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A Matrix-free Two-grid Preconditioner for Solving Boundary Integral Equations in Electromagnetism

机译：求解电磁边界积分方程的无矩阵两网格预处理器

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摘要

In this paper, we describe a matrix-free iterative algorithm based on the GMRES method for solving electromagnetic scattering problems expressed in an integral formulation. Integral methods are an interesting alternative to differential equation solvers for this problem class since they do not require absorbing boundary conditions and they mesh only the surface of the radiating object giving rise to dense and smaller linear systems of equations. However, in realistic applications the discretized systems can be very large and for some integral formulations, like the popular Electric Field Integral Equation, they become ill-conditioned when the frequency increases. This means that iterative Krylov solvers have to be combined with fast methods for the matrix-vector products and robust preconditioning to be affordable in terms of CPU time. In this work we describe a matrix-free two-grid preconditioner for the GMRES solver combined with the Fast Multipole Method. The preconditioner is an algebraic two-grid cycle built on top of a sparse approximate inverse that is used as smoother, while the grid transfer operators are defined using spectral information of the preconditioned matrix. Experiments on a set of linear systems arising from real radar cross section calculation in industry illustrate the potential of the proposed approach for solving large-scale problems in electromagnetism.
机译：在本文中，我们描述了一种基于GMRES方法的无矩阵迭代算法，用于解决以积分公式表示的电磁散射问题。对于这类问题，积分方法是微分方程求解器的一种有趣替代方法，因为它们不需要吸收边界条件，并且它们仅与辐射对象的表面啮合，从而产生了密集且较小的线性方程组。但是，在实际应用中，离散系统可能会很大，对于某些积分公式，例如流行的电场积分方程，当频率增加时，它们会变得不适。这意味着必须将迭代Krylov求解器与矩阵向量乘积的快速方法和强大的预处理结合起来，以使CPU时间负担得起。在这项工作中，我们描述了结合快速多极方法的GMRES求解器的无矩阵两网格预处理器。预处理器是在稀疏近似逆的基础上构建的代数两网格循环，它用作平滑器，而网格转移算子是使用预处理矩阵的频谱信息定义的。工业上由实际雷达横截面计算产生的一组线性系统的实验说明了该方法解决电磁学中大规模问题的潜力。

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