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On the Inverse Source Problem with Boundary Data at Many Wave Numbers

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Abstract

We review recent  results on inverse source problems for the Helmholtz type equations from boundary measurements at multiple wave numbers combined with new results including uniqueness of obstacles. We consider general elliptic differential equations of the second order and arbitrary observation sites. We present some new results and outline basic ideas of their proofs. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. To derive the increasing stability we utilize sharp bounds of the analytic continuation for higher wave numbers, the Huygens’ principle, and boundary energy estimates in the initial boundary value problems for hyperbolic equations. Some numerical examples, based on a recursive Kaczmarz-Landweber iterative algorithm, shed light on theoretical results.

Dedicated to Masahiro Yamamoto for his sixtieth birthday.

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Acknowledgements

This research is supported in part by the Emylou Keith and Betty Dutcher Distinguished Professorship and the NSF grant DMS 15-14886. S. Lu is supported by National Key Research and Development Program of China (No. 2017YFC1404103), NSFC (No. 11925104), and Program of Shanghai Academic/Technology Research Leader (19XD1420500).

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS, 67260-0033, USA

    Victor Isakov

  2. Shanghai Key Laboratory for Contemporary Applied Mathematics, Key Laboratory of Mathematics for Nonlinear Sciences and School of Mathematical Science, Fudan University, Shanghai, 200433, China

    Shuai Lu

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  1. Victor Isakov
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  2. Shuai Lu
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Correspondence to Shuai Lu .

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Editors and Affiliations

  1. Shanghai Key Laboratory for Contemporary Applied Mathematics, Key Laboratory of Mathematics for Nonlinear Sciences and School of Mathematical Sciences, Fudan University, Shanghai, China

    Jin Cheng

  2. Shanghai Key Laboratory for Contemporary Applied Mathematics, Key Laboratory of Mathematics for Nonlinear Sciences and School of Mathematical Sciences, Fudan University, Shanghai, China

    Shuai Lu

  3. Department of Mathematical Sciences, The University of Tokyo, Tokyo, Japan

    Masahiro Yamamoto

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Isakov, V., Lu, S. (2020). On the Inverse Source Problem with Boundary Data at Many Wave Numbers. In: Cheng, J., Lu, S., Yamamoto, M. (eds) Inverse Problems and Related Topics. ICIP2 2018. Springer Proceedings in Mathematics & Statistics, vol 310. Springer, Singapore. https://doi.org/10.1007/978-981-15-1592-7_4

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