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 $A$. However, in practice, GMRES may not converge due to numerical instability. In order to improve the convergence, we propose using the pseudo-inverse for the solution of the severely ill-conditioned Hessenberg systems in GMRES. Numerical experiments on semi-definite inconsistent systems indicate that the method is efficient and robust. Finally, we further improve the convergence of the method, by reorthogonalizing the Modified Gram-Schmidt procedure.
翻译:考虑用基于边缘的有限元素方法解决大量分散的对称单线性系统 $A_bf x} = $bf b} 美元,例如,用定期边界条件对流扩散方程式的离散,以及使用基于边缘的有限元素方法对电磁场进行部分差异方程式。理论上,如果系数矩阵 $A$是范围对称,即$rm R}(A) ⁇ {rm R}(A ⁇ rm T}}} $,其中$rm R}(A) 是美元的范围空间。然而,在实践中,由于数字不稳定,GMRES可能不会趋同。为了改进趋同性,我们提议使用伪反向方法来解决GMRES中条件严重恶劣的赫森堡系统。在半确定不一致的系统上进行Numerimical实验表明,该方法既有效又稳健。最后,我们通过重新将程序改进方法的趋同性。