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Inverse Source Problem for a Wave Equation with Final Observation Data

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Mathematical Analysis of Continuum Mechanics and Industrial Applications

Part of the book series: Mathematics for Industry ((MFI,volume 26))

Abstract

In this chapter, we study the inverse problem on recovering a spatial component of the source term in a wave equation by the final observation data. Employing the analytic Fredholm theory , we establish a generic well-posedness result concerning the uniqueness of our inverse source problem . Numerically, by treating a corresponding minimization problem, we investigate the variational equation for the minimizer and develop an iterative thresholding algorithm . One- and two-dimensional numerical experiments are implemented to demonstrate the robustness and accuracy of the proposed algorithm.

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Acknowledgments

The work was supported by A3 Foresight Program “Modeling and Computation of Applied Inverse Problems,” Japan Society for the Promotion of Science (JSPS). The first author was financially supported by self-determined research funds of Central China Normal University from the colleges’ basic research and operation of MOE (No. CCNU14A05039), National Natural Science Foundation of China (Nos. 11326233, 11401241 and 11571265). The second and the third authors are partially supported by Grant-in-Aid for Scientific Research (S) 15H05740, JSPS.

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Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Daijun Jiang

  2. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Tokyo, Meguro-ku, 153-8914, Japan

    Yikan Liu & Masahiro Yamamoto

Authors
  1. Daijun Jiang
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  2. Yikan Liu
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  3. Masahiro Yamamoto
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Corresponding author

Correspondence to Yikan Liu .

Editor information

Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Tokyo, Japan

    Hiromichi Itou

  2. Inst of Science and Engineering, Kanazawa University, Kanazawa, Japan

    Masato Kimura

  3. Fujitsu Ltd., Tokyo, Japan

    Vladimír Chalupecký

  4. Hiroshima Kokusai Gakuin University, Hiroshima, Japan

    Kohji Ohtsuka

  5. Kyushu University, Fukuoka, Fukuoka, Japan

    Daisuke Tagami

  6. Asahi Glass Co., Ltd., Tokyo, Tokyo, Japan

    Akira Takada

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Jiang, D., Liu, Y., Yamamoto, M. (2017). Inverse Source Problem for a Wave Equation with Final Observation Data. In: Itou, H., Kimura, M., Chalupecký, V., Ohtsuka, K., Tagami, D., Takada, A. (eds) Mathematical Analysis of Continuum Mechanics and Industrial Applications. Mathematics for Industry, vol 26. Springer, Singapore. https://doi.org/10.1007/978-981-10-2633-1_11

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  • DOI: https://doi.org/10.1007/978-981-10-2633-1_11

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-2632-4

  • Online ISBN: 978-981-10-2633-1

  • eBook Packages: EngineeringEngineering (R0)

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