Study on inverse problems for partial differential equations using the probe and enclosure methods

• Project/Area Number: 21540162
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Gunma University
• Principal Investigator: IKEHATA Masaru 群馬大学, 大学院・工学研究科, 教授 (90202910)
• Co-Investigator(Renkei-kenkyūsha): ITOU Hiromichi 群馬大学, 大学院・工学研究科, 助教 (30400790)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 境界値逆問題 / 波の物体散乱逆問題 / enclosure met hod / 熱方程式 / 波動方程式 / Helmholtz方程式 / 粘弾性体 / 逆散乱問題 / 囲い込み法 / 粘弾性体の方程式 / 解析学 / Enclosure method / Probe method / 熟方程式 / 複素幾何光学解

Research Abstract

The most representative results obtained in this study consists of two parts. The first one shows that the enclosure method using dynamical data overa finite time interval can be applied to several inverse problems for the heat and waveequations and visco-elastic system of equations in three dimensions. And also the result raises several problems to be solved. The second one is the finding of the idea thatthe logarithmic differential of the indicator function in the enclosure method appliedto inverse obstacle scattering problems governed by the Helmholtz equation yieldsinformation two times more than the original indicator function.

Research Products

• [Journal Article] On uniqueness in theinverse obstacle problem via thepositive supersolutions of the Helmholtz equation (2012)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 28
  • Peer Reviewed
• [Journal Article] Inverse obstacle scattering with limited-aperture data (2012)
  • Author(s): Ikehata, M., Niemi, E.and Siltanen, S.
  • Journal Title: Inverse Problems and Imaging Volume: 6 Pages: 77-94
  • Peer Reviewed
• [Journal Article] The enclosure method forinverse obstacle scattering problemswith dynamical data over a finite timeinterval:II. Obstacles with a dissipative boundary or finite refractive index and back-scattering data (2012)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 28
  • Peer Reviewed
• [Journal Article] On reconstruction of a cavity in a linearized viscoelastic body from infinitely many transient boundary data (2012)
  • Author(s): Ikehata, M. and Itou, H.
  • Journal Title: Inverse Problems Volume: 28
  • Peer Reviewed
• [Journal Article] An inverse acoustic scattering problem inside a cavity with dynamical back-scattering data (2012)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 28
  • Peer Reviewed
• [Journal Article] On reconstruction of a cavity in a linearized visoelastic body from infinitely many transient boundary data (2012)
  • Author(s): Ikehata, M. and Itou, H.
  • Journal Title: Inverse Problems Volume: 28 Issue: 12 Pages: 125003-125003
  • DOI: 10.1088/0266-5611/28/12/125003
  • Peer Reviewed
• [Journal Article] On uniqueness in the inverse obstacle problem via the positive supersolutions of the Helmholtz equation (2012)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 28 Issue: 3 Pages: 035007-035007
  • DOI: 10.1088/0266-5611/28/3/035007
  • Peer Reviewed
• [Journal Article] Inverse obstacle acattering with limited-aperture data (2012)
  • Author(s): Ikehata, M., Niemi, E., Siltanen, S.
  • Journal Title: Inverse Problems and Imaging Volume: 6 Issue: 1 Pages: 77-94
  • DOI: 10.3934/ipi.2012.6.77
  • Peer Reviewed
• [Journal Article] The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval : II. Obstacles with a dissipative boundaryor finite refractive index and back-scattering data (2012)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 28 Issue: 4 Pages: 045010-045010
  • DOI: 10.1088/0266-5611/28/4/045010
  • Peer Reviewed
• [Journal Article] The framework of the enclosure method with dynamical data and its applications (2011)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 27
  • Peer Reviewed
• [Journal Article] Inverse obstacle scattering problems with a single incident wave and the logarithmic differential of the indicator function in the enclosure method (2011)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 27
  • Peer Reviewed
• [Journal Article] Mittag-Leffler's function, Vekua transform and an inverse obstacle scattering problem (2010)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 26
  • Peer Reviewed
• [Journal Article] The probe and enclosure methods for inverse obstacle scattering problems (2010)
  • Author(s): Ikehata, M
  • Journal Title: The past and present., RIMS Kokyuroku Volume: No. 1702 Pages: 1-22
• [Journal Article] A note on the enclosure method for an inverse obstacle scattering problem with a single point source (2010)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 26
