相关论文: Admissible and attainable convergence behavior wit…

It is well-established that any non-increasing convergence curve is possible for GMRES and a family of pairs $(A,b)$ can be constructed for which GMRES exhibits a given convergence curve with $A$ having arbitrary spectrum. No analog of this…

The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right hand side is generally…

A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple right-hand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the…

The objective of this paper is to understand the superlinear convergence behavior of the GMRES method when the coefficient matrix has clustered eigenvalues. In order to understand the phenomenon, we analyze the convergence using the…

Given a matrix order $n$, a restart parameter $m$ ($m < n$), a decreasing positive sequence $f(0) > f(1) > ... > f(q) \geq 0$, where $q < n/m$, it is shown that there exits an $n$-by-$n$ matrix $A$ and a vector $r_0$ with $\|r_0\|=f(0)$…

A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple right-hand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the…

We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted $N\times N$ matrix whose entries are independent complex Gaussians. When the right hand side of this linear system is independent of this…

Work on generalizing the deflated, restarted GMRES algorithm, useful in lattice studies using stochastic noise methods, is reported. We first show how the multi-mass extension of deflated GMRES can be implemented. We then give a deflated…

Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or…

The convergence of the GMRES linear solver is notoriously hard to predict. A particularly enlightening result by [Greenbaum, Pt\'ak, Strako\v{s}, 1996] is that, given any convergence curve, one can build a linear system for which GMRES…

We introduce an iterative method named GPMR for solving 2x2 block unsymmetric linear systems. GPMR is based on a new process that reduces simultaneously two rectangular matrices to upper Hessenberg form and that is closely related to the…

We prove that the cycle-convergence of the restarted GMRES applied to a system of linear equations with a normal coefficient matrix is sublinear.

A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases…

Spectral clustering is one of the most popular unsupervised machine learning methods. Constructing similarity matrix is crucial to this type of method. In most existing works, the similarity matrix is computed once for all or is updated…

In this paper, we develop a new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides.The proposed method builds on a recently…

The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…

We derive a sharp upper bound on the residuals produced by the flexible GMRES (FGMRES) method. The bound shows that FGMRES exhibits two phases of convergence depending on the residual tolerance of the inner preconditioner. For small…

We show that any admissible harmonic Ritz value set is possible for prescribed GMRES residual norms, which is a complement for the results in [Duintjer Tebbens and Meurant, {\it SIAM J. Matrix Anal. Appl.}, 33 (2012), no. 3, pp. 958--978].

Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact in the non-Hermitian case introduces restrictions; e.g., initial…

Modern experimental methods enable the creation of self-assembly building blocks with tunable interactions, but optimally exploiting this tunability for the self-assembly of desired structures remains an important challenge. Many studies of…