| Project/Area Number |
18K03420
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Aichi Prefectural University |
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Principal Investigator |
Shirota Kenji 愛知県立大学, 情報科学部, 教授 (90302322)
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2022: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2021: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2020: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2019: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2018: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
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| Keywords | 波動方程式 / 逆問題解析 / 係数同定問題 / 位相最適化手法 / 順問題解析 / クロネッカー積構造行列 / BiCG系解法 / 抽象勾配法 / 波動場 / 弾性波動場 / 逆問題 / 数値解法 / H1勾配法 / ソース項同定問題 / 数値解析 / 非適切問題 / 正則化解法 / 反復解法 / 係数同定 / 波動方程式族 |
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| Outline of Final Research Achievements |
In this research, we considered about the numerical method for the coefficient identification problem in the wave type partial differential equations. We adopted the adjoint method to find the unknown coefficients. In order to identify the unknown coefficients, we applied the H1 type method proposed for the SIMP type topology optimization to our problem. By the numerical experiments, we showed the effectiveness and future works of our algorithm. Moreover, we studied about the numerical method to solve the initial-boundary value problem in scalar wave equation. We applied the finite difference type method and the spectral collocation method to the discretization in space and time direction, respectively. We introduced the matrix equation which is equivalent to the discretized equation. The stabilized GPBiCG method for the Kronecker type coefficient matrix was proposed to solve numerically the matrix equation. We showed the effectiveness of our method by the numerical experiments.
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| Academic Significance and Societal Importance of the Research Achievements |
波動方程式族の係数同定問題に対する数値解法の研究は，計算時間や観測データの問題から，国内外とも少なく，また位相最適化と逆問題解析は両方とも非適切問題に対する研究にも関わらず，交流が少ない．その現状において本研究の成果は，波動方程式族の係数同定問題が実用問題で有効可能性，位相最適化手法が逆解析に有効な可能性を示したものであり，その学術的意義は小さくない．また，時間発展型偏微分方程式の順問題に対する数値解法として，GPGPUで容易に高速化可能な手法の提案は，時間発展型線形偏微分方程式の高速計算解法開発の新たな方向性を示したものであり，その学術的意義は大きいと考える．
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