The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields

Abstract

Context. Since the 1990's, Yan and colleagues have formulated a kind of boundary integral formulation for the linear or non-linear solar force-free magnetic fields with finite energy in semi-infinite space, and developed a computational procedure by virtue of the boundary element method (BEM) to extrapolate the magnetic fields above the photosphere. Aims. In this paper, the generalized minimal residual method (GMRES) is introduced into the BEM extrapolation of the solar forcefree magnetic fields, in order to efficiently solve the associated BEM system of linear equations, which was previously solved by the Gauss elimination method with full pivoting. Methods. Being a modern iterative method for non-symmetric linear systems, the GMRES method reduces the computational cost for the BEM system from O(N 3 ) to O(N 2 ), where N is the number of unknowns in the linear system. Intensive numerical experiments are conducted on a well-known analytical model of the force-free magnetic field to reveal the convergence behaviour of the GMRES method subjected to the BEM systems. The impacts of the relevant parameters on the convergence speed are investigated in detail. Results. Taking the Krylov dimension to be 50 and the relative residual bound to be 10 −6 (or 10 −2 ), the GMRES method is at least 1000 (or 9000) times faster than the full pivoting Gauss elimination method used in the original BEM extrapolation code, when N is greater than 12 321, according to the CPU timing information measured on a common desktop computer (CPU speed 2.8 GHz; RAM 1 GB) for the model problem.

The simpler GMRES method combined with finite volume method for simulating viscoelastic flows on triangular grid

“…We set Krylov dimension m to be 3 (discussed below). The setting of e ¼ 10 À6 is typical for iterative solvers [24,[37][38][39]. From Table 2, we find the number of iterations of the SGMRES(m) is smaller than that of the GMRES(m), thus leading to the assembling time of the SGMRES(m) shorter than that of the GMRES(m).…”

“…Unfortunately, there is no theoretical guide on determining a proper or optimal m value for the GMRES-like solvers [43]. Generally, the typical settings of m = 20 or 30 are often used for the restarted GMRES-like solvers in practical applications [39,44,45]. However, it seems desirable to determine a nearly optimal setting for the Krylov dimension m by virtue of trials, so that the concerned iterative solvers may achieve the best performance under the specific circumstance [43].…”

An Overview of Recent Developments and Applications of the GMRES Method

“…GMRES 算法还可用于求解诸如最优控制、滤波估计、去耦、降阶等控制理论中的微分 Riccati 方程 [27] 。在 大特征值问题和边值问题中会出现多元线性系统， 线性控制、 滤波理论、 图像修复等方面包含了著名的 Lyapunov 矩阵方程、Sylvester 矩阵方程和 Stein 矩阵方程，这些方程同样是典型的多元线性系统问题，全局 GMRES 算法 正好为这些问题的解决提供了一个很好的工具，不同的数值实验更显示出该方法收敛行为方面的优势 [28,29] 。 GMRES 算法还用于求解 Toeplitz 方程、Helmholtz 方程和 Navier-Stokes 方程等，预处理 GMRES 并行算法也得 到了很好的应用 [30][31][32][33] 。在太阳物理的研究中，我国科学家颜毅华于 1995 年首次推导出太阳常 alpha 无力场的边 界积分表示， 并用边界元方法实现了数值求解 [34] ； Li 等人 2007 年对颜毅华的算法进行了改进， 他们引入 GMRES 算法来解决边界元方程组；由此，对 10,000 阶以上的矩阵，用 GMRES 算法使得计算效率提高了 1000~9000 倍 [35] 。 [20] 实型 Laplace 变换的线性方程组 光谱延迟修正技术 [21] 微分代数方程的初始值问题 控制、光辐射和流体力学 [23][24][25][26] 近海水域控制方程、光学辐射传输方程、计算流体力学 Euler 方程 控制理论 [27] 微分 Riccati 方程 大型奇异值问题 [28] 广义希尔维斯特矩阵方程 太阳物理研究 …”