Numerical Methods for Transmission Eigenvalues

透射本征值问题近年来在散射和逆散射领域引起了广泛的研究。虽然表述简单，但偏微分方程的任何标准理论都不能涵盖这个问题。迄今为止，对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。本文提出了求解亥姆霍兹方程和麦克斯韦方程传输特征值的鲁棒数值方法。具体而言，将开展以下研究课题。1)亥姆霍兹方程的迭代求解方法。基于四阶重表述，用有限元法求解相应的广义特征值问题。然后用迭代法求相关代数函数的根，得到传输特征值。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。然后利用旋度符合边元计算传输特征值。3)各向异性Maxwell方程组的迭代求解方法。这种方法又是基于对各向异性麦克斯韦方程组的传输特征值问题的四阶重新表述。本文将利用相关的广义麦克斯韦特征值问题建立一个根为传输特征值的代数方程。然后应用迭代法求代数方程的根。它是亥姆霍兹方程迭代方法的推广。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。提出的研究将是对亥姆霍兹方程和麦克斯韦方程的传输特征值的开创性数值研究。这些结果对透射本征值数学理论的发展具有重要意义，并可用于比较反散射理论中的各种估计。提出的研究将提供数学家和工程师可靠的工具来计算传输特征值。这将为研究各向异性介质的反散射问题，如反电磁散射问题提供新的方法。由于透射特征值可以用来估计散射物体的材料性质，因此本研究在无损检测、地球物理应用、医学成像等方面具有潜在的应用前景。例如，可以从传输本征值的位置检测电介质中是否存在空腔。数值结果将分发给数学家，用于传输特征值的分析研究和工程师，用于未知物体的检测和重建。此外，建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.