Numerical Methods for Transmission Eigenvalues

传输本征值问题近年来在散射和逆散射领域引起了许多研究者的关注。尽管简单地说，这个问题并不是任何标准的偏微分方程理论所涵盖的。到目前为止，传输本征值的数值处理非常有限。有效的数值方法将加深对问题的理解，并为数学家和工程师计算传输本征值提供工具。这一建议旨在为Helmholtz方程和Maxwell方程的传输本征值提供稳健的数值方法。特别是，将开展以下研究课题。1)Helmholtz方程的迭代方法。基于四阶重构法，一个相关的广义特征值问题将用有限元方法求解。然后，应用迭代方法搜索相关代数函数的根，得到传输本征值。2)麦克斯韦方程的连续有限元方法。麦克斯韦方程的传输本征值问题将首先写成适当的弱形式。然后用符合旋度的边缘单元来计算传输本征值。3)各向异性麦克斯韦方程的迭代方法。这种方法同样是基于各向异性麦克斯韦方程的传输本征值问题的四阶重新表述。利用广义麦克斯韦本征值问题建立一个以传输本征值为根的代数方程。然后可以用迭代方法来搜索代数方程的根。它是Helmholtz方程迭代方法的推广。然而，麦克斯韦方程的情况要复杂得多，需要额外的技术处理。该研究将是对亥姆霍兹方程和麦克斯韦方程传输本征值的开创性的数值研究。所得结果对发展传输本征值的数学理论具有重要意义，并可用于比较逆散射理论中的各种估计。这项拟议的研究将为数学家和工程师提供可靠的工具来计算传输特征值。这将为研究反散射问题，如各向异性介质的反电磁散射问题提供新的方法。由于传输本征值可以用来估计散射体的材料特性，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，可以根据传输本征值的位置来检测介质中是否存在空穴。数值结果将被分发给数学家用于传输本征值的分析研究和工程师用于未知对象的检测和重建。此外，拟议项目的成功完成将提高大学的研究能力，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.