论文标题
使用伪为对称奇异系统的pseudoinverse
GMRES using pseudoinverse for range symmetric singular systems
论文作者
论文摘要
考虑求解大型稀疏范围对称奇异线性系统$ a {\ bf x} = {\ bf b} $，例如，使用基于边缘的有限元元素的对流扩散方程式，具有周期性边界条件的对流扩散方程和部分电磁场的部分微分方程。 In theory, the Generalized Minimal Residual (GMRES) method converges to the least squares solution for inconsistent systems if the coefficient matrix $A$ is range symmetric, i.e. $ {\rm R}(A)= {\rm R}(A^{ \rm T } )$, where $ {\rm R}(A)$ is the range space of $A$. 我们得出了GMRE的必要条件，以确定不一致且一致范围对称系统的最小二乘解决方案，除了计算Hessenberg矩阵元素外，假设精确算术。 在实践中，由于数值不稳定，转基因可能不会收敛。为了改善收敛性，我们建议使用伪源来解决GMRES中严重条件的Hessenberg系统的解决方案。对不一致系统的数值实验表明该方法是有效且稳健的。最后，我们通过重新定位修改的革兰氏阴性程序来进一步提高该方法的收敛性。
Consider solving large sparse range symmetric singular linear systems $ A {\bf x}= {\bf b} $ which arise, for instance, in the discretization of convection diffusion equations with periodic boundary conditions, and partial differential equations for electromagnetic fields using the edge-based finite element method. In theory, the Generalized Minimal Residual (GMRES) method converges to the least squares solution for inconsistent systems if the coefficient matrix $A$ is range symmetric, i.e. $ {\rm R}(A)= {\rm R}(A^{ \rm T } )$, where $ {\rm R}(A)$ is the range space of $A$. We derived the necessary and sufficient conditions for GMRES to determine a least squares solution of inconsistent and consistent range symmetric systems assuming exact arithmetic except for the computation of the elements of the Hessenberg matrix. In practice, GMRES may not converge due to numerical instability. In order to improve the convergence, we propose using the pseudoinverse for the solution of the severely ill-conditioned Hessenberg systems in GMRES. Numerical experiments on inconsistent systems indicate that the method is effective and robust. Finally, we further improve the convergence of the method by reorthogonalizing the Modified Gram-Schmidt procedure.
