论文标题
对身份矩阵的低级别和小型扰动的GMRE分析
Analysis of GMRES for Low-Rank and Small-Norm Perturbations of the Identity Matrix
论文作者
论文摘要
在许多应用程序中，线性系统出现在系数矩阵中采用特殊表单$ {\ bf i} + {\ bf k} + {\ bf e} $，其中$ {\ bf i} $是dimension $ n $，$ n $，$ n $ {\ rm等级的标识矩阵（ $ \ | {\ bf e} \ | \leqε<1 $。线性系统的gmres收敛速率具有$ {\ bf i} + {\ bf k} $和$ {\ bf i} + {\ bf e} $的表单矩阵的矩阵，但仅由众所周知的理论保证，但仅由相对较弱的融合限制，但仅针对形式的$ {\ bf i} + bf i} + {\ bf i} E} $当前存在。在本文中，我们通过考虑$ {\ bf i} + {\ bf k} $的伪谱来探索线性系统的收敛属性。当大约求解线性系统$（{\ bf i} + {\ bf k} + {\ bf e}）{\ bf x} = {\ bf x} = {\ bf b} $时扰动。特别是，虽然远离原点的聚类频谱通常是快速转基因收敛的良好指标，但当某些特征值不足时，收敛可能会很慢。我们表明，最多可以有$ 2p $ eigenvalues的$ {\ bf i} + {\ bf k} $对小扰动敏感。当使用gmres求解$（{\ bf i} + {\ bf k} _j + {\ bf e} _j）{\ bf x} _j = {\ bf x} _j = {\ bf b} _j $ and and and and nin nin nin nin nar n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n an nin nlin，我们会提出数值结果。
In many applications, linear systems arise where the coefficient matrix takes the special form ${\bf I} + {\bf K} + {\bf E}$, where ${\bf I}$ is the identity matrix of dimension $n$, ${\rm rank}({\bf K}) = p \ll n$, and $\|{\bf E}\| \leq ε< 1$. GMRES convergence rates for linear systems with coefficient matrices of the forms ${\bf I} + {\bf K}$ and ${\bf I} + {\bf E}$ are guaranteed by well-known theory, but only relatively weak convergence bounds specific to matrices of the form ${\bf I} + {\bf K} + {\bf E}$ currently exist. In this paper, we explore the convergence properties of linear systems with such coefficient matrices by considering the pseudospectrum of ${\bf I} + {\bf K}$. We derive a bound for the GMRES residual in terms of $ε$ when approximately solving the linear system $({\bf I} + {\bf K} + {\bf E}){\bf x} = {\bf b}$ and identify the eigenvalues of ${\bf I} + {\bf K}$ that are sensitive to perturbation. In particular, while a clustered spectrum away from the origin is often a good indicator of fast GMRES convergence, that convergence may be slow when some of those eigenvalues are ill-conditioned. We show there can be at most $2p$ eigenvalues of ${\bf I} + {\bf K}$ that are sensitive to small perturbations. We present numerical results when using GMRES to solve a sequence of linear systems of the form $({\bf I} + {\bf K}_j + {\bf E}_j){\bf x}_j = {\bf b}_j$ that arise from the application of Broyden's method to solve a nonlinear partial differential equation.
