Development of the theory of diffusion equations for analysis on data separation

• Project/Area Number: 20K20342
• Project/Area Number (Other): 18H05323 (2018-2019)
• Research Category: Grant-in-Aid for Challenging Research (Pioneering)
• Allocation Type: Multi-year Fund (2020); Single-year Grants (2018-2019)
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: 儀我 美一 東京大学, 大学院数理科学研究科, 特任教授 (70144110)
• Co-Investigator(Kenkyū-buntansha): 石毛 和弘 東京大学, 大学院数理科学研究科, 教授 (90272020) ; 行木 孝夫 北海道大学, 理学研究院, 教授 (40271712) ; 黒田 紘敏 北海道大学, 理学研究院, 准教授 (80635657)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Project Status: Granted (Fiscal Year 2022)
• Budget Amount: ¥25,870,000 (Direct Cost: ¥19,900,000、Indirect Cost: ¥5,970,000); Fiscal Year 2023: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2022: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2021: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2020: ¥9,620,000 (Direct Cost: ¥7,400,000、Indirect Cost: ¥2,220,000); Fiscal Year 2019: ¥10,140,000 (Direct Cost: ¥7,800,000、Indirect Cost: ¥2,340,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: クリスタライン曲率流 / 小林・ワレン・カーターエネルギー / 輪郭線 / 分数冪時間微分方程式 / 全変動流方程式 / 拡散型偏微分方程式 / クラスタリング / 二相分離 / 拡散方程式 / 調和写像流

Outline of Research at the Start

データ分離問題は、機械学習における基本的な問題である。これまでは、統計的手法や、離散数学的手法を用いて、何らかの評価関数を最小にするものを求める方法が主であった。しかしデータ数が増えると、離散的方法は計算量が増えて困難になる。そこで連続的空間でその威力を発揮した偏微分方程式を用いる方法を導入したい。特に、単に評価関数の最小を求めるだけではなく、最小にどのような意味でどれくらい近いかを解明するため、非平衡非線形現象を記述する拡散方程式を用い、それによりデータ分離問題に新たな視点を与えたい。

Outline of Annual Research Achievements

創薬の問題では、莫大な分子構造の中から期待される機能を持ったものを選ばなくてはならない。分子構造の類似性がある物質から選ぶことは至難の業となっている。一方で、分子の重量等の表現空間から機械学習的手法で選ぶことは重要である。本研究では熱方程式を応用して、極めて有効な手法を提案できた。これはデータ分離問題である。分離線の構築が鍵となり、画像から必要部分の輪郭線抽出問題と類似の側面がある。特にノイズ除去には全変動流方程式やクリスタライン曲率流方程式が用いられる。全変動流方程式については４階の場合の理論を整備し、クリスタライン曲率流方程式については、これまでの成果および現状をサーベイした。
一方で、将来のデータ分離や画像処理のために、材料科学の多粒界モデルとして知られている小林・ワレン・カーターモデルについて、そのエネルギーの鋭敏界面モデルを求め、その収束問題を議論した。

Current Status of Research Progress

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

Reason

長く続くコロナ禍で、対面での研究会の開催は困難であったが、オンライン会合を重ね、当初の成果をあげることができた。

Strategy for Future Research Activity

小林・ワレン・カーターエネルギーの鋭敏界面極限は空間１次元の場合は既に求められているが、その高次元版はただちに構築することが困難であったので、うまく収束が示せる位相を探索する。またその勾配流の特異極限は、通常の勾配流にはならないことがわかっているが、その具体的な形を導出する。

