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Volume 12, Issue 2
A New Variational Approach for Inverse Source Problems

Qiya Hu, Shi Shu & Jun Zou

DOI: 10.4208/nmtma.OA-2017-0124

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 331-347.

Published online: 2018-12

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  • Abstract

We propose a new variational approach for recovering a general source profile in an elliptic system, using measurement data from the interior of the physical domain. The solution of the ill-posed inverse source problem is achieved by solving only one well-posed direct elliptic problem, resulting in the same computational cost as the one for the direct problem, and hence making the whole solution process of the inverse problem much less expensive than most existing methods. The resulting approximate solution is shown to be stable with respect to the change of the noise in the observation data, and a desired error estimate is also established in terms of the mesh size and the noise level in observation data. Numerical experiments are presented to confirm the theoretical predictions.

  • Keywords

Variational approach, inverse source problem, stability, convergence.

  • AMS Subject Headings

65N30, 65N55

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{NMTMA-12-331, author = {Qiya Hu, Shi Shu and Jun Zou}, title = {A New Variational Approach for Inverse Source Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {12}, number = {2}, pages = {331--347}, abstract = {

We propose a new variational approach for recovering a general source profile in an elliptic system, using measurement data from the interior of the physical domain. The solution of the ill-posed inverse source problem is achieved by solving only one well-posed direct elliptic problem, resulting in the same computational cost as the one for the direct problem, and hence making the whole solution process of the inverse problem much less expensive than most existing methods. The resulting approximate solution is shown to be stable with respect to the change of the noise in the observation data, and a desired error estimate is also established in terms of the mesh size and the noise level in observation data. Numerical experiments are presented to confirm the theoretical predictions.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2017-0124}, url = {http://global-sci.org/intro/article_detail/nmtma/12899.html} }
TY - JOUR T1 - A New Variational Approach for Inverse Source Problems AU - Qiya Hu, Shi Shu & Jun Zou JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 331 EP - 347 PY - 2018 DA - 2018/12 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2017-0124 UR - https://global-sci.org/intro/article_detail/nmtma/12899.html KW - Variational approach, inverse source problem, stability, convergence. AB -

We propose a new variational approach for recovering a general source profile in an elliptic system, using measurement data from the interior of the physical domain. The solution of the ill-posed inverse source problem is achieved by solving only one well-posed direct elliptic problem, resulting in the same computational cost as the one for the direct problem, and hence making the whole solution process of the inverse problem much less expensive than most existing methods. The resulting approximate solution is shown to be stable with respect to the change of the noise in the observation data, and a desired error estimate is also established in terms of the mesh size and the noise level in observation data. Numerical experiments are presented to confirm the theoretical predictions.

Qiya Hu, Shi Shu and Jun Zou. (2018). A New Variational Approach for Inverse Source Problems. Numerical Mathematics: Theory, Methods and Applications. 12 (2). 331-347. doi:10.4208/nmtma.OA-2017-0124
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