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Reconstruction algorithms of an inverse source problem for the Helmholtz equation

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Abstract

In this paper, we study an inverse source problem for the Helmholtz equation from measurements. The purpose of this paper is to reconstruct the salient features of the hidden sources within a body. We propose three stable reconstruction algorithms to detect the number, the location, the size, and the shape of the hidden sources along with compact support from a single measurement of near-field Cauchy data on the external boundary. This problem is nonlinear and ill-posed; thus, we should consider regularization techniques in reconstruction algorithms. We give several numerical experiments to demonstrate the viability of our proposed reconstruction algorithms.

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Funding

The research of Ji-Chuan Liu was supported by the NSF of China (11601512, 11326236, and 11501562) and the Fundamental Research Funds for the Central Universities (2014QNA57).

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Authors and Affiliations

  1. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Ji-Chuan Liu

  2. Teaching Section of Mathematic, Department of Basic, Army Logistics University of PLA, Chongqing, 400000, People’s Republic of China

    Xiao-Chen Li

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  1. Ji-Chuan Liu
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  2. Xiao-Chen Li
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Correspondence to Ji-Chuan Liu.

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Liu, JC., Li, XC. Reconstruction algorithms of an inverse source problem for the Helmholtz equation. Numer Algor 84, 909–933 (2020). https://doi.org/10.1007/s11075-019-00786-8

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