1997 Fiscal Year Final Research Report Summary
Numerical Analysis for Ill-posed Problems Related with Engineering
| Project/Area Number |
07309021
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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 |
広領域
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
ISO Yuusuke Graduate School of Informatics, Kyoto University, Professor, 大学院・情報学研究科, 教授 (70203065)
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| Co-Investigator(Kenkyū-buntansha) |
KUBO Shiro Graduate School of Engineering, Osaka University, Professor, 大学院・工学研究科, 教授 (20107139)
YAMAMOTO Masahiro Graduate School of Mathematical science, University of Tokyo Associate Professor, 大学院・数理科学研究科, 助教授 (50182647)
NISHIMURA Naoshi Graduate School of Engineering, Kyoto University, Associate Professor, 大学院・工学研究科, 助教授 (90127118)
KUBO Masayoshi Graduate School of Informatics, Kyoto University, Lecturer, 大学院・情報学研究科, 講師 (10273616)
NISHIDA Takaaki Graduate School of science, Kyoto University, Professor, 大学院・理学研究科, 教授 (70026110)
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| Project Period (FY) |
1995 – 1997
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| Keywords | Numerical Analysis / Ill-posed Problems / Inverse Problems / Uniqueness Theory / Stability Theory |
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| Research Abstract |
We treat mainly, in the present research, inverse problems as our aimed ill-posed problems. We mean "ill-posed problems" by the not well-posed problems in the sense of Hadamard ; the aimed differential equations are not well-posed in the sense of Hadamard and solutions have almost no continuity in connection with the given data. The ill-posedness implies instability of numerical solutions in numerical analysis for our problems, and it means impossibility of reliable numerical computation by usual numerical methods applied to the problems.
Inverse Problems , e. g. non-destructive test, are popular in the recent engineering, but almost all of them are ill-posed in the above sense. It is almost impossible to give good numerical solutions for the problem by usual numerical techniques, and we need some new idea to treat them. We give some results from this view point in the present research.
Under the circumastance stated above, we restrict ourselves for theoretical approach for numerical analysis for ill-posed problems. Dr. Masahiro Yamamoto, who is one of the investigators of our research, proposed generalized well-posedness for ill-posed problems, and he shows the uniqueness of the solution for an aimed problem implies its stability from the functional analysis. It means stability of numerical solution in a weak sense, and we put the research of uniqueness in inverse problems as one of the main topics in the research. Dr.Masahiro Kubo gives a very important result for the uniqueness in the hyperbolic inverse problems. In addition. to the uniqueness, other investigators of the present research study application of Tikhonov regularization methods, the filter methods proposed by Dr.Tosaka and multi-precision techniques proposed 5y Dr.Iso, and each method is regarded as a suitable new method in numerical analysis for ill-posed problems.
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