A comprehensive research to develop a stable and high accurate numerical method for the problems of coefficient identification in linear wave equations

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03420
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Aichi Prefectural University
• Principal Investigator: Shirota Kenji 愛知県立大学, 情報科学部, 教授 (90302322)
• Project Period (FY): 2018-04-01 – 2024-03-31
• 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

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.

Free Research Field

応用数値解析

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．