Landweber iterative regularization method for reconstructing the unknown source of the modified Helmholtz equation

Dun-Gang Li; Fan Yang; Ping Fan; Xiao-Xiao Li; Can-Yun Huang; Dun-Gang Li; Fan Yang; Ping Fan; Xiao-Xiao Li; Can-Yun Huang

doi:10.3934/math.2021598

AIMS Mathematics

2021, Volume 6, Issue 9: 10327-10342. doi: 10.3934/math.2021598

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Research article

Landweber iterative regularization method for reconstructing the unknown source of the modified Helmholtz equation

School of Science, Lanzhou University of Technology, Lanzhou, Gansu, 730050, People's Republic of China

Received: 17 February 2021 Accepted: 05 July 2021 Published: 14 July 2021
MSC : 35R25, 35R30, 47A52

This paper considers the inverse problem of determining an unknown source which depends only one spatial variable on modified Helmholtz equation. This problem is well known to be severely ill-posed, the solution (if it exists) does not depend continuously on the data. Landweber iterative regularization method is used to solve this inverse source problem. The Hölder type error estimates are obtained between the exact solution and regularization solutions under an a priori and an a posteriori regularization parameters choice rules, respectively. Numerical examples are provided to show the effectiveness of the proposed method.
- modified Helmholtz equation,
- ill-posed problem,
- posteriori regularization parameter choice rule,
- Landweber iterative method
Citation: Dun-Gang Li, Fan Yang, Ping Fan, Xiao-Xiao Li, Can-Yun Huang. Landweber iterative regularization method for reconstructing the unknown source of the modified Helmholtz equation[J]. AIMS Mathematics, 2021, 6(9): 10327-10342. doi: 10.3934/math.2021598

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Abstract

This paper considers the inverse problem of determining an unknown source which depends only one spatial variable on modified Helmholtz equation. This problem is well known to be severely ill-posed, the solution (if it exists) does not depend continuously on the data. Landweber iterative regularization method is used to solve this inverse source problem. The Hölder type error estimates are obtained between the exact solution and regularization solutions under an a priori and an a posteriori regularization parameters choice rules, respectively. Numerical examples are provided to show the effectiveness of the proposed method.

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