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2006 Fiscal Year Final Research Report Summary

Numerical harmonic analysis and its applications to image analysis and computational mathematics

Research Project

Project/Area Number 16340041
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionTOKYO METROPOLITAN UNIVERSITY

Principal Investigator

OKADA Masami  Tokyo Metropolitan University, Graduate School of Science and Technology, Professor, 理工学研究科, 教授 (00152314)

Co-Investigator(Kenkyū-buntansha) SAKAI Makoto  Tokyo Metropolitan University, Graduate School of Science and Technology, Professor, 理工学研究科, 教授 (70016129)
KURATA Kazuhiro  Tokyo Metropolitan University, Graduate School of Science and Technology, Professor, 理工学研究科, 教授 (10186489)
TAKAKURA Shouichirou  Tokyo Metropolitan University, Graduate School of Science and Technology, Professor, 理工学研究科, 教授 (10183435)
MIYACHI Akihiko  Tokyo Woman's Christian University, Dept of Math, Professor, 文理学部, 教授 (60107696)
NAKAMURA Yoshimasa  Kyoto Univ., GSI, Dept of Applied Math. and Physics, Professor, 大学院情報学研究科, 教授 (50172458)
Project Period (FY) 2004 – 2006
Keywordsspline / sampling / computational harmonic analysis / numerical analysis / KP equation / difference scheme
Research Abstract

Outline :
We have constructed new modified spline functions in multi-dimensions and have applied them to numerical analysis of nonlinear partial differential equations and image analysis. Also, we invited from abroad totally 10 active researchers in three years to organize international meetings. The details are summarized as follows.
1.Computational harmonic analysis :
We succeeded in construction of basis functions with "minimal" supports which enables rigorous interpolation and good sampling approximation of functions. Basis functions defined in more general domains are also investigated, i.e. in the case of finite intervals, several typical domains in the plane, e.t.c. To our surprise, almost radial basis functions have been obtained in several dimensions.
2.Application lo numerical analysis of PDE :
As an application, we have found a new method of computiong a series of finite difference schemes of higher orders to partial differential oparators. Also, we computed stable numerical solution to the important KP equation whose analytical solutions are difficult to find. The study of multi-scale phenomena and image analysis is ongoing.
3.Harmonic analysis of complex singularity :
In a joint work with oversea researchers, we developed the theory of harmonic analysis on a complex variety with singularities. This work will hopefully inspire us in future works related to singularities.

  • Research Products

    (11 results)

All 2006

All Journal Article (10 results) Book (1 results)

  • [Journal Article] Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method.2006

    • Author(s)
      Ide, Takanori, Okada, Masami
    • Journal Title

      J. Comput. Appl. Math. 194・No.2

      Pages: 425-459

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrodinger equation.2006

    • Author(s)
      Kurata, Kazuhiro, Watanabe, Tatsuya
    • Journal Title

      Commun. Pure Appl. Anal. 5・No.3

      Pages: 595-608

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Weighted Hardy spaces on an interval and Poisson integrals associated with ultraspherical series.2006

    • Author(s)
      Miyachi, Akihiko
    • Journal Title

      J. Funct. Anal. 239・No.2

      Pages: 446-496

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Transplantation theorem for Jacobi series in weighted Hardy spaces. II.2006

    • Author(s)
      Miyachi, Akihiko
    • Journal Title

      Math. Ann. 336・No.1

      Pages: 111-153

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] New numerical integrator for the Stackel system conserving the same number of constants of motion as the degree of freedom.2006

    • Author(s)
      Minesaki, Yukitaka, Nakamura, Yoshimasa
    • Journal Title

      J. Phys. A 39・No.30

      Pages: 9453-9476

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method.2006

    • Author(s)
      Ide, Takanori, Okada, Masami
    • Journal Title

      J.Comput.Appl.Math. 194・No.2

      Pages: 425-459

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrodinger equation.2006

    • Author(s)
      Kurata, Kazuhiro, Watanabe, Tatsuya
    • Journal Title

      Commun.Pure Appl.Anal. 5・No.3

      Pages: 595-608

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Weighted Hardy spaces on an interval and Poisson integrals associated with ultraspherical series.2006

    • Author(s)
      Miyachi, Akihiko
    • Journal Title

      J.Funct.Anal. 239・No.2

      Pages: 446-496

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Transplantation theorem for Jacobi series in weighted Hardy spaces. II.2006

    • Author(s)
      Miyachi, Akihiko
    • Journal Title

      Math.Ann. 336・No.1

      Pages: 111-153

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] New numerical integrator for the Stackel system conserving the same number of constants of motion as degree of freedom.2006

    • Author(s)
      Minesaki, Yukitaka, Nakamura, Yoshimasa
    • Journal Title

      J.Phys.A 39・No.30

      Pages: 9453-9476

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] Harmonic analysis and its applications.2006

    • Author(s)
      Akihiko Miyachi, Eiichi Nakai, Masami Okada
    • Total Pages
      122
    • Publisher
      Yokohama Publishers, Yokohama
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2008-05-27  

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