含有腔体的介质电磁反散射中新的特征值的分析与计算

在各项异性介质的电磁反散射问题中，测量的散射数据一般不能唯一决定介质参数，利用携带介质本质信息的某些特征值作为目标特征是进行介质无损探测的一个可能的方式，而采用人工参数替代物理波数作为特征值将具有允许特征值为复数和无需多频数据的优点。本项目拟结合反散射理论、电磁学理论、特殊算子理论和数值方法，研究含有腔体的介质电磁反散射中新的特征值。本质上，新的特征值问题是非均匀介质下的Maxwell方程在无界域上关于人工参数的非自伴特征值问题。我们将讨论特征值的离散性、分布区域和存在性，建立支撑某些抽样类算法计算特征值的数学理论，研究由特征值的变化识别介质参数改变的机制，考察特征值问题中的调节参数对特征值关于介质参数变化的敏感度的影响，证明特征值关于介质参数和调节参数的稳定性。本项目的宗旨是对新的特征值提供系统的理论和数值研究，最终为介质的无损探测提供合适的目标特征。

In inverse electromagnetic anisotropic medium scattering the parameters of the medium may not be uniquely determined by the measured scattered data. A possible approach for nondestructive testing of a medium is to use some eigenvalues carrying essential information about the medium as target signatures. Moreover, using some artificial parameter instead of the physical wave number playing the role of eigenvalues will have the advantages that the eigenvalues can be complex and that multi-frequency data are not necessary. In this project, we plan to study new eigenvalues in inverse electromagnetic medium scattering for a cavity by combining inverse scattering theory, electromagnetic theory, theory about some special operators with numerical methods. Essentially, a new eigenvalue problem is a non-selfadjoint eigenvalue problem about the artificial parameter in an unbounded domain for Maxwell’s equations in an inhomogeneous medium. We will discuss the discreteness, distribution and existence of the eigenvalues, build the mathematical theory supporting some sampling methods to compute the eigenvalues, study the mechanism for identifying the change of the medium by the shifts of eigenvalues, investigate how the tunning parameter in the eigenvalue problem effects the sensitivity of the eigenvalues to changes in the parameters of the medium and prove the stabilities of the eigenvalues about the parameters of the medium and the tunning parameter, respectively. The aim of this project is to provide systematic study on theory and computations for new eigenvalues and finally provide appropriate target signatures for nondestructive testing of the medium.