Solution structure and stability for a two-parameter acoustic inverse scattering problem

Jing Xu; Jingzhi Li; Yan Chang; Tian Niu; Jing Xu; Jingzhi Li; Yan Chang; Tian Niu

doi:10.3934/era.2026039

Electronic Research Archive

2026, Volume 34, Issue 2: 844-865. doi: 10.3934/era.2026039

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Solution structure and stability for a two-parameter acoustic inverse scattering problem

1.
School of Statistics and Mathematics & Research Center for Regional Digital Economy and Digital Governance, Inner Mongolia University of Finance and Economics, Hohhot 010070, China
2.
Department of Mathematics & National Center for Applied Mathematics Shenzhen & SUSTech, International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
3.
School of Mathematics & School of Physics, Harbin Institute of Technology, Harbin 150001, China
4.
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

Received: 03 December 2025 Revised: 20 January 2026 Accepted: 21 January 2026 Published: 26 January 2026

This paper investigated the structure and stability of solutions for simultaneously recovering density and bulk modulus in acoustic inverse scattering. The nonlinear scattering model was linearized via the Born approximation, and the inverse problem was casted as a $ 2 \times 2 $ block operator equation. The off-diagonal blocks of this operator matrix captured the cross-coupling interactions between the two physical parameters. By employing the maximal Tseng generalized inverse, we developed an analytical formulation that remained valid even when diagonal blocks were non-invertible, overcoming limitations of traditional Schur complement methods. To address the inherent ill-posedness, we incorporated Tikhonov regularization into this linearized system, deriving explicit solution representations and establishing a rigorous stability estimate. Our analysis revealed that the strength of the off-diagonal interaction imposed a fundamental lower bound on the regularization parameter. As the parameter approached this bound, the reconstruction error exhibited singular behavior.
- acoustic inverse scattering,
- multi-parameter inversion,
- maximal Tseng generalized inverse,
- Tikhonov regularization,
- stability estimate
Citation: Jing Xu, Jingzhi Li, Yan Chang, Tian Niu. Solution structure and stability for a two-parameter acoustic inverse scattering problem[J]. Electronic Research Archive, 2026, 34(2): 844-865. doi: 10.3934/era.2026039

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Abstract

This paper investigated the structure and stability of solutions for simultaneously recovering density and bulk modulus in acoustic inverse scattering. The nonlinear scattering model was linearized via the Born approximation, and the inverse problem was casted as a $ 2 \times 2 $ block operator equation. The off-diagonal blocks of this operator matrix captured the cross-coupling interactions between the two physical parameters. By employing the maximal Tseng generalized inverse, we developed an analytical formulation that remained valid even when diagonal blocks were non-invertible, overcoming limitations of traditional Schur complement methods. To address the inherent ill-posedness, we incorporated Tikhonov regularization into this linearized system, deriving explicit solution representations and establishing a rigorous stability estimate. Our analysis revealed that the strength of the off-diagonal interaction imposed a fundamental lower bound on the regularization parameter. As the parameter approached this bound, the reconstruction error exhibited singular behavior.

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