The development and analysis of sweeping preconditioners for scattering problems

本研究的主要目标是发展有效的迭代方法，用于计算声和电磁散射问题的数值近似解。散射问题的任何有限元近似的一个基本问题是有效地实现远场边界条件。在这项研究中，这种情况将近似使用有限元与“完美匹配层”（PML）远场近似或有限元边界积分公式。这项研究将开发创新的新的预处理技术的解决方案的代数系统所产生的上述近似技术。 预条件子的构造使用H-矩阵近似结合扫描算法。 电磁散射问题在新技术的开发中起着重要作用，其应用例如天线和微波工程、遥感、隐形车辆设计、电磁和等离子体隐身、生物医学工程以及电磁和地震地下成像，并且在定位和监测油气藏中尤其不可或缺。 能够有效地解决声和电磁散射问题的模拟涉及复杂的几何形状，往往与精细尺度inhomogenities。本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真实和强大。

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.