Research Products

• [Int'l Joint Research] ダルムシュタット工科大学(ドイツ)
• [Int'l Joint Research] ワルシャワ大学(ポーランド)
• [Int'l Joint Research] ワルシャワ大学(ポーランド)
• [Journal Article] Crystalline surface diffusion flow for graph-like curves (2023)
  • Author(s): Giga Mi-Ho、Giga Yoshikazu
  • Journal Title: Discrete and Continuous Dynamical Systems Volume: 43 Issue: 3&4 Pages: 1436-1468
  • DOI: 10.3934/dcds.2022160
  • Peer Reviewed / Open Access
• [Journal Article] Continuity of Derivatives of a Convex Solution to a Perturbed One-Laplace Equation by p-Laplacian (2022)
  • Author(s): Giga Yoshikazu、Tsubouchi Shuntaro
  • Journal Title: Archive for Rational Mechanics and Analysis Volume: 244 Issue: 2 Pages: 253-292
  • DOI: 10.1007/s00205-022-01763-7
  • Peer Reviewed / Open Access
• [Journal Article] Exhaustive existence and non-existence results for Hardy-Henon equations in R^n (2022)
  • Author(s): Giga Yoshikazu、Ngo Quoc Anh
  • Journal Title: Partial Differential Equations and Applications Volume: 3 Issue: 6
  • DOI: 10.1007/s42985-022-00190-3
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Motion by crystalline-like mean curvature: A survey (2022)
  • Author(s): Giga Yoshikazu、Pozar Norbert
  • Journal Title: Bulletin of Mathematical Sciences Volume: 12 Issue: 02 Pages: 1-68
  • DOI: 10.1142/s1664360722300043
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] 熱拡散方程式 (HDE) モデルによる新薬探索支援AIツールの開発 (2022)
  • Author(s): 今村 恵子、日高 中、儀我 美保、儀我 美一、井上 治久
  • Journal Title: 革新的AI創薬 Volume: - Pages: 193-197
  • Peer Reviewed
• [Journal Article] 相の構造を考慮した拡散界面エネルギーとその特異極限 (2022)
  • Author(s): 儀我 美一、上坂 正晃
  • Journal Title: 応用数理 Volume: 32 Pages: 186-197
  • Peer Reviewed
• [Journal Article] Crystalline flow starting from a general polygon (2022)
  • Author(s): Giga Mi-Ho、Giga Yoshikazu、Kuroda Ryo、Ochiai Yusuke
  • Journal Title: Discrete & Continuous Dynamical Systems Volume: 42 Issue: 4 Pages: 2027-2027
  • DOI: 10.3934/dcds.2021182
  • Peer Reviewed / Open Access
• [Journal Article] Normal Trace for a Vector Field of Bounded Mean Oscillation (2022)
  • Author(s): Giga Yoshikazu、Gu Zhongyang
  • Journal Title: Potential Analysis Volume: － Issue: 1 Pages: 409-434
  • DOI: 10.1007/s11118-021-09973-6
  • Peer Reviewed
• [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] A finer singular limit of a single-well Modica?Mortola functional and its applications to the Kobayashi?Warren?Carter energy (2021)
  • Author(s): Giga Yoshikazu、Okamoto Jun、Uesaka Masaaki
  • Journal Title: Advances in Calculus of Variations Volume: － Issue: 1 Pages: 163-182
  • DOI: 10.1515/acv-2020-0113
  • Peer Reviewed / Open Access
• [Journal Article] The hydrostatic approximation for the primitive equations by the scaled Navier?Stokes equations under the no-slip boundary condition (2021)
  • Author(s): Furukawa Ken、Giga Yoshikazu、Kashiwabara Takahito
  • Journal Title: Journal of Evolution Equations Volume: 21 Issue: 3 Pages: 3331-3373
  • DOI: 10.1007/s00028-021-00674-6
  • Peer Reviewed / Open Access
• [Journal Article] The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$ (2021)
  • Author(s): Giga Yoshikazu、Gries Mathis、Hieber Matthias、Hussein Amru、Kashiwabara Takahito
  • Journal Title: Journal of Evolution Equations Volume: 21 Issue: 4 Pages: 4145-4169
  • DOI: 10.1007/s00028-021-00716-z
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions (2021)
  • Author(s): Y. Giga, F. Onoue and K. Takasao
  • Journal Title: Differential and Integral Equations Volume: 34 Pages: 21-126
  • Peer Reviewed
• [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] The hydrostatic Stokes semigroup and well-posedness of the primitive equations on spaces of bounded functions (2020)
  • Author(s): Giga Yoshikazu、Gries Mathis、Hieber Matthias、Hussein Amru、Kashiwabara Takahito
  • Journal Title: Journal of Functional Analysis Volume: 279 Issue: 3 Pages: 108561-108561
  • DOI: 10.1016/j.jfa.2020.108561
  • NAID: 120006383828
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Rigorous justication of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equations (2020)
  • Author(s): K. Furukawa, Y. Giga, M. Hieber, A. Hussein, T. Kashiwabara and M. Wrona
  • Journal Title: Nonlinearity Volume: 33 Issue: 12 Pages: 6502-6516
  • DOI: 10.1088/1361-6544/aba509
