Article
Mathematics, Applied
Fang Zeng, Shixu Meng
Summary: This paper examines the inverse electromagnetic scattering of a cavity surrounded by an inhomogeneous medium in three dimensions. The authors propose a method using scattered electric field measurements to uniquely determine the cavity shape and reconstruct it using the linear sampling method. Numerical examples are provided to demonstrate the effectiveness of the algorithm.
Article
Mathematics, Applied
Xuqing Zhang, Jiayu Han
Summary: The modified transmission eigenvalue problem of elastic waves, arising from inverse scattering theory, is analyzed in this study. The well-posedness of the problem is first analyzed, followed by a rigorous error analysis of conforming finite element methods. Two types of multigrid schemes are then established. Numerical research is conducted on the effect of material flaws on the modified elastic transmission eigenvalues, and the multigrid schemes are applied for parallel computing. Numerical experiments validate the theoretical convergence order and the efficiency of the proposed schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Jianli Xiang, Guozheng Yan
Summary: This study focuses on recovering refractive indices and transmission coefficients through acoustic far-field measurement. The study establishes the uniqueness of a constant refractive index and piecewise constant refractive index.
IMA JOURNAL OF APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Rafael Ceja Ayala, Isaac Harris, Andreas Kleefeld, Nikolaos Pallikarakis
Summary: In this paper, we analyze the transmission eigenvalue problem with two conductivity parameters. Assuming an isotropic scatterer, we study the scattering of a plane wave as the underlying physical model. Previous studies focused on one conductive boundary parameter, while we consider the case of two parameters. We prove the existence and discreteness of transmission eigenvalues and examine their dependence on physical parameters. Additionally, we establish the monotonicity of the first transmission eigenvalue with respect to the parameters and investigate the limiting procedure when the second boundary parameter vanishes. Finally, we conduct extensive numerical experiments to validate our theoretical work.
APPLICABLE ANALYSIS
(2023)
Article
Mathematics
Wei-Chen Chang, Tiexiang Li, Wen-Wei Lin, Jenn-Nan Wang
Summary: This study investigates the interior transmission eigenvalues for elastic scattering in an inhomogeneous medium with an obstacle, aiming to provide an efficient numerical algorithm to compute as many positive eigenvalues as possible. The problem is simplified to a generalized eigenvalue problem using the finite element method, and the Jacobi-Davidson algorithm is applied for solution. Special attention is needed in Case 1 due to the presence of a large number of zero eigenvalues that depend on discretization size.
RESEARCH IN THE MATHEMATICAL SCIENCES
(2021)
Article
Mathematics, Applied
Yalin Zhang
Summary: This study focuses on the exterior transmission eigenvalue problem for anisotropic media with spherical symmetry assumptions, discussing the existence and asymptotic distribution of exterior transmission eigenvalues, and investigating the inverse problem of determining the refractive index from two sets of spectral data under appropriate conditions.
APPLICABLE ANALYSIS
(2021)
Article
Mathematics, Applied
Ou Yunhui, Zeng Fang
Summary: This paper focuses on the interior inverse scattering problem of reconstructing the shape of an elastic cavity. It proves a reciprocity relation for the scattered elastic field and a uniqueness theorem for the inverse problem. The decomposition method is then employed to determine the boundary of the cavity and convergence results are presented. Numerical examples are provided to demonstrate the viability of the method.
APPLICABLE ANALYSIS
(2022)
Article
Mathematics
Peter Monk, Virginia Selgas
Summary: This paper studies a class of modified interior transmission eigenvalues for solving fluid-solid interaction problems. By introducing an artificial diffusivity constant, the eigenvalues are modified and their location in the complex plane is determined using an operator. Numerical results show that some eigenvalues can still be determined from far field data even with noise interference.
RESEARCH IN THE MATHEMATICAL SCIENCES
(2022)
Article
Mathematics, Applied
Shixi Wang, Hai Bi, Yanjun Li, Yidu Yang
Summary: Based on the work of Monk and Selgas (2022), we investigate the finite element method for the modified transmission eigenvalues in inverse scattering of a fluid-solid interaction problem. We provide a comprehensive error analysis, including a priori and a posteriori error estimates, under minimal regularity assumptions on the solution. We also propose adaptive computations.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Yalin Zhang, Jia Zhao
Summary: This paper considers the exterior transmission eigenvalues for spherical symmetry media and spherically symmetric eigenfunctions. We present the number and asymptotic distribution (described by subscript numbers) of these eigenvalues in the complex plane under various coefficient conditions.
Article
Mathematics, Applied
Jianli Xiang, Guozheng Yan
Summary: This paper addresses the inverse scattering problem of time-harmonic acoustic waves by a mixed-type scatterer. The well-posedness of the direct problem is established by the variational method, and the factorization method is used to simultaneously recover the support of the inhomogeneous medium and the shape of the impenetrable obstacle. Numerical examples are provided to demonstrate the feasibility and effectiveness of the inverse algorithm.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Jingni Xiao
Summary: This paper studies the corner scattering for the operator del . gamma(x) del + k (2) rho(x) in R-2, and proves that any admissible incident field will be scattered by concave corners by constructing a new type of complex geometric optics solutions. The existence of non-scattering waves when gamma - I has a jump across the corner is also briefly discussed.
Article
Physics, Multidisciplinary
D. Gintides, S. Giogiakas, L. Mindrinos
Summary: The paper discusses the scattering of an incident electromagnetic wave on a cylinder and solves the inverse problem by introducing a novel numerical scheme for reconstructing the refractive index.
Article
Geochemistry & Geophysics
Dongjin Bai, Xiaolong Dong, Saibun Tjuatja, Di Zhu, Zijin Zhang
Summary: Reported radiometric measurements show that microwave emission features of layered irregular and inhomogeneous medium are characterized by incoherent and coherent radiative transfer processes. In this article, a comprehensive layer emission model (CLEM) based on the scattering operator formulation is presented to consider medium and boundary scattering as well as coherent boundary interaction in general layered medium. CLEM integrates wave-based coherent multiple reflection operators and intensity-based multiple scattering processes to comprehensively describe rough boundary scattering, volume scattering, medium/boundary interaction, and coherent boundary interaction. Simulations and analyses based on CLEM are conducted to evaluate coherent boundary interaction and different impacting factors for ice- and snow-covered ground cases, and validation is performed with emission observations of snow-covered terrain and frozen soil.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
(2023)
Article
Mathematics
Evagelia S. Athanasiadou
Summary: The study successfully reconstructed the shape of the buried object using a modified linear sampling method, without requiring any a priori knowledge of the material properties. Additionally, the impedance characteristics of the object's surface were determined.
