| Project/Area Number |
18540110
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Gunma University |
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Principal Investigator |
MATSUURA Tsutomu Gunma University, Graduate School of Engineering, associate professor (80181692)
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| Co-Investigator(Kenkyū-buntansha) |
SAITOH Saburo Gunma University, Graduate School of Engneering, professor (10110397)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,820,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2007: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2006: ¥2,000,000 (Direct Cost: ¥2,000,000)
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| Keywords | inverse problem / reproducing kernel theory / algorithm / visualization / real inversion of the Laplace transform / Tikhonov regularization / multiple-precision arithmetic / 実ラプラス逆変換 |
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| Research Abstract |
We succeeded to put reproduction kernel theory and regularization theory together Using our new method we can solve some inverse problems which have not been able to be solved historically. (1) The inverse problem of heat conduction is known as a difficult problem historically. We applied Tikhonov regularization for this inverse problem and gave some concrete algorithms for solving it. And we confirmed the effectiveness of those algorithms by numerical experiments. In addition, I contrived visual of the solution to clarify utility of this manner of solving. We think that it may be said that the inverse problem of the heat conduction has been settled by these our contributions. (2) We made the calculation environment of the multiple-precision arithmetic from the both sides of hardware and software. And we applied these implements for development of computation algorithm on real inversion of Laplace transform. Furthermore we verified that we can obtain numerical solutions of this typical ill-posed problem with satisfactory accuracy by our new method. (3) We verified theoretically and numerically that the usage of sinc function and Fredholm integral equation of the second kind are very effective for real inversion of Laplace transform. And we demonstrated numerically that double exponential method is very powerful for solving this ill-posed problem. (4) We pointed out theoretically that singular value decomposition is available and effective for solving real inverse problem of Laplace transform. And we showed that multiple-precision arithmetic is essential for this calculation. Furthermore we established this method and necessary numerical tables and applied for a patent of this study in conjunction with Kyoto University.
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