The development and analysis of sweeping preconditioners for scattering problems

这项研究的主要目的是发展有效的迭代方法来计算声学和电磁散射问题的数值近似。在散射问题的任何有限元近似中，一个基本问题是远场边界条件的有效实现。在这项研究中，这一条件将使用具有完全匹配层(PML)远场近似的有限元或有限元边界积分公式来近似。这项研究将为由上述逼近技术产生的代数系统的解开发创新的新的预条件技术。预条件矩阵的构造采用H-矩阵逼近和扫描算法相结合的方法。电磁散射问题在新技术的发展中起着重要的作用，在天线和微波工程、遥感、隐身飞行器设计、电磁和等离子隐身、生物医学工程、电磁和地震地下成像等方面都有应用，在定位和监测油气藏方面尤为重要。能够有效地解决声学和电磁散射问题的模拟涉及复杂的几何结构，通常具有细微尺度的不均匀。这项研究的目标是显著降低与解决上述问题相关的计算成本，从而使涉及电磁散射的系统的计算机设计技术更加现实和健壮。

The main objective of this research is the development of efficient iterative methods for computing numerical approximations to the solutions of acoustic and electromagnetic scattering problems. A fundamental issue in any finite element approximation to a scattering problem is the efficient implementation of the far field boundary condition. In this research, this condition will be approximated using either finite elements with a ``Perfectly Matched Layer'' (PML) far field approximation or finite element boundary integral formulations. The research will develop innovative new preconditioning techniques for the solution of the algebraic systems resulting from the above-mentioned approximation techniques. The preconditioners are constructed using H-matrix approximation combined with a sweeping algorithm. Electromagnetic scattering problems play an important role in the development of new technologies with applications in, for example, antenna and microwave engineering, remote sensing, stealth vehicle design, electromagnetic and plasmonic cloaking, biomedical engineering, and electromagnetic and seismic subsurface imaging, and are especially indispensable in locating and monitoring gas and oil reservoirs. Simulations capable of efficiently solving acoustic and electromagnetic scattering problems involve complicated geometry, often, with fine scale inhomegenities. It is the goal of this research to significantly reduce the computational costs associated with solving above problems, thus enabling more realistic and robust computer design technology for systems involving electromagnetic scattering.