New High-Accurate Numerical Methods for Inverse Problems by the Direct Computations of Integral Equations of the First Kind

• Project/Area Number: 26400198
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kyoto University
• Principal Investigator: Fujiwara Hiroshi 京都大学, 情報学研究科, 准教授 (00362583)
• Co-Investigator(Kenkyū-buntansha): 滝口 孝志 防衛大学校(総合教育学群、人文社会科学群、応用科学群、電気情報学群及びシステム工学群), 総合教育学群, 准教授 (50523023)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: 高精度数値計算 / 数値的不安定性 / 逆問題 / 非適切問題 / 第一種積分方程式 / 多倍長計算 / 数値解析 / 高精度計算

Outline of Final Research Achievements

We showed reliable and high-accurate numerical computations to inverse scattering problem which was known as a typical ill-posed problems in the sense of Hadamard. High-accurate discretization methods play an essential role in our method, and particularly multiple-precision arithmetic is required to reduce rounding errors. We also developed multiple-precision arithmetic environment on MATLAB which was widely used in scientific and engineering computation. The environment is faster than the official multiple-precision arithmetic environment VPA.

Research Products

• [Journal Article] Convergence of Finite Difference Schemes Applied to the Cauchy Problems of Quasi-linear Partial Differential Equations of the Normal Form (2018)
  • Author(s): Higashimori Nobuyuki, Fujiwara Hiroshi, Iso Yuusuke
  • Journal Title: Advances in Difference Equations and Discrete Dynamical Systems Volume: - Pages: 113-124
  • DOI: 10.1007/978-981-10-6409-8_6
  • ISBN: 9789811064081, 9789811064098
  • Peer Reviewed
• [Journal Article] Multiple-Precision Arithmetic Environment in MATLAB and Its Application to Reliable Computation of Fractional Order Derivatives (2018)
  • Author(s): Hiroshi Fujiwara
  • Journal Title: マス・フォア・インダストリ研究 Volume: 8 Pages: 83-120
  • Open Access
• [Journal Article] Direct Numerical Computation of the Stationary Radiative Transport Equation on Modern Parallel Architectures (2017)
  • Author(s): Fujiwara Hiroshi、Oishi Naoya
  • Journal Title: Proceedings of the The 16th International Symposium on Distributed Computing and Applications to Business, Engineering and Sciences Volume: 1 Pages: 29-32
  • DOI: 10.1109/dcabes.2017.62
  • Peer Reviewed
• [Journal Article] Semi-discrete finite difference schemes for the nonlinear Cauchy problems of the normal form (2017)
  • Author(s): Higashimori Nobuyuki, Fujiwara Hiroshi
  • Journal Title: Proceedings of the Japan Academy, Series A, Mathematical Sciences Volume: 93 Issue: 9 Pages: 99-104
  • DOI: 10.3792/pjaa.93.99
  • NAID: 40021392001
  • Peer Reviewed / Open Access
• [Journal Article] 生体内の光伝播の数値計算のための定常輻射輸送方程式の上流差分スキームの数値解析 (2017)
  • Author(s): 藤原宏志
  • Journal Title: 計算数理工学会レビュー Volume: 2017-2 Pages: 21-32
  • NAID: 40021333956
• [Journal Article] 定常輻射輸送方程式の速度方向区分定数近似の数値解析 (2016)
  • Author(s): 藤原宏志
  • Journal Title: 計算数理工学論文集 Volume: 16 Pages: 91-96
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] 再生核をもちいたサンプリング値からの原函数再構成とその高精度数値計算 (2016)
  • Author(s): 藤原宏志，齋藤三郎
  • Journal Title: 数理解析研究所講究録 Volume: 1980 Pages: 207-219
  • Acknowledgement Compliant
• [Journal Article] メモリアクセスを考慮した輻射輸送方程式の並列計算の検討 (2015)
  • Author(s): 藤原宏志
  • Journal Title: 計算数理工学論文集 Volume: 15 Pages: 61-66
  • NAID: 40020679511
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] 線型移流方程式を解くCIP法の安定性 (2015)
  • Author(s): 田中 大毅, 藤原 宏志, 西田 孝明, 磯 祐介
  • Journal Title: 日本応用数理学会論文誌 Volume: 25 Pages: 91-104
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] How the computerized tomography was practicalized (2015)
  • Author(s): Takiguchi, T.
  • Journal Title: Bulletin of JSSAC Volume: 21 Pages: 50-57
• [Journal Article] Unique continuation of microlocally analytic functions and their structure (2012)
