Related papers: GMRES convergence bounds for eigenvalue problems
The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…
The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of $G=\operatorname{SL}(n)$ .…
In this paper, we analyze the convergence of the preconditioned GMRES method for the first order finite element discretizations of the Helmholtz equation in media with losses. We consider a Laplace preconditioner, an inexact Laplace…
In a multidimensional infinite layer bounded by two hyperplanes, the inhomogeneous Helmholtz equation with a polynomial right-hand side is considered. It is shown that the Dirichlet and Dirichlet-Neumann boundary-value problems with…
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…
As the need for computational power and efficiency rises, parallel systems become increasingly popular among various scientific fields. While multiple core-based architectures have been the center of attention for many years, the rapid…
An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on…
The standard approach for finding eigenvalues and eigenvectors of matrix polynomials starts by embedding the coefficients of the polynomial into a matrix pencil, known as linearization. Building on the pioneering work of Nakatsukasa and…
We introduce a unified framework for computing approximately-optimal preconditioners for solving linear and non-linear systems of equations. We demonstrate that the condition number minimization problem, under structured transformations…
Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree…
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero…
This paper introduces a preconditioned method designed to comprehensively address the saddle point system with the aim of improving convergence efficiency. In the preprocessor construction phase, a technical approach for solving the…
We introduce the Generalized Preconditioned Locally Harmonic Residual (GPLHR) method for solving standard and generalized non-Hermitian eigenproblems. The method is particularly useful for computing a subset of eigenvalues, and their eigen-…
We propose an eigensolver and the corresponding package, GCGE, for solving large scale eigenvalue problems. This method is the combination of damping idea, subspace projection method and inverse power method with dynamic shifts. To reduce…
Circulant preconditioners are commonly used to accelerate the rate of convergence of iterative methods when solving linear systems of equations with a Toeplitz matrix. Block extensions that can be applied when the system has a block…
In this contribution, we study the numerical behavior of the Generalized Minimal Residual (GMRES) method for solving singular linear systems. It is known that GMRES determines a least squares solution without breakdown if the coefficient…
Direct and inverse initial-boundary problems on a bounded interval for systems of quasilinear evolution equations with general nonlinearities are considered. In the case of inverse problems conditions of integral overdetermination are…
Combined perturbation bounds are presented for eigenvalues and eigenspaces of Hermitian matrices or singular values and singular subspaces of general matrices. The bounds are derived based on the smooth decompositions and elementary…
An apriori bound for the condition number associated to each of the following problems is given: general linear equation solving, minimum squares, non-symmetric eigenvalue problems, solving univariate polynomials, solving systems of…
In this paper, a full (nested) multigrid scheme is proposed to solve eigenvalue problems. The idea here is to use the multilevel correction method to transform the solution of eigenvalue problem to a series of solutions of the corresponding…