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Reconstruction methods in inverse boundary value problems

Research Project

Project/Area Number 13640115
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionGunma University

Principal Investigator

TANUMA Kazumi  Gunma University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (60217156)

Co-Investigator(Kenkyū-buntansha) SAITOH Saburoh  Gunma University, Faculty of Engineering, Professor, 工学部, 教授 (10110397)
ASHINO Ryuichi  Osaka Kyoiku University, Faculty of education, Associate Professor, 教育学部, 助教授 (80249490)
NAKAMURA Gen  Hokkaido University, Faculty of Science, Professor, 大学院・理学研究科, 教授 (50118535)
AMANO Kazuo  Gunma University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90137795)
IKEHATA Masaru  Gunma University, Faculty of Engineering, Professor, 工学部, 教授 (90202910)
片山 良一  大阪教育大学, 教育学部, 教授 (10093395)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2001: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordsinverse boundary value problems / impedance tomography / reconstruction formula / crystallographic texture / surface impedance tensor / Rayleigh wave / initial stress / acoustoelastic coefficient / texture / 音弾性 / Strohの定式化 / Dirichlet to Neumann map / 逆問題 / 導電方程式 / 弾性体方程式 / 非等方弾性体 / 再構成
Research Abstract

1.We considered the problem of determining conductivity of the medium from the measurements of the electric potential on the boundary and the corresponding current flux across the boundary, that is, from the Dirichlet to Neumann map. Three kinds of formulas for reconstructing conductivity and its normal derivative from the localized Dirichlet to Neumann map were obtained. They are the formulas for pointweise reconstruction, reconstruction in a weak form, and reconstruction in the form of Fourier transform. In these formulas, the normal derivative of the conductivity at the boundary is reconstructed from the localized Dirichlet to Neumann map without any information about the conductivity itself, which will give an efficient method in their numerical realization. Since our reconstruction formulas involve limiting process, we also gave the estimates for their convergences.
2.The surface impedance tensor is a Hermitian second-order tensor which, for a homogeneous elastic half-space, maps the displacements given at the surface to the tractions needed to sustain them. We accounted for the effects of crystallographic texture in orthorhombic aggregates of cubic crystallites only up to terms linear in the texture coefficients and gave an explicit formula for the terms in the surface impedance tensor up to those linear in the texture coefficients. By the same method, we gave a dispersion formula for the velocity of Rayleigh waves propagating textured polycrystals with initial stress, which shows how the initial stress and the crystallographic texture have an influence, within terms linear in them, on the angular dependency of the Rayleigh waves velocity. This formula includes terms that describe the effects of texture on the acoustoelastic coefficients.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report

Research Products

(27 results)

All Other

All Publications (27 results)

  • [Publications] Gen Nakamura: "Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map"Inverse Problems. 17. 405-419 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Gen Nakamura: "Direct determination of the derivatives of conductivity at the boundary from the localized Dirichlet-to-Neumann map"Communications of the Korean Mathematical Society. 16. 415-425 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Habib Ammari: "Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion"Journal of Elasticity. 67. 97-129 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazumi Tanuma: "Surface impedance tensors of textured polycrystals"Journal of Elasticity. 67. 131-147 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazumi Tanuma: "Angular dependence, of Rayleigh-wave velocity in prestressed polycrystalline media with monoclinic texture"Journal of Elasticity. 69. 181-214 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Gen Nakamura: "Formulas for reconstructing conductivity and its normal derivatives at the boundary from the localized Dirichlet to Neumann map"Recent Development in Theories & Numerics, International Conference on Inverse Problems, World Scientific. 192-201 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Ryuichi Ashino: "Smooth tight frame wavelets and image microanalysis in the Fourier domain"Computer Math.Applc.. 45. 1551-1579 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Ikehata: "Reconstruction formula for identifying cracks"Journal of Elasticity. 71. 59-72 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 田沼 一実: "係数同定のための逆問題"数学解析の望ましい姿を探って(九州大学出版会). 117-126 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Gen Nakamura, Kazumi Tanuma: "Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map"Inverse Problems. Vol.17, No.3. 405-419 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Gen Nakamura, Kazumi Tanuma: "Direct determination of the derivatives of conductivity at the boundary from the localized Dirichlet-to-Neumann map"Communications of the Korean Mathematical Society. Vol.16, No.3. 415-425 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Habib Ammari, Hyeonbae Kang, Gen Nakamura, Kazumi Tanuma: "Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion"Journal of Elasticity. Vol.67, No.2. 97-129 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazumi Tanuma, Chi-Sing Man, Mojia Huang, Gen Nakamura: "Surface impedance tensors of textured polycrystals"Journal of Elasticity. Vol.67, No.2. 131-147 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazumi Tanuma, Chi-Sing Man: "Angular dependence of Rayleigh wave velocity in prestressed polycrystalline media with monoclinic texture"Journal of Elasticity. Vol.69. 181-214 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Gen Nakamura, Kazumi Tanuma: "Formulas for reconstructing conductivity and its normal derivatives at the boundary from the localized Dirichlet to Neumann map"Recent Development in Theories & Numerics, International Conference on Inverse Problems, World Scientific. 192-201 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Ryuichi Ashino, Stevan J.Desjardins, Christopher Heil, Michihiro Nagase, Remi Vaillancourt: "Smooth tight frame wavelets and image microanalysis in the Fourier domain"Computer Math.Applic.. Vol.45, No.10-11. 1551-1579 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Ikehata, G.Nakamura: "Reconstruction formula for identifying cracks"Journal of Elasticity. Vol.71. 59-72 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazumi Tanuma: "Inverse problems for reconstructing coefficients"Towards better shapes of mathematical analysis (Kyushu University Press). 117-126 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazumi Tanuma: "Angular Dependence of Rayleigh-Wave Velocity in Prestressed Polycrystalline Media with Monoclinic Texture"Journal of Elasticity. 69. 181-214 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 田沼一実: "係数同定のための逆問題"数学解析の望ましい姿を探って(九州大学出版会). 117-126 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kazuni Tamura: "Surface impedance tensors of textured polycrystals"Journal of Elasticity. (印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Habib Ammari: "Complex asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion"Journal of Elasticity. (印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Gen Nakamura: "Formulas for reconstructing conductivity and its normal derivative at the boundary from localized Dirichlet to Neumann"Proceedings for the International Conference on Inverse Problems 2002 at City Univ. Hong Kong. (印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] G. Nakamura: "Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map"Inverse Problems. 17. 405-419 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Gen Nakamura: "Direct determination of the derivatives of conductivity at the boundary from the localized Dirichlet to Neumann map"Communications of the Korean Mathematical Society. 16・3. 415-425 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Ashino: "Microfiltering with multiwavelets"Computers Math. Applic.. 41・1-2. 111-133 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y. Katayama: "The structure of the automorphism group of an approximately finite dimensional factor"Mathematical Physics in Mathematics and Physics, Field Inst. Commun. A. M. S.. 30. 249-260 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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