  • Author(s): Takashi Takiguchi
  • Journal Title: Complex Variables and Elliptic Equations Volume: (published on-line first) Issue: 11 Pages: 1507-1519
  • DOI: 10.1080/17476933.2012.744402
  • Peer Reviewed
• [Presentation] Numerical challenges to the radiative transport equation for near-infrared light propagation in tissue (2018)
  • Author(s): H. Fujiwara
  • Organizer: Numerical Analysis: Applications to Biomedical Problems and Foundations
  • Int'l Joint Research
• [Presentation] 連立線型方程式に対する定常反復解法の並列計算に関する一考察 (2017)
  • Author(s): 藤原宏志
  • Organizer: 数値解析学の最前線 ---理論・方法・応用---
  • Invited
• [Presentation] Quantitative simulation of light emitting by optical fibers in the radiative transport equation (2017)
  • Author(s): H. Fujiwara
  • Organizer: A3 Workshop on Applied Inverse Problems and Related Topics
  • Int'l Joint Research
• [Presentation] Discretization of a conservation law of the stationary radiative transport equation (2017)
  • Author(s): Hiroshi Fujiwara
  • Organizer: Numerical Methods and Analysis for Structures and Sigularites in Fulids II
  • Place of Presentation: 名古屋大学 理学南館 (愛知県名古屋市)
  • Int'l Joint Research
• [Presentation] Semi-discrete numerical scheme of the 2D stationary radiative transport equation on highly-parallel architectures (2017)
  • Author(s): Hiroshi Fujiwara
  • Organizer: Korea-Japan Joint Seminar on Numerical Analysis and Scientific Computation
  • Place of Presentation: Jeju, 韓国
  • Int'l Joint Research
• [Presentation] 光散乱積分の局所化による生体内の光伝播の数理モデルの高速直接計算法 (2017)
  • Author(s): 藤原宏志
  • Organizer: 第19回ヒト脳機能マッピング学会
  • Place of Presentation: 京都大学 百周年時計台記念館 (京都府京都市)
• [Presentation] Multiple-precision arithmetic on MATLAB and its application to the real inverse Laplace transform (2016)
  • Author(s): Hiroshi Fujiwara
  • Organizer: Recent development in inverse problems for partial differential equations and its applications (偏微分方程式の逆問題とその応用の新展開)
  • Place of Presentation: 数理解析研究所，京都
  • Year and Date: 2016-01-28
  • Int'l Joint Research
• [Presentation] An inverse problem to develop CT for concrete structures (2016)
  • Author(s): Takashi Takiguchi
  • Organizer: Recent development in inverse problems for partial differential equations and its applications (偏微分方程式の逆問題とその応用の新展開)
  • Place of Presentation: 数理解析研究所，京都
  • Year and Date: 2016-01-28
  • Int'l Joint Research
• [Presentation] Multiple-precision arithmetic on 64-bit MATLAB based on exflib (2016)
  • Author(s): Hiroshi Fujiwara
  • Organizer: Taiwan-Japan Joint Workshop on Inverse Problems 2016
  • Place of Presentation: 京都大学 工学部総合校舎 (京都府京都市)
  • Int'l Joint Research
• [Presentation] Numerical Computation of the 2D Radiative Transport Equation by Upwind Finite Volume Method (2016)
  • Author(s): Hiroshi Fujiwara
  • Organizer: SIAM: East Asian Section Conference 2016
  • Place of Presentation: Macau, 中国
  • Int'l Joint Research
• [Presentation] Direct computation of the radiative transfer equation for near-infrared light propagation in biological tissue (2016)
  • Author(s): Hiroshi Fujiwara and Naoya Oishi
  • Organizer: The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Orlando, 米国
  • Int'l Joint Research
• [Presentation] 2次元定常輻射輸送方程式の上流有限体積法による数値解析 (2016)
  • Author(s): 藤原宏志、東森信就
  • Organizer: 日本数学会2016年度秋季総合分科会 応用数学分科会
  • Place of Presentation: 関西大学千里山キャンパス第4学舎4号館 (大阪府吹田市)
• [Presentation] Semi-discrete upwind finite volume methods for the 2D stationary radiative transport equation by GPGPU (2016)
  • Author(s): Hiroshi Fujiwara
  • Organizer: Taiwan-Japan Joint Workshop on Numerical Alaysis and Scientific Computation
  • Place of Presentation: Taipei, 中華民国
  • Int'l Joint Research
• [Presentation] 移流・異常拡散方程式の基本解について (2016)
