2004 Fiscal Year Final Research Report Summary
Inverse Problems for the Family of Wave Equations
| Project/Area Number |
14340038
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | HOKKAIDO UNIVERSITY |
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Principal Investigator |
NAKAMURA Gen Hokkaido Univ., Grad.School of Sci., Professor, 大学院・理学研究科, 教授 (50118535)
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| Co-Investigator(Kenkyū-buntansha) |
OZAWA Tohuru Hokkaido Univ., Grad.School of Sci., Professor, 大学院・理学研究科, 教授 (70204196)
JINBO Shuichi Hokkaido Univ., Grad.School of Sci., Professor, 大学院・理学研究科, 教授 (80201565)
TONEGAWA Yoshiro Hokkaido Univ., Grad.School of Sci., Asso.Prof., 大学院・理学研究科, 助教授 (80296748)
TSUTAYA Kimitoshi Hokkaido Univ., Grad.School of Sci., Asso.Prof., 大学院・理学研究科, 助教授 (60250411)
GIGA Yoshikazu Tokyo Univ., Grad.School of Sci., Professor, 大学院・数理科学研究科, 教授 (70144110)
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| Project Period (FY) |
2002 – 2004
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| Keywords | inverse problem / probe method / singular source method / linear sampling method / enclosure method / no response test / complex geometric optic solution / oscillating-decaying solution |
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| Research Abstract |
We studied identifying the discontinuity of the medium such as inclusions, cavities, cracks and the physical property of the medium. For identifying the discontinuity for the medium, we improved and adopted the probe method and enclosure method. Especially, we studied the behavior of the reflected solution and the unique continuation property which are essential for the probe method, and we accomplished the probe method. As for the enclosure method, we enlarged its application by replacing the complex geometric optic solution which is difficult to construct and localize by introducing the osciallating-decaying solution. We also showed that the reconstruction methods for the inverse boundary value problem such as the probe method, singular source method, no response test are unified into the no response test, and the probe method and singular source method are the same methods. For the inverse scattering problem, we solved the difficulty of the linear sampling method by proposing two new reconstruction methods. Moreover, we succeeded in establishing the probe method for the one space dimensional parabolic equation and giving the theoretical frame work for Shirota's computational method for identifying the discontinuity of the coefficient for the wave equation.
As for identifying the physical property of the medium, we studied two inverse problems for identifying the residual stress and the damage of steel-concrete connected beam. We gave the dispersion formula of the speed of the Rayleigh wave and applied it for the former inverse problem. For the latter problem, we established identifying the damage from the frequency response function which is a practical measured data. We also studied identifying the coefficient for the nonlinear wave equation and succeeded in observing that we can identify the linear and the quadratic part of the coefficient by linearizing the Dirichlet to Neumann map.
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