A quasi-boundary method for solving an inverse diffraction problem

Zhenping Li; Xiangtuan Xiong; Jun Li; Jiaqi Hou; Zhenping Li; Xiangtuan Xiong; Jun Li; Jiaqi Hou

doi:10.3934/math.2022618

AIMS Mathematics

2022, Volume 7, Issue 6: 11070-11086. doi: 10.3934/math.2022618

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Research article

A quasi-boundary method for solving an inverse diffraction problem

1.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
2.
Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China

Received: 17 October 2021 Revised: 18 March 2022 Accepted: 21 March 2022 Published: 06 April 2022
MSC : 35R25, 35R30, 47A52

In this paper, we deal with the reconstruction problem of aperture in the plane from their diffraction patterns. The problem is severely ill-posed. The reconstruction solutions of classical Tikhonov method and Fourier truncated method are usually over-smoothing. To overcome this disadvantage of the classical methods, we introduce a quasi-boundary regularization method for stabilizing the problem by adding a-priori assumption on the exact solution. The corresponding error estimate is derived. At the continuation boundary $ z = 0 $, the error estimate under the a-priori assumption is also proved. In theory without noise, the proposed method has better approximation than the classical Tikhonov method. For illustration, two numerical examples are constructed to demonstrate the feasibility and efficiency of the proposed method.
- inverse diffraction problem,
- ill-posed,
- regularization,
- quasi-boundary method,
- Tikhonov method
Citation: Zhenping Li, Xiangtuan Xiong, Jun Li, Jiaqi Hou. A quasi-boundary method for solving an inverse diffraction problem[J]. AIMS Mathematics, 2022, 7(6): 11070-11086. doi: 10.3934/math.2022618

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Abstract

In this paper, we deal with the reconstruction problem of aperture in the plane from their diffraction patterns. The problem is severely ill-posed. The reconstruction solutions of classical Tikhonov method and Fourier truncated method are usually over-smoothing. To overcome this disadvantage of the classical methods, we introduce a quasi-boundary regularization method for stabilizing the problem by adding a-priori assumption on the exact solution. The corresponding error estimate is derived. At the continuation boundary $ z = 0 $, the error estimate under the a-priori assumption is also proved. In theory without noise, the proposed method has better approximation than the classical Tikhonov method. For illustration, two numerical examples are constructed to demonstrate the feasibility and efficiency of the proposed method.

References

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