  • Author(s): 滝口孝志
  • Organizer: 第1７回地下環境水文学に関する研究集会
  • Place of Presentation: 筑波大学総合研究棟B (茨城県つくば市)
• [Presentation] Development of Multiple-Precision Arithmetic on MATLAB for Numerical Computations of Ill-Posed Problems (2015)
  • Author(s): Hiroshi Fujiwara
  • Organizer: 2015 Japan-Taiwan Joint Workshop on Inverse Problem
  • Place of Presentation: National Taiwan University, Taipei, Taiwan
  • Year and Date: 2015-11-21
  • Int'l Joint Research
• [Presentation] Development of ulrtasonic tomography for concrete structures (2015)
  • Author(s): Noriyuki Mita and Takashi Takiguchi
  • Organizer: Mathematical Backgrounds and Future Progress of Practical Inverse Problems (実用逆問題の背景にある数理と新展開)
  • Place of Presentation: 九州大学マスフォアインダストリ研究所, 伊都
  • Year and Date: 2015-11-13
  • Int'l Joint Research
• [Presentation] On mathematical background of practicalization of computerized tomography at its initial stage (2015)
  • Author(s): Takashi Takiguchi
  • Organizer: 解析学の耳袋
  • Place of Presentation: プラサ ヴェルデ, 沼津
  • Year and Date: 2015-10-30
• [Presentation] 球面上の回転対称な高精度積分則と3次元輸送方程式の高速計算への応用 (2015)
  • Author(s): 藤原宏志，磯祐介
  • Organizer: 第28回 計算力学講演会
  • Place of Presentation: 横浜国立大学，横浜
  • Year and Date: 2015-10-11
• [Presentation] On the idea for practicalization of the computerized tomography at its initial stage (2015)
  • Author(s): Takashi Takiguchi
  • Organizer: RIMS研究集会「微分方程式の逆問題とその周辺」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-01-26
• [Presentation] Basic properties of concrete and its non destructive testing (2014)
  • Author(s): Noriyuki Mita and Takashi Takiguchi
  • Organizer: Collaboration between theory and practice in inverse problems
  • Place of Presentation: 九州大学マスファアインダストリ研究所
  • Year and Date: 2014-12-18
• [Presentation] On uniqueness problems for the Radon transform (2014)
  • Author(s): Takashi Takiguchi
  • Organizer: RIMS研究集会「Several aspects of microlocal analysis」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2014-10-23
• [Presentation] 球面上の高精度数値積分則の 3 次元輻射輸送方程式への応用 (2014)
  • Author(s): 藤原宏志
  • Organizer: 連続体力学の数理 CoMFoS14
  • Place of Presentation: 京都大学学友会館
  • Year and Date: 2014-10-11
• [Presentation] 高精度積分則と GPU による 3 次元輻射輸送方程式の大規模計算の高速化 (2014)
  • Author(s): 藤原宏志，大石直也
  • Organizer: RIMS研究集会「新時代の科学技術を牽引する数値解析学」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2014-10-09
• [Presentation] 球面上の高精度積分則とある非適切問題への応用 (2014)
  • Author(s): 藤原宏志
  • Organizer: 第63回理論応用力学講演会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2014-09-28
• [Presentation] An inverse problem for the heat equation in view of practical application (2014)
  • Author(s): Takashi Takiguchi
  • Organizer: Inverse Problems from Theory to Application
  • Place of Presentation: At-Bristol, Great Britain
  • Year and Date: 2014-08-26
  • Invited
• [Presentation] An Accurate Quadrature Rule on the Sphere for the Fast Computation of the Radiative Transport Equation (2014)
  • Author(s): Hiroshi Fujiwara
  • Organizer: EASIAM 2014
  • Place of Presentation: Ambassador City Jomtien, Pattaya, Chonburi, Thailand
  • Year and Date: 2014-06-23 – 2014-06-25
• [Remarks] extend precision floating-point arithmetic library
  • URL: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib
• [Remarks] extend precision floating-point arithmetic library
  • URL: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib/
• [Remarks] exflib (multiple-precision arithemtic library)
  • URL: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib/exflib-index.html
• [Funded Workshop] Mathematical Backgrounds and Future Progress of Practical Inverse Problems (2015)
  • Place of Presentation: Institute of Mathematics for Industry (IMI), Kyushu University
  • Year and Date: 2015-11-10