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- W4307206561 abstract "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 $|{bf E}| leq epsilon < 1$. 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 $epsilon$ 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." @default.
- W4307206561 created "2022-10-30" @default.
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- W4307206561 date "2022-10-21" @default.
- W4307206561 modified "2023-09-27" @default.
- W4307206561 title "Analysis of GMRES for Low-Rank and Small-Norm Perturbations of the Identity Matrix" @default.
- W4307206561 doi "https://doi.org/10.48550/arxiv.2210.12053" @default.
- W4307206561 hasPublicationYear "2022" @default.
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