Numerical Methods for Transmission Eigenvalues

传输特征值问题最近吸引了散射和逆散射领域的许多研究人员。尽管表述简单，但任何标准偏微分方程理论均未涵盖该问题。迄今为止，传输特征值的数值处理非常有限。有效的数值方法将增强对问题的理解，并为数学家和工程师提供计算传输特征值的工具。该提案旨在为亥姆霍兹方程和麦克斯韦方程的传输特征值提供稳健的数值方法。具体而言，将开展以下研究课题。 1) 亥姆霍兹方程的迭代方法。基于四阶重构，相关的广义特征值问题将通过有限元方法求解。然后将应用迭代方法来搜索相关代数函数的根，该根最终成为传输特征值。 2) 麦克斯韦方程组的连续有限元法。首先将麦克斯韦方程组的传输特征值问题写成合适的弱形式。然后，旋度一致的边缘元素将用于计算传输特征值。 3) 各向异性麦克斯韦方程组的迭代方法。该方法再次基于各向异性麦克斯韦方程组的传输特征值问题的四阶重构。相关的广义麦克斯韦特征值问题将用于建立一个代数方程，其根是传输特征值。然后可以应用迭代方法来搜索代数方程的根。它是亥姆霍兹方程迭代方法的扩展。然而，麦克斯韦方程组的情况要困难得多，需要额外的技术处理。所提出的研究将是亥姆霍兹方程和麦克斯韦方程组传输特征值的开创性数值研究。这些结果对于传输特征值数学理论的发展非常重要，并且可用于比较逆散射理论中的各种估计。 拟议的研究将为数学家和工程师提供计算传输特征值的可靠工具。它将带来研究逆散射问题的新方法，例如各向异性介质的逆电磁散射问题。由于透射特征值可用于估计散射物体的材料特性，因此所提出的研究在无损检测、地球物理应用、医学成像等方面具有潜在的用途。例如，可以从透射特征值的位置检测电介质中空腔的存在。数值结果将分发给数学家进行传输特征值的分析研究和工程师进行未知物体的检测和重建。此外，该项目的成功完成将增强大学的研究能力，并为研究生提供宝贵的研究机会。

The transmission eigenvalue problem has attracted many researchers in the scattering and inverse scattering communities recently. Although simply stated, the problem is not covered by any standard theory of partial differential equations. Numerical treatment of transmission eigenvalues is very limited to date. Effective numerical methods will enhance the understanding of the problem and provide tools for mathematicians and engineers to compute transmission eigenvalues. This proposal aims at robust numerical methods for transmission eigenvalues for the Helmholtz equation and the Maxwell's equations. In particular, the following research topics will be carried out. 1) Iterative methods for the Helmholtz equation. Based on a fourth order reformulation, an associated generalized eigenvalue problem will be solved by the finite element method. Then iterative methods will be applied to search roots of a related algebraic function which turn out to be the transmission eigenvalues. 2) Continuous finite element method for the Maxwell's equations. The transmission eigenvalue problem of the Maxwell's equations will be written in a suitable weak form first. Then the curl conforming edge elements will be used to compute the transmission eigenvalues. 3) Iterative methods for the anisotropic Maxwell's equations. This approach is again based on a forth order reformulation of the transmission eigenvalue problem of the anisotropic Maxwell's equations. An associated generalized Maxwell's eigenvalue problem will be used to set up an algebraic equation whose roots are the transmission eigenvalues. Then iterative methods can be applied to search the roots of the algebraic equation. It is an extension of the iterative methods for the Helmholtz equation. However, the case for the Maxwell's equations is much more difficult and require additional technical treatment.The proposed research will be a pioneer numerical study on transmission eigenvalues for the Helmholtz equation and the Maxwell's equations. The results are important for the development of mathematical theory for transmission eigenvalues and can be used to compare various estimates in inverse scattering theory. The proposed research will provide mathematicians and engineers reliable tools to compute transmission eigenvalues. It will lead to new methods for studying the inverse scattering problems such as inverse electromagnetic scattering problem for anisotropic media. Since transmission eigenvalues can be used to estimate material properties of the scattering object, the proposed research has potential usage in non-destructive testing, geophysical applications, medical imaging, etc. For example, it is possible to detect the presence of cavities in the dielectric from the location of the transmission eigenvalues. The numerical results will be disseminated to mathematician for analytical study of transmission eigenvalues and engineers for detection and reconstruction of unknown objects. In addition, successful accomplishment of the proposed project will enhance the research capacity of the university and provide graduate students valuable research opportunities.