  • Peer Reviewed
• [Journal Article] On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval (2010)
  • Author(s): Ikehata, M. and Kawashita, M.
  • Journal Title: Inverse Problems Volume: 26
  • Peer Reviewed
• [Journal Article] The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval (2010)
  • Author(s): Ikehata, M.
  • Journal Title: Inverse Problems Volume: 26
  • Peer Reviewed
• [Journal Article] 逆問題における不連続性の抽出のための解析的方法-探針法10年- (2010)
  • Author(s): 池畠優
  • Journal Title: 数学 Volume: 62(3) Pages: 289-314
  • NAID: 130004558915
  • Peer Reviewed
• [Journal Article] A note on the enclosure method for an inverse obstacle scattering problem with a single point source (2010)
  • Author(s): Ikehata, M.
  • Journal Title: Inverae Problems Volume: 26
  • Peer Reviewed
• [Journal Article] On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval (2010)
  • Author(s): Ikehata, M., Kawashita, M.
  • Journal Title: Inverse Problems Volume: 26
  • Peer Reviewed
• [Journal Article] Mittag-Leffler's function, Vekua transform and an inverse obstacle scattering problem (2010)
  • Author(s): Ikehata.M.
  • Journal Title: Inverse Problems 26
  • Peer Reviewed
• [Journal Article] The probe and enclosure methods for inverse obstacle scattering problems. The past and present.
  • Author(s): Ikehata.M.
  • Journal Title: RIMS Kokyuroku (in press)
• [Presentation] The enclosure method for the wave equation using dynamical scattering data (2012)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「偏微分方程式の逆問題解析とその周辺に関する研究」
  • Place of Presentation: 京都大学数理解析研究所, 京都
  • Year and Date: 2012-11-19
• [Presentation] Extracting the geometry of an obstacle from the bistatic scattering data (2012)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「偏微分方程式の解の幾何」
  • Place of Presentation: 京都大学数理解析研究所, 京都
  • Year and Date: 2012-11-08
• [Presentation] Extracting the geometry of an acoustic enclosure from dynamical back-scattering data: an inverse problem for the wave equation (2012)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「幾何学的偏微分方程式に対する保存則と正則性特異性の研究」
  • Place of Presentation: 京都大学数理解析研究所, 京都
  • Year and Date: 2012-06-14
• [Presentation] Inverse obstacle scattering with dynamical data over a finite time interval (2012)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「偏微分方程式の逆問題解析とその周辺分野に関する研究」
  • Place of Presentation: 京都大学数理解析研究所, 京都
  • Year and Date: 2012-01-24
• [Presentation] Inverse obstacle scattering with dynamical data over a finite time interval (2012)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「偏微分方程式の逆問題解析とその周辺分野に関する研究」
  • Place of Presentation: 京都大学数理解析研究所(京都府)(招待講演)
  • Year and Date: 2012-01-24
• [Presentation] The enclosure method ininverse obstacle scattering (2011)
  • Author(s): Ikehata, M.
  • Organizer: Finish-Japanese-Korean workshop on inverse problems and the 17th Inverse days of the Finnish Inverse ProblemSociety
  • Place of Presentation: University of Helsinki, Finland
  • Year and Date: 2011-12-14
• [Presentation] The enclosure method in inverse obstacle scattering (2011)
  • Author(s): Ikehata, M.
  • Organizer: Finish-Japanese-Korean workshop on inverse problems and the 17 th Inverse Days of the Finnish Inverse Problem Society
  • Place of Presentation: University of Helsinki (Helsinki, Finland)(招待講演)
  • Year and Date: 2011-12-14
• [Presentation] 波の物体散乱の逆問題における直接的方法 (2011)
  • Author(s): 池畠優
  • Organizer: 公開談話会
  • Place of Presentation: 名古屋大学, 愛知
  • Year and Date: 2011-10-24
• [Presentation] 波の物体散乱の逆問題における直接的方法 (2011)
  • Author(s): 池畠優
  • Organizer: 公開談話会
  • Place of Presentation: 名古屋大学大学院多元数理科学研究科(愛知県)(招待講演)
  • Year and Date: 2011-10-24
• [Presentation] The enclosure method for inverse obstacle scattering problems with dynamical back-scattering data over a finite time interval (2011)
  • Author(s): 池畠優
  • Organizer: 数理科学セミナー
  • Place of Presentation: 松本市浅間温泉みやま荘, 長野
  • Year and Date: 2011-09-26
• [Presentation] The enclosure method for inverse obstacle scattering problems with dynamical back-scattering data over a finite time interval (2011)
  • Author(s): 池畠優
  • Organizer: 数理科学セミナー
  • Place of Presentation: 松本市浅間温泉みやま荘(長野県)
  • Year and Date: 2011-09-26
• [Presentation] 有限観測時間におけるデータを用いた熱および波動方程式に対する逆問題と囲い込み法 (2010)
  • Author(s): 池畠優
  • Organizer: 日本数学会2010 年度秋期総合分科会函数方程式論分科会特別講演
  • Place of Presentation: 名古屋大学, 愛知
  • Year and Date: 2010-09-23
• [Presentation] 有限観測時間におけるデータを用いた熱および波動方程式に対する逆問題と囲い込み法 (2010)
