Inverse source problem for the hyperbolic equation with a time-dependent principal part
In this paper, we investigate the inverse problem on determining the spatial component of the source term in the hyperbolic equation with a time-dependent principal part. Based on a Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of...
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| Published in: | Journal of Differential Equations Vol. 262; no. 1; pp. 653 - 681 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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05.01.2017
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| ISSN: | 0022-0396, 1090-2732 |
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| Abstract | In this paper, we investigate the inverse problem on determining the spatial component of the source term in the hyperbolic equation with a time-dependent principal part. Based on a Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to reformulate the inverse problem into a related optimization problem, for which we develop an iterative thresholding algorithm by using the corresponding adjoint system. Numerical examples up to three spatial dimensions are presented to demonstrate the accuracy and efficiency of the proposed algorithm. |
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| AbstractList | In this paper, we investigate the inverse problem on determining the spatial component of the source term in the hyperbolic equation with a time-dependent principal part. Based on a Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to reformulate the inverse problem into a related optimization problem, for which we develop an iterative thresholding algorithm by using the corresponding adjoint system. Numerical examples up to three spatial dimensions are presented to demonstrate the accuracy and efficiency of the proposed algorithm. |
| Author | Liu, Yikan Jiang, Daijun Yamamoto, Masahiro |
| Author_xml | – sequence: 1 givenname: Daijun surname: Jiang fullname: Jiang, Daijun email: jiangdaijun@mail.ccnu.edu.cn organization: School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China – sequence: 2 givenname: Yikan surname: Liu fullname: Liu, Yikan email: ykliu@ms.u-tokyo.ac.jp organization: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan – sequence: 3 givenname: Masahiro surname: Yamamoto fullname: Yamamoto, Masahiro email: myama@ms.u-tokyo.ac.jp organization: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan |
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| Cites_doi | 10.1093/imamat/1.4.309 10.1137/0110046 10.1016/S0021-7824(99)80010-5 10.1080/00036811.2012.738359 10.1088/0266-5611/28/9/095008 10.1080/03605302.2013.771659 10.1088/1742-6596/12/1/001 10.1088/0266-5611/12/6/013 10.1088/0266-5611/19/1/309 10.1081/PDE-100106139 10.3934/ipi.2015.9.163 10.3934/ipi.2007.1.29 10.1080/00036811.2013.829263 10.1137/15M1018836 10.1090/conm/268/04314 10.1109/TIP.2007.909319 10.1002/cpa.20042 10.1016/j.acha.2007.10.005 10.1088/0266-5611/14/5/009 10.1088/0266-5611/17/4/310 10.1088/0266-5611/11/2/013 10.1557/PROC-398-425 10.1070/SM1987v058n01ABEH003103 10.1137/080716542 |
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| Title | Inverse source problem for the hyperbolic equation with a time-dependent principal part |
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