压缩感知在电磁场积分方程快速计算中的应用

This project is mainly on the compressive sensing (CS) theory and its application in computational electromagnetics (CEM), to form a new excitation source in which much information from different incident angles is included. Numerical analysis of electromagnetic problems will be researched under this new excitation. The main contents can be summarized as follows: Theoretically, take fast computation analysis of electromagnetic integral equations for example, a new model based on CS will be constructed to solve given linear promblems, which will greatly improve the computational efficiency. Technically, the three key techniques,such as sparse representation, measurement matrix and reconstruction algorithm, will be investigated and optimized to improve the accuracy and reduce the number of measurements. In application, the new theoretic model will be used to solve electromagnetic scattering problems, by optimization and matching on techniques, the scattering analysis over a the entire angle range will be accomplished with only a few incident source illuminated. Further more, in conjugation with advanced frequency-sweep analysis method and preprocessing techniuqes, an systemic and efficient scheme for obtaining scattering characteristics in both spatial and frequency domains is formed. In particular, owing to its generality in handling an invariant linear system, the proposed method can also be applied in other related engineering problems.

本项目主要将压缩感知理论（CS）引入到计算电磁学中，构建一种富含空间信息的新型电磁场激励源，并在此激励下开展相关电磁场问题的计算分析，研究内容概括如下： (1)以电磁场积分方程快速求解为例，基于压缩感知理论构建一种求解特定线性问题的新模型，以提高系统随参数变化时的分析效率； (2)围绕电磁散射问题的快速、精确计算，对压缩感知理论中的稀疏表示、观测矩阵及恢复算法等进行研究和优化，以大幅降低观测数目并提高计算精度； (3)将该理论模型应用于电磁散射问题的计算，通过技术优化和匹配，以所构建激励源的数次照射，获得目标全空间的电磁电磁特性响应。在此基础之上，借助于先进的宽频算法和预处理技术，形成一种系统、高效的目标频空特性分析方法。 基于本项目所提理论框架的普适性，其研究成果将在其他线性系统的分析中获得应用。

电磁场积分方程方法因计算精度高、无需扩大计算区域、计算效率高等优势，在电磁散射、电磁辐射、电磁兼容等众多领域有着广泛的应用，但是在分析含参数激励（如多入射角度）问题时，除常规的差值和逼近方法外尚无十分有效的快速分析方法。本项目基于压缩感知理论构建一种富含空间信息的新型电磁场激励源，并对压缩感知理论中的稀疏表示、观测矩阵及恢复算法等关键技术进行研究和优化，最终通过该激励源的数次照射，获得目标全空间的电磁特性响应。项目实施以来，在以下五个方面取得了较好进展：第一，开展了压缩感知求解宽角度电磁散射问题的理论框架研究，对典型散射目标和天线结构项等进行了分析；第二，对激励向量的所包含的先验知识进行了有效提取和分析，构建出若干新型稀疏变换方法，在同等计算精度的情况下，大幅减少了观测次数；第三，对观测矩阵开展了研究，提出了一种随机观测方法，大幅提高了观测效率；第四，对恢复算法开展了系统的研究和实验分析，给出了相对较优的恢复算法；第五，将研究方案推广到频域有限元、时域有限元等其他计算电磁学方法当中并取得初步效果。