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On boundary detachment phenomena for the total variation flow with dynamic boundary conditions (2020)
  • Author(s): Giga Yoshikazu、Nakayashiki Ryota、Rybka Piotr、Shirakawa Ken
  • Journal Title: Journal of Differential Equations Volume: 269 Issue: 12 Pages: 10587-10629
  • DOI: 10.1016/j.jde.2020.07.015
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Discrete schemes for time-fractional fully nonlinear evolution equations and their convergence (2020)
  • Author(s): Y. Giga, Q. Liu, H. Mitake
  • Journal Title: Asymptot. Anal. Volume: accepted Issue: 1-2 Pages: 1-12
  • DOI: 10.3233/asy-191583
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A new numerical scheme for discrete constrained total variation flows and its convergence (2020)
  • Author(s): Y. Giga, K. Sakakibara, K. Taguchi and M. Uesaka
  • Journal Title: Numer. Math. Volume: 146 Issue: 1 Pages: 181-217
  • DOI: 10.1007/s00211-020-01134-y
  • Peer Reviewed / Open Access
• [Journal Article] Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term (2020)
  • Author(s): Y. Giga and N. Pozar
  • Journal Title: SN Partial Differ. Equ. Appl. Volume: 1 Issue: 6 Pages: 1-26
  • DOI: 10.1007/s42985-020-00040-0
  • Peer Reviewed / Open Access
• [Journal Article] On the Helmholtz decompositions of vector fields of bounded mean oscillation and in real Hardy spaces over the half space (2020)
  • Author(s): Y. Giga and Z. Gu
  • Journal Title: Adv. Math. Sci. Appl. Volume: 29 Pages: 87-128
  • Peer Reviewed
• [Journal Article] Prediction of compound bioactivities using heat-diffusion equation (2020)
  • Author(s): T. Hidaka, K. Imamura, T. Hioki, T. Takagi, Y. Giga, M.-H. Giga, Y. Nishimura, Y. Kawahara, S. Hayashi, T. Niki, M. Fushimi and H. Inoue
  • Journal Title: Patterns Volume: 1 Issue: 9 Pages: 100140-100140
  • DOI: 10.1016/j.patter.2020.100140
  • Peer Reviewed / Open Access
• [Journal Article] Numerical computations of split Bregman method for fourth order total variation flow (2020)
  • Author(s): Y. Giga and Y. Ueda
  • Journal Title: J. Comput. Phys. Volume: 405 Pages: 109114-109114
  • DOI: 10.1016/j.jcp.2019.109114
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Mathematical structures for epilepsy: High-frequency oscillation and interictal epileptic slow (red slow). (2020)
  • Author(s): Namiki, T., Tsuda, I., Tadokoro, S., Kajikawa, S., Kunieda, T., Matsumoto, R., Matsuhashi, M., Ikeda, A.
  • Journal Title: Neurosci Res Volume: - Pages: 178-187
  • DOI: 10.1016/j.neures.2019.11.008
  • Peer Reviewed
• [Journal Article] Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms (2019)
  • Author(s): Y. Giga, H. Mitake and H. V. Tran
  • Journal Title: Discrete Contin. Dyn. Syst. Ser. B Volume: Online First Issue: 10 Pages: 3983-3999
  • DOI: 10.3934/dcdsb.2019228
  • NAID: 120006643619
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Quantitative description and classification of protein structures by a novel robust amino acid network: interaction selective network (ISN) (2019)
  • Author(s): Shohei KONNO, Takao NAMIKI, Koichiro ISHIMORI
  • Journal Title: Scientific Reports Volume: 9 Issue: 1 Pages: 16654-16654
  • DOI: 10.1038/s41598-019-52766-6
  • Peer Reviewed / Open Access
• [Journal Article] Relation between spatio-temporal patterns generated by two-dimensional cellular automata and a singular function (2019)
  • Author(s): A. Kawaharada and T. Namiki
  • Journal Title: International Journal of Networking and Computing Volume: 9 Issue: 2 Pages: 354-369
  • DOI: 10.15803/ijnc.9.2_354
  • NAID: 130007680124
  • ISSN: 2185-2839, 2185-2847
• [Presentation] The level-set mean curvature flow equation versus the total variation flow equation (2022)
  • Author(s): Y. Giga
  • Organizer: Catch-all Mathematical Colloquium of Japan
  • Int'l Joint Research / Invited
• [Presentation] Continuity of derivatives of a convex solution to a total variation equation perturbed by p-Laplacian (2022)
  • Author(s): Y. Giga
  • Organizer: UWA Analysis Seminars
  • Int'l Joint Research / Invited
• [Presentation] Motion by crystalline-like mean curvature (2022)
  • Author(s): Y. Giga
  • Organizer: School of Mathematical Sciences, Peking University
  • Int'l Joint Research / Invited
