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

逆等效源解决方案确定等效源分布，使得它们能够再现在样本位置集合中观察到的辐射场或散射场。一旦有了源，就可以获得关于辐射或散射物体(S)的诊断信息，还可以计算新观测位置的辐射或散射场，特别是远场。在本项目的第一部分中，研究并实现了多种超有效的求解等效面源反问题(IESSP)的技术，这些技术基于多层快速多极子方法(MLFMM)中的分层传播平面波表示的概念，以及基于分布球谐展开的无网格场展开。特别是，可以在不损失任何精度的情况下实现基于非常定向的高斯光束的平面波谱的平移算子，并且可以生成具有朝向解域的定向辐射的简化球谐集，这导致了更健壮、更灵活和更有效的求解方法。在这些非常强大的技术之上，可以实现用于迭代求解过程的新的预条件和起始向量估计技术。在本继续提案中，将进一步探讨其中一些技术，但主要重点应放在IESSP解算器的超高效解算技术上，该解算器的观测数据位于多层平面地面之上。例如，这样的场景对于汽车天线和散射测量很重要，但超高效的求解技术对于求解辐射和散射问题的积分方程式也具有巨大的潜力。多层平面地面的存在给算法带来了新的挑战，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，必须使用更适合于多层地面环境的平面波表示法，例如基于Weyl恒等式的平面波表示法。

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.