  • Author(s): Z. Li, Y. Liu and M. Yamamoto
  • Journal Title: Inverse Prob. Imaging Volume: 17 Pages: 1-22
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Fractional calculus and time-fractional differential equations: revisit and construction of a theory (2022)
  • Author(s): M. Yamamoto
  • Journal Title: Mathematics, FraMathematics, Special issue Fractional Integrals and Derivatives: “ True ”versus“ False ” Volume: special issue Pages: 2227698-2227698
  • Peer Reviewed / Open Access
• [Journal Article] Least square formulation for ill-posed inverse problems and applications (2022)
  • Author(s): E. Chung, K. Ito, M. Yamamoto
  • Journal Title: Appl. Anal. Volume: 101 Pages: 52475261-52475261
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation (2022)
  • Author(s): X. Jing, M. Yamamoto
  • Journal Title: Inverse Problems and Imaging Volume: 16 Pages: 11991217-11991217
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A Carleman estimate and an energy method for a first-order symmetric hyperbolic system (2022)
  • Author(s): G. Floridia, H. Takase, M. Yamamoto
  • Journal Title: Inverse Problems and Imaging Volume: 16 Pages: 11631178-11631178
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Carleman estimate for the Navier-Stokes equations and applications (2022)
  • Author(s): O.Y. Imanuvilov, L. Lorenzi, M. Yamamoto
  • Journal Title: Inverse Problems Volume: 38 Pages: 08500630-08500630
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Continuation of solutions to elliptic and parabolic equations on hyperplanes and application to inverse source problems (2022)
  • Author(s): J. Cheng, M. Yamamoto
  • Journal Title: Inverse Problems Volume: 38 Pages: 08500523-08500523
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Uniqueness and numerical reconstruction for inverse problems dealing with interval size search (2022)
  • Author(s): J. Apraiz, J. Cheng, A. Doubova, E. Fernandez-Cara, M. Yamamoto
  • Journal Title: Inverse Problems and Imaging Volume: 16 Pages: 569594-569594
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate (2022)
  • Author(s): O.Y. Imanuvilov, Y. Kian, M. Yamamoto
  • Journal Title: J. Inverse and Ill-posed Problems Volume: 30 Pages: 191203-191203
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Identification of time-varying source term in time-fractional evolution equations (2022)
  • Author(s): Y. Kian, E. Soccorsi, X.Qi, M. Yamamoto
  • Journal Title: Commun. Math. Sci. Volume: 20 Pages: 5384-5384
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Some inverse problems for the Burgers equation and related systems (2022)
  • Author(s): J. Apraiz, A. Doubova, E. Fernandez-Cara, M. Yamamoto
  • Journal Title: Commun. Nonlinear Sci. Numer. Simul. Volume: 107 Pages: 10611323-10611323
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On the maximum principle for the multi-term fractional transport equation (2022)
  • Author(s): Y. Lucko, A. Suzuki, M. Yamamoto
  • Journal Title: J. Math. Anal. Appl. Volume: 505 Pages: 12557914-12557914
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Stability estimate for a semilinear elliptic inverse problem (2021)
  • Author(s): Mourad Choulli. Guanghui Hu, Masahiro Yamamoto
  • Journal Title: NoDEA Nonlinear Differential Equations Appl. Volume: 28
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Recovering the weight function in distributed order fractional equation from interior measurement (2021)
  • Author(s): Jijun Liu, C.L. Sun, Masahiro Yamamoto
  • Journal Title: Appl. Numer. Math. Volume: 168 Pages: 84-103
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Carleman estimates and controllability for a degenerate structured population model (2021)
  • Author(s): Genni Fragnelli, Masahiro Yamamoto
  • Journal Title: Appl. Math. Optim. Volume: 84 Pages: 999-1044
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The uniqueness of inverse problems for a fractional equation with a single measurement (2021)
  • Author(s): Kian, Yavar; Li, Zhiyuan; Liu, Yikan; Yamamoto, Masahiro
  • Journal Title: Math. Ann. Volume: 380 Pages: 1465-1495
  • NAID: 120007116481
