Combining Fast Multipole Techniques and an Approximate Inverse Preconditioner for Large Electromagnetism Calculations
Combining Fast Multipole Techniques and an Approximate Inverse Preconditioner for Large Electromagnetism Calculations
复制标题
DOI：
10.1137/040603917
复制
发表时间：
2005-10
期刊：
影响因子：
--
通讯作者：
B. Carpentieri;I. Duff;L. Giraud;G. Sylvand
中科院分区：
文献类型：
--
作者：
B. Carpentieri;I. Duff;L. Giraud;G. Sylvand
边界元法已成为电磁学中求解麦克斯韦方程组的一种常用工具。从线性代数的角度来看，这导致了大型密集复杂线性系统的解决方案，其中未知数与被照明物体表面上定义的网格边缘相关联。在本文中，我们通过预处理Krylov求解器解决这些线性系统的迭代解。我们的主要重点是设计一个有效的并行预条件。在这方面，我们考虑一个近似的逆方法的基础上Frobenius范数最小化。该预处理器是从稀疏近似的密集系数矩阵，和模式的预处理器和系数矩阵计算先验使用几何信息从网格。我们描述了这样的预处理可以自然地实现在一个并行代码，实现了多极技术的矩阵向量积计算。我们调查的数值可扩展性，我们的预处理现实的工业测试问题，并表明，它表现出一些限制非常大的问题的大小接近一百万个未知数。为了提高这些大问题的鲁棒性，我们提出了一个嵌入式迭代方案，结合嵌套GMRES求解器与不同的快速多极子计算。我们通过大量的数值实验表明，这种新的计划是非常强大的负担得起的内存和CPU成本非常大的和具有挑战性的问题的解决方案。
The boundary element method has become a popular tool for the solution of Maxwell's equations in electromagnetism. From a linear algebra point of view, this leads to the solution of large dense complex linear systems, where the unknowns are associated with the edges of the mesh defined on the surface of the illuminated object. In this paper, we address the iterative solution of these linear systems via preconditioned Krylov solvers. Our primary focus is on the design of an efficient parallelizable preconditioner. In that respect, we consider an approximate inverse method based on the Frobenius-norm minimization. The preconditioner is constructed from a sparse approximation of the dense coefficient matrix, and the patterns both for the preconditioner and for the coefficient matrix are computed a priori using geometric information from the mesh. We describe how such a preconditioner can be naturally implemented in a parallel code that implements the multipole technique for the matrix-vector product calculation. We investigate the numerical scalability of our preconditioner on realistic industrial test problems and show that it exhibits some limitations on very large problems of size close to one million unknowns. To improve its robustness on those large problems we propose an embedded iterative scheme that combines nested GMRES solvers with different fast multipole computations. We show through extensive numerical experiments that this new scheme is extremely robust at affordable memory and CPU costs for the solution of very large and challenging problems.