2006 Fiscal Year Final Research Report Summary
Mathematical analysis for inverse problems in mathematical scienes and establishment of numerical methods
| Project/Area Number |
15340027
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | The University of Tokyo |
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Principal Investigator |
YAMAMOTO Masahiro The University of Tokyo, Graduate School of Mathematical Scenes, Associate Professor (50182647)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAMURA Gen Hokkaido University, Graduate School of Natural Sciences, Professor (50118535)
SAITOH Saburo Gunma University, Faculty of Engineering, Professor (10110397)
ISO Yuusuke Kyoto University, Graduate School of Informatics, Professor (70203065)
IKAWA Mitsuru Kyoto University, Graduate School of Natural Sciences, Professor (80028191)
NISHIDA Takaaki Waseda University, Graduate School of Natural Sciences, Professor (70026110)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Inverse problem / mathematical analysis / numerical analysis / instability / regularization |
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| Research Abstract |
In inverse problems, one has to determine physical properties of the interior of an object by available data on the boundary and to identify causes from results, and researches for various inverse problems become important in many fields such as mathematical sciences and industrie. The development of numerical methods as well as the mathemaical analyses for inverse problems become more requested, because the importance of the inverse problem is better recognized and computers and observation equipments are improved rapidly. It is more necessary for one to improve numerical methods which are free from the conventional manners for the forward problem. However even the mathematical researches for inverse problems on which such relevant numerical methods should be based, are not yet sufficiently done. One of important inverse problems in industry is for the risk management : one aims at the optimal control of a plant by means of suitable evaluation of the interior states of the plant, and has intrinsic instability. With physically acceptable a priori conditions, one can recover stability, which is called the conditional stability. Therefore one has to choose suitabl stabilizing methods, and should not apply conventional methods. For reasonable numerical performances, one has to choose numerical methods guaranteeing the accuracy which corresponds to the degree of conditional stability of the original inverse problem. Thus one must establish conditional stability results. In this research, we have developed numerical methods on the basis of exploited mathematical researches for inverse problems. Our mathematical results are remarkable by various mathematical knowledge such as complex analysis and partial differential equations. Moreover the link with the industry is deeper and one patent was applied.
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