MINRES and MINERR Are Better than SYMMLQ in Eigenpair Computations

Dul, Franciszek

doi:10.1137/s106482759528226x

Cited by 10 publications

(8 citation statements)

References 20 publications

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“…Thus, for systems with symmetric and definite matrix, the Conjugate Gradient (CG) is an efficient method and is often a very good choice. For cases where the matrix is symmetric but indefinite, methods such as the Symmetric LQ (SYMMLQ) or Minimum Residual (MINRES) (Paige and Saunders 1975) are both suitable choices, although for some applications, especially when the matrix is close to singularity, as often happens for the inverse iteration, the MINRES method has been considered to be slightly better (Dul 1998). Nonetheless, from our experience, we have also determined that SYMMLQ works sufficiently well even for these problems.…”

Section: Modern Iterative Linear Solversmentioning

confidence: 93%

Computation of linear and nonlinear stationary states of photonic structures using modern iterative solvers

2007

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In this paper we describe efficient methods to obtain the stationary states of linear and nonlinear photonic systems, which have gained particular interest in the field of integrated and nonlinear optics. While the methods presented are directly applicable to optical physics, they are also general and should be of interest in a broad range of phenomena presently under study in other areas of physics and engineering. The strategy consists in combining the use of classical methods, such as inverse iteration or the Newton method, together with modern, nonstationary linear solvers, such as SYMMLQ or GMRES, in order to obtain efficient numerical computations to problems involving large matrices. We have selected several example problems in order to discuss the practical implementation details, not normally described in the present literature. Moreover, the problems we have selected provide a backdrop to contrast and motivate the use of different methods for systems which are symmetric and non-symmetric, single and multi-component, and also real and complex. Information relative to numerical performance of the different algorithms, including a survey for a nonsymmetric problem, which requires the adjustment of a restarting parameter for the GMRES algorithm, is also presented.

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Section: Modern Iterative Linear Solversmentioning

confidence: 93%

Computation of linear and nonlinear stationary states of photonic structures using modern iterative solvers

2007

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“…In one of our sample problems with appropriate tuned preconditioner, there is little difference between the two methods, but for ill-conditioned problems without a preconditioner, as shown in [3], SYMMLQ might not even be able to improve the eigenvalue residual in a reasonable number of iterations.…”

Section: Comparison Of Symmlq and Minres Used In Rqimentioning

confidence: 94%

“…With extensive numerical tests, Dul in [3] claimed that MINRES improves eigenvector approximation to some prescribed level in considerably fewer iterations than SYMMLQ. Rigorous analysis and comparison of the two methods is not seen in the literature.…”

Section: Comparison Of Symmlq and Minres Used In Rqimentioning

confidence: 99%

“…Reference [3] also shows that the curve of eigenvalue residuals of MINRES iterates is generally smooth, whereas that of SYMMLQ iterates tends to be oscillatory. This phenomenon can be explained qualitatively by the fact the interior eigenvalues are susceptible to being impersonated by non-converged Ritz values.…”

Section: Comparison Of Symmlq and Minres Used In Rqimentioning

confidence: 99%

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Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration

Xue¹,

Elman²

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. We present a detailed convergence analysis of preconditioned MINRES for approximately solving the linear systems that arise when Rayleigh Quotient Iteration is used to compute the lowest eigenpair of a symmetric positive definite matrix. We provide insight into the "slow start" of MINRES iteration in both a qualitative and quantitative way, and show that the convergence of MINRES mainly depends on how quickly the unique negative eigenvalue of the preconditioned shifted coefficient matrix is approximated by its corresponding harmonic Ritz value. By exploring when the negative Ritz value appears in MINRES iteration, we obtain a better understanding of the limitation of preconditioned MINRES in this context and the virtue of a new type of preconditioner with "tuning". Comparison of MINRES with SYMMLQ in this context is also given. Finally we show that tuning based on a rank-2 modification can be applied with little additional cost to guarantee positive definiteness of the tuned preconditioner.Key words. Rayleigh Quotient Iteration, harmonic Ritz value, MINRES, tuned preconditioner 1. Introduction. There has been considerable interest in recent years in developing and analyzing eigensolvers with inner-outer structure for computing eigenvalues of matrices closest to some specified value. These algorithms usually involve at each step (outer iteration) a shift-invert matrix-vector product implemented by solving the shifted linear system iteratively (inner iteration). The use of inner iteration becomes mandatory if the matrices are too large for factorization-based exact shift-invert matrix-vector products to be practical. Inexact inverse iteration is the most simple algorithm of this type and the best understood one. Early papers on the convergence of inexact inverse iteration with fixed shift include [8] and [9], where the main concern is to choose a decreasing sequence of stopping tolerances for inner solvers to maintain linear convergence of the outer iteration. Analysis of inexact Rayleigh Quotient Iteration (RQI) in [13] and [18] shows how the inexactness of the inner solve can affect the convergence of the outer iteration. More recent work focuses on improving the convergence of inner iterations as well as the relation between the inner and outer iterations. Reference [16] introduces some new perspectives on preconditioning in this setting, namely, that faster convergence of inner iterations can be obtained by modifying the right hand side of the preconditioned linear system. Refined analysis of this approach in [1], [2] and [5] shows how different formulations of the linear system, with variable shift and different inner stopping criteria, can affect the convergence of the inner and outer iterations. An alternative preconditioning approach called "tuning" is analyzed in [6] for non-symmetric eigenvalue problems and in [7] for symmetric problems. A preconditioner with tuning is a low rank modification of an ordinary preconditioner. Tuning forces the preconditioning operator to behave in the same way as...

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“…Our numerical results show that the technique can be used to handle both simple and multiple eigenvalues. Note that linear systems with multiple right-hand sides that appear in the corrector step of Algorithm 4.2 below can be effectively solved using the conjugate gradient method in Reference [31] or the MINRES in Reference [32]. The two-grid finite element discretization scheme for the SE problem (47) with boundary condition (48) is described as follows.…”

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confidence: 99%

A time‐independent approach for computing wave functions of the Schrödinger–Poisson system

Chien

Jeng

Zou

2007

Numerical Linear Algebra App

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SUMMARYWe describe a two-grid finite element discretization scheme for computing wave functions of the Schrödinger-Poisson (SP) system. To begin with, we compute the first k eigenpairs of the Schrödinger-Poisson eigenvalue (ESP) problem on the coarse grid using a continuation algorithm, where the nonlinear Poisson equation is solved iteratively. We use the k eigenpairs obtained on the coarse grid as initial guesses for computing their counterparts of the ESP on the fine grid. The wave functions of the SP system can be easily obtained using the formula of separation of variables. The proposed algorithm has the following advantages. (i) The initial approximate eigenpairs used in the fine grid can be obtained with low computational cost. (ii) It is unnecessary to discretize the partial derivative of the wave function with respect to the time variable in the SP system. (iii) The major computational difficulties such as closely clustered eigenvalues that occur in the SP system can be effectively computed. Numerical results on the ESP and the SP system are reported. In particular, the rate of convergence of the proposed algorithm is O(h 4 ).

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MINRES and MINERR Are Better than SYMMLQ in Eigenpair Computations

Cited by 10 publications

References 20 publications

Computation of linear and nonlinear stationary states of photonic structures using modern iterative solvers

Computation of linear and nonlinear stationary states of photonic structures using modern iterative solvers

Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration

A time‐independent approach for computing wave functions of the Schrödinger–Poisson system

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