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Uniqueness in determining fractional orders of derivatives and initial values (2021)
  • Author(s): Yamamoto, Masahiro
  • Journal Title: Inverse Problems Volume: 37
  • Peer Reviewed
• [Journal Article] Conditional stability for an inverse coefficient problem of a weakly coupled time-fractional diffusion system with half order by Carleman estimate (2021)
  • Author(s): Ren, Caixuan; Huang, Xinchi; Yamamoto, Masahiro
  • Journal Title: J. Inverse Ill-posed Problems Volume: 29 Pages: 635-651
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Global stability result for parabolic Cauchy problems (2021)
  • Author(s): Choulli, Mourad; Yamamoto, Masahiro
  • Journal Title: J. Inverse Ill-posed Problems Volume: 29 Pages: 895-915
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation (2021)
  • Author(s): Cannarsa, Piermarco; Doubova, Anna; Yamamoto, Masahiro
  • Journal Title: Inverse Problems Volume: 37
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Case studies for solutions of real-world problems by mathematical thinking from steel industry to environmental issues (2023)
  • Author(s): M. Yamamoto
  • Organizer: Dipartimento di Scienze di Base e Applicate per l’Ingegneria Seminario di Analis Universit`a di Roma La Sapienza
  • Invited
• [Presentation] Fractional calculus and time-fractional differential equations: revisit and construction of a theory (2022)
  • Author(s): M. Yamamoto
  • Organizer: Differential Equations and their Applications, Weekly Online Seminar V. I. Romanovskiy Institute of Mathematics, Uzbekistan Tashkent
  • Invited
• [Presentation] Minicourse: Inverse problems for timefractional diffusion-wave equations (2022)
  • Author(s): M. Yamamoto
  • Organizer: Summer School-Workshop on Analysis, Control & Inverse Problems for Diffusive Systems withe Application to Natural and Social Sciences Bari
  • Int'l Joint Research / Invited
• [Presentation] Mathematics as foundation for social cooperation and case studies from steel industry to environmental issue (2022)
  • Author(s): M. Yamamoto
  • Organizer: Classe di Scienze Fisiche, Matematiche e Naturali Messina
  • Invited
• [Presentation] Comparison principles and time fractiona diffusion-wave equations (2022)
  • Author(s): Masahiro Yamamoto
  • Organizer: Computations, Analysis and Applications of PDEs with Nonlocal and Singular Operators, Singapore
• [Presentation] Stability for inverse problems by Carleman estimates and applications to fluid dynamics (2021)
  • Author(s): Masahiro Yamamoto
  • Organizer: Inverse and Ill-posed Problems: Theory and Numerics, Russia
  • Int'l Joint Research / Invited
• [Presentation] Carleman estimates and inverse problems for transport equation (2021)
  • Author(s): Masahiro Yamamoto
  • Organizer: Analysis and Numerics of Design, Control and Inverse Problems, Italy
  • Int'l Joint Research / Invited
• [Presentation] Direct and inverse problems for time-fractional partial differential equation-monotone method (2021)
  • Author(s): Masahiro Yamamoto
  • Organizer: Analysis, Control, and Numerics for PDE Models of Interests to Physics and Life Sciences, Italy
  • Int'l Joint Research / Invited
• [Presentation] Unique existence of solutions for some time fractional partial differential equations and some inverse problems: recent results (2021)
  • Author(s): Masahiro Yamamoto
  • Organizer: Nonlocal Diffusion Problems, Nonlocal Interface Evolution, Poland
  • Int'l Joint Research / Invited
• [Presentation] Recent results on direct and inverse problems for time-fractional partial differential equations (2021)
  • Author(s): Masahiro Yamamoto
  • Organizer: Workshop on Inverse Problems for Partial Differential Equations, China
  • Int'l Joint Research / Invited
• [Presentation] Inverse problems for transport equations: stability and uniqueness (2021)
  • Author(s): Masahiro Yamamoto
  • Organizer: Joint Fudan - RICAM Seminar on Inverse Problems, Austria
  • Int'l Joint Research / Invited
• [Presentation] Inverse problems for transport equations of first order by Carleman estimate (2021)
  • Author(s): Masahiro Yamamoto
  • Organizer: Eurasian Conference on Applied Mathematics, Novosibirsk, Russia
  • Int'l Joint Research / Invited