2002 Fiscal Year Final Research Report Summary
Discovery of new methods that may yield reconstruction algorithms in inverse problems
| Project/Area Number |
13640152
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Gunma University |
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Principal Investigator |
IKEHATA Masaru Gunma University, Faculty of Engineering Professor, 工学部, 教授 (90202910)
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| Co-Investigator(Kenkyū-buntansha) |
TANUMA Kazumi Gunma University, Faculty of Engineering Associate Professor, 工学部, 助教授 (60217156)
OHE Takashi Okayama University of Science, Faculty of Informatics, Associate Professor, 総合情報学部, 助教授 (90258210)
NAKAMURA Gen Hokkaido University, Graduate school of sciences, 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)
SAITHO Saburou Gunma University, Faculty of Engineering Professor, 工学部, 教授 (10110397)
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| Project Period (FY) |
2001 – 2002
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| Keywords | inverse problem / inverse boundary value problem / enclosure method / Dirichlet-to-Neumann map / probe method / discontinuity surface / inclusion / crack |
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| Research Abstract |
1. A numerical implementation of the enclosure method and its regularization We made a numerical implementation of an extraction formula of the convex hull of polygonal cavities or inclusions from a single set of the Cauchy data of a solution of the governing equation. In order to explain the numerical results we considered how to modify the formula when the data contain error and gave a modified formula.
2. Development of the probe method
We considered inverse problems for the mixed type boundary value problems. By studying the detailed behavior of the so-called reflected solutions, we found that the probe method discovered by the head investigator can be applied to those problems. In particular, we gave an application of the De Giorgi-Nash-Moser theorem to the probe method.
3. A generalization of the enclosure method
We discovered a method that is based on the analyticity and asymptotic behavior of Mittag-Leffler's function and yields the visible parts of the boundary of unknown inclusions from the Dirichlet-to-Neumann map.
Moreover we applied the method to a mathematical model for alternative current.
4 . An application of the enclosure method to an inverse problem for the multilayered material
We considered the problem of extracting information about unknown inclusions embedded in a background layered material that has different constant conductivities across finitely many parallel planes, from the Dirichlet-to-Neumann map. For the purpose we constructed the exponentially growing solutions of the governing equation for the back ground material and studied their asymptotic behavior. Using the property of those solutions, we gave an extraction formula of the convex hull of the inclusions .
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