Numerical Methods for Transmission Eigenvalues

传输本征值问题是近年来散射和逆散射领域的研究热点。虽然简单地说，这个问题是不涵盖任何标准理论的偏微分方程。传输特征值的数值处理是非常有限的日期。有效的数值方法将有助于加深对问题的理解，并为数学家和工程师提供计算传输特征值的工具。该建议的目的是在强大的数值方法传输本征值的亥姆霍兹方程和麦克斯韦方程。具体而言，将开展以下研究课题。1)Helmholtz方程的迭代方法。基于一个四阶变换，一个相应的广义特征值问题将用有限元法求解。然后用迭代法求出相应代数函数的根，这些根就是传输本征值。2)麦克斯韦方程组的连续有限元法。麦克斯韦方程组的传输本征值问题将首先写成适当的弱形式。然后利用符合旋度的棱边元计算透射本征值。3)各向异性麦克斯韦方程组的迭代方法。这种方法再次是基于四阶的各向异性麦克斯韦方程的传输本征值问题的重新制定。一个相关的广义麦克斯韦特征值问题将被用来建立一个代数方程，其根是传输特征值。然后用迭代法求解方程的根。它是Helmholtz方程迭代法的推广。然而，麦克斯韦方程组的情况下是更加困难的，需要额外的技术处理。拟议的研究将是一个先驱的数值研究的传输本征值的亥姆霍兹方程和麦克斯韦方程。结果是重要的传输本征值的数学理论的发展，并可用于比较各种估计逆散射理论。 该研究为数学家和工程师提供了计算传输特征值的可靠工具。这将为研究诸如各向异性介质的电磁逆散射问题等逆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对象的材料特性，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，它是可能的，以检测腔体的存在，在电介质中的传输本征值的位置。数值结果将分发给数学家进行分析研究的传输特征值和工程师的检测和重建的未知对象。此外，成功完成拟议项目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.