On fast reconstruction of periodic structures with partial scattering data

John Daugherty; Nate Kaduk; Elena Morgan; Dinh-Liem Nguyen; Peyton Snidanko; Trung Truong; John Daugherty; Nate Kaduk; Elena Morgan; Dinh-Liem Nguyen; Peyton Snidanko; Trung Truong

doi:10.3934/era.2024303

Electronic Research Archive

2024, Volume 32, Issue 11: 6481-6502. doi: 10.3934/era.2024303

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On fast reconstruction of periodic structures with partial scattering data

1.
College of Professional Programs, Southwest Baptist University, Bolivar, MO 65613, USA
2.
Department of Mathematical Sciences, Kent State University, Kent, OH 44240, USA
3.
Department of Mathematics, Marist College, Poughkeepsie, NY 12601, USA
4.
Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA
5.
Department of Mathematics, Bradley University, Peoria, IL 61625, USA
6.
Department of Mathematics and Physics, Marshall University, Huntington, WV 25755, USA

Received: 21 August 2024 Revised: 12 November 2024 Accepted: 19 November 2024 Published: 29 November 2024

This paper presents a numerical method for solving the inverse problem of reconstructing the shape of periodic structures from scattering data. This inverse problem is motivated by applications in the nondestructive evaluation of photonic crystals. The numerical method belongs to the class of sampling methods that aim to construct an imaging function for the shape of the periodic structure using scattering data. By extending the results of Nguyen, Stahl, and Truong [Inverse Problems, 39:065013, 2023], we studied a simple imaging function that uses half the data required by the numerical method in the cited paper. Additionally, this imaging function is fast, simple to implement, and very robust against noise in the data. Both isotropic and anisotropic cases were investigated, and numerical examples were presented to demonstrate the performance of the numerical method.
- inverse scattering,
- periodic structure,
- imaging function,
- partial data
Citation: John Daugherty, Nate Kaduk, Elena Morgan, Dinh-Liem Nguyen, Peyton Snidanko, Trung Truong. On fast reconstruction of periodic structures with partial scattering data[J]. Electronic Research Archive, 2024, 32(11): 6481-6502. doi: 10.3934/era.2024303

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Abstract

This paper presents a numerical method for solving the inverse problem of reconstructing the shape of periodic structures from scattering data. This inverse problem is motivated by applications in the nondestructive evaluation of photonic crystals. The numerical method belongs to the class of sampling methods that aim to construct an imaging function for the shape of the periodic structure using scattering data. By extending the results of Nguyen, Stahl, and Truong [Inverse Problems, 39:065013, 2023], we studied a simple imaging function that uses half the data required by the numerical method in the cited paper. Additionally, this imaging function is fast, simple to implement, and very robust against noise in the data. Both isotropic and anisotropic cases were investigated, and numerical examples were presented to demonstrate the performance of the numerical method.

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