A Study on the High-Accuracy Numerical Reconstruction Methods for the Solution of Some Inverse Problems for Partial Differential Equations

2008 Fiscal Year Final Research Report

• Project/Area Number: 18540150
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Okayama University of Science
• Principal Investigator: OHE Takashi Okayama University of Science, 理学部, 准教授 (90258210)
• Co-Investigator(Renkei-kenkyūsha): OHNAKA Kohzaburo 大阪大学, 大学院・工学研究科, 准教授 (60127199) ; IKEHATA Masaru 群馬大学大学院, 工学研究科, 教授 (90202910) ; YAMATANI Katsu 名城大学, 都市情報学部, 准教授 (80293611)
• Project Period (FY): 2006 – 2008
• Keywords: 逆問題 / 数値解析 / 数値解法 / 応用数学 / 数理工学 / 画像処理 / 医用工学

Research Abstract

Laplace方程式やHelmholtz方程式などの偏微分方程式の逆問題について、その解の直接的・数値的再構成法の開発を行い、数値実験による有効性の検証を行なった。特に(1)囲い込み法に基づく方法、(2)代用電荷法に基づく方法、(3)重みつき境界積分に基づく方法について開発を行なった。いくつかの解法については安定性解析や正則化手法に関する理論解析を行ない、有効性に関する理論的保証を与えた。

Research Products

• [Journal Article] A Direct Identification Method of a Point Source for 3-dimensional Scalar Wave Equation (2008)
  • Author(s): H. Inui and K. Ohnaka
  • Journal Title: Information Vol.11 Pages: 171-178
  • Peer Reviewed
• [Journal Article] Enclosure method for an inverse crack problem and the Mittag-Leffler function (2008)
  • Author(s): M. Ikehata and T. Ohe
  • Journal Title: Inverse Problems Vol.24, 015006 Pages: 27
  • Peer Reviewed
• [Journal Article] On an Inverse Source Problem for 3-dimensional Wave Equation (2008)
  • Author(s): K. Ohnaka and H. Inui
  • Journal Title: Joint Research on Environmental Science and Technology for the Earth, Annual Report of FY 2007, M. Ike and P. H. Viet (eds.) Pages: 544-547
  • Peer Reviewed
• [Journal Article] 高階導関数に着目した段階的なDCT係数予測と画像圧縮への応用 (2008)
  • Author(s): 芦澤恵太, 山谷克
  • Journal Title: 電子情報通信学会論文誌(A) Vol.J91-A Pages: 808-816
  • Peer Reviewed
• [Journal Article] A direct identification method for current dipoles from electromagnetic fields (2007)
  • Author(s): K. Yamatani, T. Ohe and K. Ohnaka
  • Journal Title: Journal of Computational and Applied Mathematics Vol.201 Pages: 164-174
  • Peer Reviewed
• [Journal Article] 多近傍情報による予測と残差直交変換の階層化およびその画像圧縮への応用 (2007)
  • Author(s): 芦澤恵太, 山谷克, 斎藤直樹
  • Journal Title: 日本応用数理学会論文誌 Vol.17 Pages: 239-257
  • Peer Reviewed
• [Journal Article] On a numerical solution of the Cauchy problem for the Laplace equation by the fundamental solutions method (2007)
  • Author(s): Takashi Ohe, Katsu Yamatani, Kohzaburo Ohnaka
  • Journal Title: Proceedings on Applied Mathematics and Mechanics Vol.7 Pages: 2040035-2040036
  • Peer Reviewed
• [Journal Article] Improvement of DCT-based compression algorithms using Poisson's equation (2006)
  • Author(s): K. Yamatani, N. Saito
  • Journal Title: IEEE Transactions on Image Processing Vol.15 Pages: 3672-3689
  • Peer Reviewed
• [Presentation] Laplace方程式のCauchy問題に対する代用電荷法の正則化について (2008)
  • Author(s): 大江貴司, 大中幸三郎
  • Organizer: 日本応用数理学会環瀬戸内応用数理研究部会第12回シンポジウム
  • Place of Presentation: 山形大学理学部
  • Year and Date: 2008-10-12
• [Presentation] 多近傍情報を用いたDCT係数予測 (2007)
  • Author(s): 芦澤恵太,山谷克
  • Organizer: 日本応用数理学会2007年度年会
  • Place of Presentation: 北海道大学
  • Year and Date: 2007-09-17
• [Presentation] 変化する強度を持つ複数点波源に対する同定 (2007)
  • Author(s): 乾裕一,大中幸三郎
  • Organizer: 日本応用数理学会2007年度年会
  • Place of Presentation: 北海道大学
  • Year and Date: 2007-09-15
• [Presentation] On a numerical solution of the Cauchy problem for he Laplace equation by the fundamental solutions method (2007)
  • Author(s): T. Ohe, K. Yamatani and K. Ohnaka
  • Organizer: The 6th International Congress on Industrial and Applied Mathematics
  • Place of Presentation: University of Zurich, Switzerland
  • Year and Date: 2007-07-20
• [Presentation] A numerical method for inverse crack problem using the Mittag-Leffler function (2007)
  • Author(s): T. Ohe and M. Ikehata
  • Organizer: 研究集会「数理科学セミナー」
  • Place of Presentation: 筑波大学
  • Year and Date: 2007-02-10
• [Presentation] 円弧上で与えられたCauchy条件に対するLaplace方程式の数値解法について (2006)
  • Author(s): 大江貴司, 大中幸三郎
  • Organizer: 日本応用数理学会2006年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2006-09-16
• [Presentation] Inverse crack problem and the Mittag-Leffler function (2006)
  • Author(s): M. Ikehata and T. Ohe
  • Organizer: Inverse Problems in Applied Sciences --- towards the break through ---
  • Place of Presentation: 北海道大学
  • Year and Date: 2006-07-06
• [Presentation] A Numerical Method for Finding Current Sources for Time-Harmonic Maxwell's Equations (2006)
  • Author(s): K. Ohnaka
  • Organizer: The Sixth East Asia PDE Conference
  • Place of Presentation: Wuhan University, China
  • Year and Date: 2006-05-16