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画像処理の数理における実解析的手法の探索

Research Project

Project/Area Number 15654023
Research Category

Grant-in-Aid for Exploratory Research

Allocation TypeSingle-year Grants
Research Field Basic analysis
Research InstitutionTohoku University (2004-2005)
Kyushu University (2003)

Principal Investigator

小川 卓克  東北大学, 大学院・理学研究科, 教授 (20224107)

Co-Investigator(Kenkyū-buntansha) 服部 哲弥  東北大学, 大学院・理学研究科, 教授 (10180902)
木村 正人  九州大学, 大学院・数理学研究院, 助教授 (70263358)
後藤 俊一  北海道教育大学, 教育学部, 助教授 (30225651)
石井 克幸  神戸大学, 海事科学部, 助教授 (40232227)
松本 敏隆  広島大学, 理学部, 助手 (20229561)
長藤 かおり  九州大学, 大学院・数理学研究院, 助教授 (40326426)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywords平均曲率流 / 数値アルゴリズム / 粘性解 / 鉄磁性モデル / Schroedinger写像 / 特異摂動問題 / 無限フラクタル格子 / integrated semigroup / Navier-Stokes方程式 / 調和写像流 / Euler方程式 / 解の正則性条件 / Besov空間 / 対数型臨界Sobolev不等式 / 粘性消滅極限 / 複素Ginzburg-Landau方程式 / 半導体素子方程式
Research Abstract

研究実績は以下のとおり.
研究代表者の小川は研究分担者の石井克幸と協力者の後藤陽子と共同で平均曲率流方程式を等高面の方法で考え、そのBence-Merrimen-Osher型の数値解析アルゴリズムの解への収束を、半線形熱方程式の解に対する特異摂動の観点から考え、粘性解の方法により証明した。平均曲率流方程式は特に画像処理の際のノイズ消去に有効に用いられるがその場合のBMOアルゴリズムの有効性が示せた.
また,小川は単独で,鉄磁性体の2次元ising型spinモデル(シグマ模型)に対する連続体近似を考え、その半線形化方程式のエネルギー空間における可解性を新しいゲージ変換を考えることにより与えた。また関連して、粘性が入る場合に鉄磁性体モデルとSchrodinger写像、調和写像熱流との相関を議論し、それぞれ係数が極限と成る場合の状況について、ゲージ変換による議論、単調性公式による議論により特異性の発生について考察した。
分担者の服部はプレシルピンスキーガウケットと呼ばれる無限フラクタル格子上の単純ランダムウォークと自己回避確率連鎖を連続的に内挿する自己抑制・吸引的確率連鎖の族を構成し,変位の指数を与えた.
分担者の木村はパラメータを含む移流項を持つ楕円型方程式の第一固有値の特異摂動問題を考察した.移流の代表速度を表すパラメータが無限大に近づくとき起こる固有値の指数減衰現象について,空間1次元の場合に精密な漸近挙動評価を得た.
分担者の松本は生成作用素の定義域が稠密でないanalytic semigroupおよび、integrated semigroupの時間に依存しない非線形摂動を考察し、汎関数を用いた一般的な増大条件の下で、弱解を与える発展作用素が存在するための必要十分条件を、方程式に対する陰的差分近似の存在によって与えた。

Report

(3 results)
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (27 results)

All 2006 2005 2004 Other

All Journal Article (16 results) Book (2 results) Publications (9 results)

  • [Journal Article] Ferromagnetic spin model and the Schrodinger map on two dimensions, "Nonlinear Dispersive Equations" (T.Ozawa and Y.Tsutsumi eds),2006

    • Author(s)
      T.Ogawa
    • Journal Title

      GAKUTO International Ser.Math.Sci.Appl., (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The uniform boundedness of the radial solution for drift-diffusion system,2006

    • Author(s)
      M.Kurokiba, T.Nagai, T.Ogawa
    • Journal Title

      Comm.Pure Appl.Anal. (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Ferromagnetic spin model and the Landau-Lifschiz equation "Harmonic Analysis and Nonlinear Partial Differential Equations",2006

    • Author(s)
      T.Ogawa
    • Journal Title

      数理解析研究所講究録 (in press)

    • NAID

      110004663190

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Optimal rate of convergence of the Bence-Merriman-Osher algorithm for motion by mean curvature,2006

    • Author(s)
      K.Ishii
    • Journal Title

      数理解析研究所講究録 (in press)

    • NAID

      120000949789

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Abstract Cauchy problems for quasi-linear evolution equations with non-densely defined operators,2006

    • Author(s)
      T.Matsumoto, N.Tanaka
    • Journal Title

      Taiwanese J.Math. (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Convergence of the approximation scheme of the motion by mean curvature in the Bence-Merriman-Osher algorithm,2005