• [Presentation] The fourth-order total variation flows in R^n (2022)
  • Author(s): Y. Giga
  • Organizer: School of Mathematical Sciences, Peking University
  • Int'l Joint Research / Invited
• [Presentation] On a singular limit of a single-well Modica-Mortola functional and its applications (2022)
  • Author(s): Y. Giga
  • Organizer: ESI workshop: Free Boundary Problems and related Evolution Equations
  • Int'l Joint Research / Invited
• [Presentation] Motion by crystalline-like mean curvature (2022)
  • Author(s): Y. Giga
  • Organizer: SIAM Conference on Analysis of Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] 有界領域上の有界平均振動ベクトル場のヘルムホルツ分解 (2021)
  • Author(s): 儀我 美一
  • Organizer: 応用解析研究会
  • Invited
• [Presentation] 相転移を伴う熱流体現象の数理モデリング 第1話 - 第10話 (2021)
  • Author(s): 儀我 美一
  • Organizer: 社会連携講座「冷媒熱流体の数理」
  • Invited
• [Presentation] On the Helmholtz decomposition of $BMO$ space of vector fields (2021)
  • Author(s): Y. Giga
  • Organizer: Geometric and Functional Inequalities and Applications
  • Int'l Joint Research / Invited
• [Presentation] 熱交換器 1 - 6 (2021)
  • Author(s): 儀我 美一
  • Organizer: 社会連携講座「冷媒熱流体の数理」
  • Invited
• [Presentation] On a singular limit of a single-well Modica-Mortola functional and its applications (2021)
  • Author(s): Y. Giga
  • Organizer: Nonlinear Analysis Seminar Series
  • Int'l Joint Research / Invited
• [Presentation] 単底型モディカ・モルトラ汎関数の微細な特異極限 (2021)
  • Author(s): 儀我美一
  • Organizer: Workshop on Analysis in Kagurazaka 2021
  • Invited
• [Presentation] On total variation flow type equations (2021)
  • Author(s): Y. Giga
  • Organizer: Seminar Analysis and Applications
  • Int'l Joint Research / Invited
• [Presentation] On some macroscopic models of phase transitions (2021)
  • Author(s): Y. Giga
  • Organizer: Nonlinear Science Seminar
  • Int'l Joint Research / Invited
• [Presentation] Spatially discrete total variation map flow and its time discrete approximation (2021)
  • Author(s): Y. Giga
  • Organizer: Differential Equations for Data Science
  • Int'l Joint Research / Invited
• [Presentation] 微分方程式で表現される粘性や拡散の効果 (2021)
  • Author(s): 儀我美一
  • Organizer: 談話会
  • Invited
• [Presentation] A finer singular limit of a single-well Modica-Mortola functional and its applications (2020)
  • Author(s): Y. Giga
  • Organizer: Nonlocal diffusion problems, nonlocal interface evolution
  • Int'l Joint Research / Invited
• [Presentation] On total variation flow type equations (2020)
  • Author(s): Y. Giga
  • Organizer: SN PDE Webinar
  • Int'l Joint Research / Invited
• [Presentation] A finer singular limit of a single-well Modica-Mortola functional and its applications (2020)
  • Author(s): Y. Giga
  • Organizer: Partial Differential Equations under Various Metrics
  • Int'l Joint Research / Invited
• [Presentation] On total variation flow under dynamic boundary conditions (2020)
  • Author(s): Y. Giga
  • Organizer: Qualitative Properties for Nonlinear Diffusion Equations
• [Presentation] On total variation flow type equations (2019)
  • Author(s): Y. Giga
  • Organizer: PDE seminar series
• [Presentation] On the large time behavior of solutions to birth and spread type equations (2019)
  • Author(s): Y. Giga
  • Organizer: Dynamics, Equations and Applications (DEA 2019)
• [Presentation] On total variation flow type equations (2019)
  • Author(s): Yoshikazu Giga
  • Organizer: Colloquium
  • Int'l Joint Research / Invited
• [Presentation] On large time behavior of solutions to birth and spread type equations (2019)
  • Author(s): Yoshikazu Giga
  • Organizer: Surface, Bulk, and Geometric Partial Differential Equations: Interfacial, stochastic, non-local and discrete structures
  • Int'l Joint Research / Invited
• [Presentation] 全変動流型方程式 (2019)
  • Author(s): 儀我 美一
  • Organizer: 日本数学会2019年度年会 実函数論分科会 特別講演
  • Invited
• [Presentation] 全変動流方程式とその仲間達 (2018)
  • Author(s): 儀我 美一
  • Organizer: 第22回 早稲田大学 数学・応用数理談話会
  • Invited
• [Presentation] 微分方程式の解を巡って (2018)
  • Author(s): 儀我 美一
  • Organizer: Arithmer Seminar #01
  • Invited
• [Funded Workshop] Differential Equations for Data Science 2023 (2023)
• [Funded Workshop] Interface and Free Boundaries (2022)
• [Funded Workshop] Preliminary meeting for Interfaces and Free Boundaries (2021)