Formulation and super-efficient solution of large-scale inverse equivalent surface-source problems with consideration of various side constraints

逆等效源解确定等效源分布，使得它们能够再现在样品位置集合中观察到的辐射或散射场。一旦有了源，就可以获得关于辐射或散射物体的诊断信息，也可以计算新的观测地点的辐射或散射场，特别是远场的辐射或散射场。在本项目的第一部分中，研究并实现了各种基于分层传播平面波表示概念的逆等效表面源问题（IESSPs）的超高效求解技术，这些技术来自多层快速多极方法（MLFMM），以及通过分布球面谐波展开的无网格场展开。特别是，它可以在不损失精度的情况下实现非常有方向性的基于高斯光束的平面波谱平移算子，并且可以在求解域上产生具有方向性辐射的简化球谐波集，从而实现更健壮、更灵活和更有效的求解方法。在这些非常强大的技术之上，可以实现迭代求解过程的新的预处理和开始向量估计技术。在本续文中，我们将对其中的一些技术进行进一步的研究，但主要的重点将放在多层平面地面观测数据的IESSP求解器的超高效求解技术上。这些场景对汽车天线和散射测量很重要，但超高效求解技术在求解辐射和散射问题的积分方程方面也有很大的潜力。多层平面地面的存在对算法提出了新的挑战，特别是在效率方面，因为成功的自由空间方法不能直接转移到这种情况下。特别是，有必要使用更适合多层地面环境的平面波表示，例如基于weyl恒等式的平面波表示。

Inverse equivalent source solutions determine equivalent source distributions such that they are able to reproduce radiation or scattering fields observed in a collection of sample locations. Once the sources are available, diagnostic information about the radiation or scattering object(s) can be obtained and it is also possible to compute the radiation or scattering fields in new observation locations, in particular also in the far field. In the first part of this project, a variety of super-efficient solution techniques for inverse equivalent surface source problems (IESSPs) have been investigated and realized, which are based on the concepts of hierarchical propagating plane wave representations, as known from the multilevel fast multipole method (MLFMM), and on meshless field expansions via distributed spherical harmonics expansions. In particular, it was possible to realize very directive Gaussian-beam based translation operators for the plane-wave spectra without any loss of accuracy and it was possible to generate reduced sets of spherical harmonics with directive radiation toward the solution domain, which lead to a more robust, more flexible, and more efficient solution approach. On top of these very powerful techniques, new preconditioning and start vector estimation techniques for the iterative solution process could be realized. In this continuation proposal, some of these techniques shall be further pursued, but the major focus shall be on super-efficient solution techniques for IESSP solvers with observation data above multilayered planar ground. Such scenarios are, e.g., important for automotive antenna and scattering measurements, but the super-efficient solution techniques have also great potential toward the solution of integral equations of radiation and scattering problems. The presence of multilayered planar ground imposes new challenges on the algorithms, in particular with respect to efficiency, since the successful free-space methods are not directly transferable to this scenario. In particular, it will be necessary to work with plane-wave representations which are more suitable for the multilayered ground environment, such as those based on the Weyl-identity.