格林函数随距离快速衰减，离散电磁积分方程所得满阵是欠秩矩阵，可用稀疏矩阵来近似。项目称这种特性为积分方程或算子的数据稀疏特性。已有积分方程快速算法都直接或间接的应用了此特性，但鲜见对它们的系统性研究，尤其是关于介质方面的系统研究。项目将填补这一空白，系统研究不同积分方程-重点是均匀介质的各类面积分方程和非均匀介质的体积分方程-的数据系数特性，应用ID(interpolative decomposition)和skeleton 概念挖掘这一特性的潜力，开发新的迭代和直接算法。研究不同积分方程算子、多尺度网格、不同介质参数类型及其大小变化等因素如何影响基于ID和skeleton技术的精度和效率；寻求使用ID 构造skeleton 的精度控制方式和构造skeleton 的最佳方式；开发基于ID和skeleton的预处理技术和矩阵求逆技术，实现多介质、多尺度目标的快速迭代和直接求解计算。

The matrices arising from electromagnetic integral equations are rank deficient and can be approximated by sparse matrices since Green's function decays rapid as the distance increases. This virtue is denoted by "data sparse characteristic" of integral equations in this proposal. No systematic study has been carried out on it although various fast methods have explicitly or implicitly utilized it. This proposal aims to fill this gap through a comprehensively study on it. In particular, we will investigate the data sparse characteristics of the often used surface integral equations for homogeneous dielectric targets and of the volume ones for targets with inhomogeneous dielectric. The proposal employs the interpolative decomposition (ID) and the concept of "skeleton" to generate the data sparse presentation. The performance of the ID in constructing skeletons is studied in detail with respect to different integral equations. The impacts of multiscale meshes and different permitivity and permeability parameters of the dielectrics on the performance of ID are studied as well. The accuracy control mechanism of the ID is figured out and the suitable manner for skeleton construction is developed. The fast direct factorization algorithms, as well as the efficient preconditioning techniques, based on the ID and the skeleton concept are developed, aiming to solve composite and multiscale problems iteratively or directly, where the different surface and volume integral equations may be involved in.