Transmission eigenvalue problem of internal inverse scattering for fully coated media

DING Huihui; LIU Lihan

doi:10.11714/acta.snus.ZR20250105

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Transmission eigenvalue problem of internal inverse scattering for fully coated media

更新时间：2026-01-29

- Transmission eigenvalue problem of internal inverse scattering for fully coated media
- Acta Scientiarum Naturalium Universitatis Sunyatseni Pages: 1-10(2026)
- 作者机构：
  
  重庆师范大学数学科学学院，重庆 401331
- 作者简介：
- 基金信息：
- DOI：10.11714/acta.snus.ZR20250105
  CLC： O29
- Received：13 June 2025，
  
  Revised：2025-11-26，
  
  Accepted：28 November 2025，
  
  Published Online：23 January 2026，
- 稿件说明：
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DING Huihui, LIU Lihan. Transmission eigenvalue problem of internal inverse scattering for fully coated media[J/OL]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2026, 1-10.
DOI：

DING Huihui, LIU Lihan. Transmission eigenvalue problem of internal inverse scattering for fully coated media[J/OL]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2026, 1-10. DOI： 10.11714/acta.snus.ZR20250105.

摘要

研究具有全涂层边界条件的非均匀介质内部反散射的传输特征值问题. 首先建立四阶非线性传输特征值问题，其次提出了一个带有辅助变量的等效混合公式，将非线性特征值问题转化为线性特征值问题，再利用Riesz表示定理、Rellich紧性定理构造恰当的算子，并利用Brezzi理论、柯西收敛准则、Poincaré不等式证明算子的紧性和强制性. 然后进行有限元离散化，证明了在凸域和非凸域上传输特征值的最优收敛速率，并且得到一个稀疏的广义特征值问题，通过压缩几乎所有具有巨大多重数的∞特征值，在不恶化稀疏性的情况下，显著减小了矩阵大小.

Abstract

Transmission eigenvalue problem of internal inverse scattering in inhomogeneous media with fully coated boundary conditions is investigated. First，a fourth-order nonlinear transmission eigenvalue problem is established. Second， an equivalent mixed formulation with auxiliary variables is proposed to transform the nonlinear problem into a linear eigenvalue problem. Appropriate operators are constructed using the Riesz representation theorem and the Rellich compactness theorem. Then， the compactness and coerciveness of operators are proven through the Brezzi theory， Cauchy convergence criterion， and Poincaré inequality. Subsequently， finite element discretization is performed. It is demonstrated that optimal convergence rates for transmission eigenvalues can be achieved on both convex and non-convex domains. A sparse generalized eigenvalue problem is derived， which significantly reduces matrix size by compressing nearly all ∞ eigenvalues with huge multiplicities while preserving sparsity.

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references

AN J ， SHEN J ， 2014 . Efficient spectral methods for transmission eigenvalues and estimation of the index of refraction ［J］. J Math Study ， 47 （ 1 ）： 1 - 20 .

CAKONI F ， COLTON D ， 2014a . A qualitative approach to inverse scattering theory ［M］. Cham ： Springer .

CAKONI F ， COLTON D ， MENG S X ， 2014b . The inverse scattering problem for a penetrable cavity with internal measurements ［M］//STEFANOV P， et al. Inverse problems and applications ， Providence ： American Mathematical Society .

CAKONI F ， COLTON D ， MONK P ， 2001 . The direct and inverse scattering problems for partially coated obstacles ［J］. Inverse Probl ， 17 （ 6 ）： 1997 - 2015 .

CAKONI F ， GINTIDES D ， HADDAR H ， 2010 . The existence of an infinite discrete set of transmission eigenvalues ［J］. SIAM J Math Anal ， 42 （ 1 ）： 237 - 255 .

COLTON D ， MONK P ， SUN J G ， 2010 . Analytical and computational methods for transmission eigenvalues ［J］. Inverse Probl ， 26 （ 4 ）： 045011 .

COLTON D ， PÄIVÄRINTA L ， SYLVESTER J ， 2007 . The interior transmission problem ［J］. Inverse Probl Imag ， 1 （ 1 ）： 13 - 28 . ［LinkOut］

GIORGI G ， HADDAR H ， 2012 . Computing estimates of material properties from transmission eigenvalues ［J］. Inverse Probl ， 28 （ 5 ）： 055009 .

GONG B ， SUN J G ， TURNER T ， et al ， 2020 . Finite element approximation of transmission eigenvalues for anisotropic media ［EB/OL］. arXiv： 2001.05340v1 .

HARRIS I ， 2021 . Analysis of two transmission eigenvalue problems with a coated boundary condition ［J］. Appl Anal ， 100 （ 9 ）： 1996 - 2019 .

KLEEFELD A ， PIERONEK L ， 2018 . Computing interior transmission eigenvalues for homogeneous and anisotropic media ［J］. Inverse Probl ， 34 （ 10 ）： 105007 .

LECHLEITER A ， PETERS S ， 2015 . Determining transmission eigenvalues of anisotropic inhomogeneous media from far field data ［J］. Commun Math Sci ， 13 （ 7 ）： 1803 - 1827 .

LI Y J ， YANG Y D ， BI H ， 2023 . The a priori and a posteriori error estimates for modified interior transmission eigenvalue problem in inverse scattering ［J］. Commun Comput Phys ， 34 （ 2 ）： 503 - 529 .

LIU Q ， LI T X ， ZHANG S ， 2023 . A mixed element scheme for the Helmholtz transmission eigenvalue problem for anisotropic media ［J］. Inverse Probl ， 39 （ 5 ）： 055005 .

PETERS S ， KLEEFELD A ， 2016 . Numerical computations of interior transmission eigenvalues for scattering objects with cavities ［J］. Inverse Probl ， 32 （ 4 ）： 045001 .

XIE H H ， WU X M ， 2017 . A multilevel correction method for interior transmission eigenvalue problem ［J］. J Sci Comput ， 72 （ 2 ）： 586 - 604 .

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