Volume 4, Issue 4
A Multi-Frequency Sampling Method for the Inverse Source Problems with Sparse Measurements

Xiaodong Liu & Shixu Meng

CSIAM Trans. Appl. Math., 4 (2023), pp. 653-671.

Published online: 2023-10

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  • Abstract

We consider the inverse source problems with multi-frequency sparse near field measurements. In contrast to the existing near field operator based on the integral over the space variable, a multi-frequency near field operator is introduced based on the integral over the frequency variable. A factorization of this multi-frequency near field operator is further given and analyzed. Based on such a factorization, we introduce a single-receiver multi-frequency sampling method to reconstruct a shell support of the source. Its theoretical foundation is derived from the properties of the factorized operators and a properly chosen point spread function. Numerical examples are provided to illustrate the multi-frequency sampling method with sparse near field measurements. Finally we briefly discuss how to extend the near field case to the far field case.

  • AMS Subject Headings

35P25, 45Q05, 78A46, 74B05

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COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-4-653, author = {Liu , Xiaodong and Meng , Shixu}, title = {A Multi-Frequency Sampling Method for the Inverse Source Problems with Sparse Measurements}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2023}, volume = {4}, number = {4}, pages = {653--671}, abstract = {

We consider the inverse source problems with multi-frequency sparse near field measurements. In contrast to the existing near field operator based on the integral over the space variable, a multi-frequency near field operator is introduced based on the integral over the frequency variable. A factorization of this multi-frequency near field operator is further given and analyzed. Based on such a factorization, we introduce a single-receiver multi-frequency sampling method to reconstruct a shell support of the source. Its theoretical foundation is derived from the properties of the factorized operators and a properly chosen point spread function. Numerical examples are provided to illustrate the multi-frequency sampling method with sparse near field measurements. Finally we briefly discuss how to extend the near field case to the far field case.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2022-0052}, url = {http://global-sci.org/intro/article_detail/csiam-am/22073.html} }
TY - JOUR T1 - A Multi-Frequency Sampling Method for the Inverse Source Problems with Sparse Measurements AU - Liu , Xiaodong AU - Meng , Shixu JO - CSIAM Transactions on Applied Mathematics VL - 4 SP - 653 EP - 671 PY - 2023 DA - 2023/10 SN - 4 DO - http://doi.org/10.4208/csiam-am.SO-2022-0052 UR - https://global-sci.org/intro/article_detail/csiam-am/22073.html KW - Sampling method, multi-frequency, sparse data, inverse source problems. AB -

We consider the inverse source problems with multi-frequency sparse near field measurements. In contrast to the existing near field operator based on the integral over the space variable, a multi-frequency near field operator is introduced based on the integral over the frequency variable. A factorization of this multi-frequency near field operator is further given and analyzed. Based on such a factorization, we introduce a single-receiver multi-frequency sampling method to reconstruct a shell support of the source. Its theoretical foundation is derived from the properties of the factorized operators and a properly chosen point spread function. Numerical examples are provided to illustrate the multi-frequency sampling method with sparse near field measurements. Finally we briefly discuss how to extend the near field case to the far field case.

Liu , Xiaodong and Meng , Shixu. (2023). A Multi-Frequency Sampling Method for the Inverse Source Problems with Sparse Measurements. CSIAM Transactions on Applied Mathematics. 4 (4). 653-671. doi:10.4208/csiam-am.SO-2022-0052
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