    • Author(s)
      Y.Goto, K.Ishii, T.Ogawa
    • Journal Title

      Comm.Pure Appl.Anal., 4

      Pages: 311-339

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Displacement exponents of self-repelling walks and self-attracting walks on the pre-Sierpinski gasket,2005

    • Author(s)
      K.Hattori, T.Hattori
    • Journal Title

      Journal of Mathematical Science University of Tokyo, 12

      Pages: 417-443

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 速い移流を持つStrum-Liouville問題の第1固有値の精密な漸近挙動2005

    • Author(s)
      木村正人, 永田真一
    • Journal Title

      日本応用数理学会論文誌 15

      Pages: 209-220

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Stability analysis for Allen-Cahn type equation associated with the total variation energy2005

    • Author(s)
      K.Shirakawa, M.Kimura
    • Journal Title

      Nonlinear Analysis 60

      Pages: 257-282

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Optimal rate of convergence of the Bence-Merriman-Osher algorithm for motion by mean curvature,2005

    • Author(s)
      K.Ishii
    • Journal Title

      SIAM J.Math.Anal., 4

      Pages: 841-866

    • NAID

      120000949789

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Convergence of the approximation scheme of the motion by mean curvature in the Bence-Merriman-Osher algorithm2005

    • Author(s)
      Y.Goto, K.Ishii, T.Ogawa
    • Journal Title

      Comm.Pure Appl.Anal. (in press)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Stability analysis for Allen-Cahn type equation associated-with the total variation energy2005

    • Author(s)
      K.Shirakawa, M.Kimura
    • Journal Title

      Nonlinear Analysis (in press)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The limiting uniquness criterion by vorticity to Navier-Stokes equations in Besov spaces2004

    • Author(s)
      T.Ogawa, Y.Taniuchi
    • Journal Title

      Tohoku Math.J. 56

      Pages: 65-77

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Uniquness and inviscid limit to the complex Ginzburg-Landau equation in two dimensional general domain2004

    • Author(s)
      T.Ogawa, T.Yokota
    • Journal Title

      Comm.Math.Phys. 245

      Pages: 105-121

    • Related Report
      2004 Annual Research Report
  • [Journal Article] L^p-L^q type estimate for the semi-linear dumped wave equation in two dimensions.2004

    • Author(s)
      T.Hosono, T.Ogawa
    • Journal Title

      J.Differential Equations 203

      Pages: 82-118

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Time-dependent nonlinear perturbations of analytic and integrated semigroups2004

    • Author(s)
      T.Matsumoto, S.Oharu, N.Tanaka
    • Journal Title

      Nonlinear Partial Differential Equations and Their Applications, GAKUTO Internat.Ser.Math.Sci.Appl. 20

      Pages: 430-449

    • Related Report
      2004 Annual Research Report
  • [Book] 要説 わかりやすい微分積分2005

    • Author(s)
      小川卓克
    • Total Pages
      169
    • Publisher
      サイエンス社
    • Related Report
      2005 Annual Research Report
  • [Book] ランダムウォークとくりこみ群2004

    • Author(s)
      服部哲弥
    • Total Pages
      352
    • Publisher
      共立出版
    • Related Report
      2004 Annual Research Report
  • [Publications] T.Ogawa: "Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow."SIAM J.Math.Anal.. 34. 1317-1329 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Ogawa, Y.Taniuchi: "A note on blow-up criterion to the 3-D Euler Equations in a bounded domain,"J.Math.Fluid Mech.. 5. 17-23 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Ogawa, Y.Taniuchi: "On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain.,"J.Differential Equations. 190. 36-63 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kurokiba, T.Ogawa: "Finite time blow-up of the solution for the nonlinear parabolic equation of the drift diffusion type,"Diff.Integral Equations. 16. 427-452 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Kozono, T.Ogawa, Y.Taniuchi: "Navier-Stokes equations in the Besov space near L^∞ and BMO,"Kyushu J.Math.. 57. 303-324 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Ogawa, Y.Taniuchi: "The limiting uniquness criterion by vorticity to Navier-Stokes equations in Besov spaces,"Tohoku Math.J.. 56. 65-77 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Ogawa, T.Yokota: "Uniquness and inviscid limit to the complex Ginzburg-Landau equation in two dimensional general domain,"Comm.Math.Phys.. 245. 105-121 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Nagatou: "A computer-assisted proof on the stability of the Kolmogorov flows of incompressible viscous fluid."Journal of Computational and Applied Mathematics. (in press). (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Matsumoto, S.Oharu, K.Shitaoka: "On the semigroup approach to age-dependent spatially distributed two sex models of population dynamics,"Adv.Math.Sci.Appl.. 13. 423-442 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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