  • Author(s): 池畠優
  • Organizer: 日本数学会2010年度秋季総合分科会函数方程式論分科会特別講演
  • Place of Presentation: 名古屋大学大学院多元数理科学科(愛知県)
  • Year and Date: 2010-09-23
• [Presentation] Some remarks on inverse obstacle scattering problems with a single incidentwave (2010)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「偏微分方程式の逆問題解析とその周辺分野に関する研究」
  • Place of Presentation: 京都大学数理解析研究所, 京都
  • Year and Date: 2010-06-23
• [Presentation] Some remarks on inverse obstacle scattering problems with a single incident wave (2010)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「偏微分方程式の逆問題解析とその周辺分野に関する研究」
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2010-06-23
• [Presentation] The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval (2010)
  • Author(s): 池畠優
  • Organizer: 日本数学会2010 年度年会函数方程式論分科会
  • Place of Presentation: 慶應義塾大学, 神奈川
  • Year and Date: 2010-03-24
• [Presentation] The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval (2010)
  • Author(s): 池畠優
  • Organizer: 日本数学会2010年度年会函数方程式論分科会
  • Place of Presentation: 慶應義塾大学(神奈川県)
  • Year and Date: 2010-03-24
• [Presentation] The probe and enclosure methods for inverse obstacle scattering problems. The past and present (2010)
  • Author(s): 池畠優
  • Organizer: 第27回九州における偏微分方程式研究集会
  • Place of Presentation: 九州大学西新プラザ大会議室, 福岡
  • Year and Date: 2010-01-26
• [Presentation] The probe and enclosure methods for inverse obstacle scattering problems. The past and present. (2010)
  • Author(s): 池畠優
  • Organizer: 第27回九州における偏微分方程式研究集会
  • Place of Presentation: 九州大学西新プラザ大会議室(福岡県)
  • Year and Date: 2010-01-26
• [Presentation] The enclosure method and its applications to inverse problems with dynamical data over a finite time interval (2009)
  • Author(s): 池畠優
  • Organizer: 名古屋微分方程式セミナー
  • Place of Presentation: 名古屋大学, 愛知
  • Year and Date: 2009-12-14
• [Presentation] The enclosure method and its applications to inverse problems with dynamical data over a finite time interval (2009)
  • Author(s): 池畠優
  • Organizer: 名古屋微分方程式セミナー
  • Place of Presentation: 名古屋大学(愛知県)
  • Year and Date: 2009-12-14
• [Presentation] 有限観測時間におけるデータを用いた音波の障害物による散乱の逆問題 (2009)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「現象解析と関数方程式の新展望」
  • Place of Presentation: 京都大学,京都
  • Year and Date: 2009-11-17
• [Presentation] 有限観測時間におけるデータを用いた音波の障害物による散乱の逆問題 (2009)
  • Author(s): 池畠優
  • Organizer: RIMS研究集会「現象解析と関数方程式の新展望」
  • Place of Presentation: 京都大学(京都府)
  • Year and Date: 2009-11-17
• [Presentation] Recent applications of the enclosure method to inverse problems with dynamical data over afinite time interval (2009)
  • Author(s): Masaru Ikehata
  • Organizer: Inverse Problems Seminsar
  • Place of Presentation: University of Helsinki, Finland
  • Year and Date: 2009-11-09
• [Presentation] Recent Applications of the enclosure method to inverse problems with dynamical data over a finite time interval (2009)
  • Author(s): Masaru Ikehata
  • Organizer: Inverse Problems Seminar
  • Place of Presentation: University of Helsinki(Finland)
  • Year and Date: 2009-11-09
• [Presentation] Recent applications of the enclosure method to dynamical inverse problems (2009)
  • Author(s): 池畠優
  • Organizer: 広島応用解析セミナー(第11回)
  • Place of Presentation: 広島大学工学部, 広島
  • Year and Date: 2009-09-02
• [Presentation] Recent applications of the enclosure method to dynamical inverse problems (2009)
  • Author(s): 池畠優
  • Organizer: 広島応用解析セミナー(第11回)
  • Place of Presentation: 広島大学工学部(広島県)
  • Year and Date: 2009-09-02
• [Presentation] The enclosure method for the wave equation using dynamical scattering data
  • Author(s): 池畠 優
  • Organizer: RIMS研究集会「偏微分方程式の逆問題解析とその周辺に関する研究」
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Invited
• [Presentation] Extracting the geometry of an obstacle from the bistatic scattering data
  • Author(s): 池畠 優
  • Organizer: RIMS研究集会「偏微分方程式の解の幾何」
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Invited
• [Presentation] Extracting the geometry of an acoustic enclosure from dynamical back-scattering data: an inverse problem for the wave equation
  • Author(s): 池畠 優
  • Organizer: RIMS研究集会「幾何学的偏微分方程式に対する保存則と正則性特異性の研究」
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Invited
• [Remarks]
  • URL: http://math.dept.eng.gunma-u.ac.jp/~ikehata/
• [Remarks] masaru ikehata
  • URL: http://math.dept.eng.gunma-u.ac.jp/~ikehata/
• [Remarks]
  • URL: http://math.dept.eng.gunma-u.ac.jp/~ikehata/
• [Remarks]
  • URL: http://math.dept.eng.gunma-u.ac.jp/~ikehata/
• [Remarks]
  • URL: http://math.dept.eng.gunma-u.ac.jp/~ikehata/