RECONSTRUCTION PROCEDURE FOR INVERSE PROBLEMS

逆问题的重构过程

• 批准号: 12640153
• 负责人: NAKAMURA Gen
• 金额: $ 2.56万
• 依托单位: HOKKAIDO UNIVERSITY (2001)Gunma University (2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640153/
• 关键词: INVERSE PROBLEMS; DIRICHLET TO NEUMANN MAP; UNIQUENESS; RECONSTRUCTION; DISCRETIZATION; RESIDUAL STRESS; TRANSVERSALLY ISOTROPIC; INVERSE SCATTERING; ACOUSTICAL SCATTERING; INCLUSION; PIEZOELECTRICITY; WELL POSEDNESS; CRACK; CONDUCTIVITY EQUA

The following results were obtained.(1) As a preparation of the research on the inverse problem for piezoelectricity, the well posedness of the initial boundary value problem for the equation of piezoelectricity was proven.(2) The uniqueness of identifying the coefficients of the nonlinear term of the second degree in the conductivety equation from the Dirichlet to Neumann map was proven.(3) The identification of the residual stress depending on the depth was considered for the inverse problem associated with the Lamb problem. The reconstruction procedure at the boundary was given for the general case and that of in the interior was given for a special case.(4) In order to apply the oscillating-decaying solution to inverse problem, the relation between the Cauchy data of the boundary value problem for anisotropic conductive equation with analytic conductivity and the singularity of its solution was shown.(5) The asymptotic expansion of the solution for the Lame system with inclusions as their diameter tends to zero was proven.(6) The reconstruction procedure of identifying impenetrable obstacles and their physical properties for acoustical inverse scattering problem from farfield pattern was given.(7) A reconstruction formula for identifying the material coefficients pointwisely from the localized Dirichlet to Neumann map was given.(8) The unique continuation for transversally isotropic elastic equation and isotropic elastic equation with residual stress were proven. Also, as their application Runge properties for these equations were proven.(9) An approximate identification for the inverse boundary value problem for the Schrodinger equation with potential from the discretized Dirichlet to Neumann map was proven.(10) The global uniqueness for the inverse boundary value problem for identifying the convection term of the steady state heat equation from the Dirichlet to Neuman map was proven.

得到了以下结果：(1)作为压电反问题研究的铺垫，证明了压电方程初边值问题的适定性。(2)证明了Dirichlet - Neumann映射电导率方程中二阶非线性项系数辨识的唯一性。(3)针对与Lamb问题相关的逆问题，考虑了随深度变化的残余应力识别。给出了一般情况下在边界处的重建过程，特殊情况下在内部的重建过程。(4)为了将振荡衰减解应用于反问题，给出了具有解析电导率的各向异性导电方程边值问题的柯西数据与其解的奇异性之间的关系。(5)证明了含有夹杂物的Lame体系在直径趋于零时解的渐近展开式。(6)给出了远场声反散射问题中识别不可穿透障碍物及其物理性质的重建过程。(7)给出了从局部Dirichlet - Neumann映射中逐点识别材料系数的重构公式。(8)证明了横向各向同性弹性方程和带残余应力的各向同性弹性方程的唯一延拓性。同时，作为应用，证明了这些方程的龙格性质。(9)证明了具有势的薛定谔方程边值反问题从离散Dirichlet到Neumann映射的近似辨识。（10）证明了从Dirichlet到Neuman映射识别稳态热方程对流项的反边值问题的全局唯一性。

项目成果

J.Cheng: "Uniqueness of identifying the convection term"Comm. Korean Math. Soc.. 16. 405-413 (2001)

J.Cheng：“识别对流项的唯一性”Comm。

G. Nakamura: "Uniqueness for on inverse boundary value problem for Dirac operators"Comm. Partial Differential Equations. 25. 1327-1369 (2000)

G. Nakamura：“狄拉克算子的逆边值问题的唯一性”Comm。

J.Cheng: "Stability for the inverse potential problem by finite measurements on the boundary"Inverse Problems. 17. 273-280 (2001)

J.Cheng：“边界上有限测量的逆位势问题的稳定性”逆问题。

M.Akamatsu: "Well-poseclness of initial-boundary value problem for piezoelectric equations"Applicable Analysis. (in press).

M.Akamatsu：“压电方程初始边值问题的良好性”应用分析。

G. Nakamaura, S. Saitoh, A. Syarif: "Representations and harmonic extension formulas of harmonic functions on half spaces"Complex Variables Theory Appl.. 42. 323-332 (2000)

G. Nakamaura、S. Saitoh、A. Syarif：“半空间上调和函数的表示和调和扩展公式”复变量理论应用.. 42. 323-332 (2000)