A New Shifted Block GMRES Method with Inexact Breakdowns for Solving Multi-Shifted and Multiple Right-Hand Sides Linear Systems

Sun, Donglin; Huang, Ting‐Zhu; Carpentieri, Bruno; Jing, Yan-Fei

doi:10.1007/s10915-018-0787-6

Cited by 9 publications

(2 citation statements)

References 37 publications

(75 reference statements)

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“…( 5) with N eq > 1. The shifted Krylov algorithms and related techniques are briefly explained in this paper and the details can be found in [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein. Currently, shifted Krylov algorithms are used in many computational science fields such as quantum chromodynamics [3], electronic structure calculations [4,8,18,19], excited electron calculations [20], nuclear physics [21,22], transport calculations with non-equilibrium Green's function theory [23], and nano-structured superconducting systems [24].…”

Section: Introductionmentioning

confidence: 99%

K$ω$ -- Open-source library for the shifted Krylov subspace method of the form $(zI-H)x=b$

Hoshi,

Kawamura,

Yoshimi

et al. 2020

Preprint

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We develop Kω, an open-source linear algebra library for the shifted Krylov subspace methods. The methods solve a set of shifted linear equations for a given matrix and a vector, simultaneously. The leading order of the operational cost is the same as that for a single equation. The shift invariance of the Krylov subspace is the mathematical foundation of the shifted Krylov subspace methods. Applications in materials science are presented to demonstrate the advantages of the algorithm over the standard Krylov subspace methods such as the Lanczos method. We introduce benchmark calculations of (i) an excited (optical) spectrum and (ii) intermediate eigenvalues by the contour integral on the complex plane. In combination with the quantum lattice solver HΦ, Kω can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models.

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Section: Introductionmentioning

confidence: 99%

K$ω$ -- Open-source library for the shifted Krylov subspace method of the form $(zI-H)x=b$

Hoshi,

Kawamura,

Yoshimi

et al. 2020

Preprint

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“…For the sake of balancing robustness and convergence rate, Robbé and Sadkane proposed an inexact breakdown detection for the block GMRES algorithm (denoted by IB-BGMRES) [20], which could keep and reintroduce directions associated with the almost converged parts in next iteration if necessary. We refer to [1,2,20], for relevant works on inexact breakdown detection, as well as to [23][24][25][26]28], for related variants of block Krylov subspace methods for solving linear systems with multiple right-hand sides.…”

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confidence: 99%

A Block Minimum Residual Norm Subspace Solver with Partial Convergence Management for Sequences of Linear Systems

Giraud¹,

Jing²,

Xiang³

2022

SIAM J. Matrix Anal. Appl.

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We are concerned with the iterative solution of linear systems with multiple right-hand sides available one group after another with possibly slowly-varying left-hand sides. For such sequences of linear systems, we first develop a new block minimum norm residual approach that combines two main ingredients. The first component exploits ideas from GCRO-DR [SIAM J. Sci. Comput., 28(5) (2006), pp. 1651-1674, enabling to recycle information from one solve to the next. The second component is the numerical mechanism to manage the partial convergence of the right-hand sides, referred to as inexact breakdown detection in IB-BGMRES [Linear Algebra Appl., 419 (2006), pp. 265-285], that enables the monitoring of the rank deficiency in the residual space basis expanded block-wise.Secondly, for the class of block minimum norm residual approaches, that relies on a block Arnoldi-like equality between the search space and the residual space (e.g., any block GMRES or block GCRO variants), we introduce new search space expansion policies defined on novel criteria to detect the partial convergence. These novel detection criteria are tuned to the selected stopping criterion and targeted convergence threshold to best cope with the selected normwise backward error stopping criterion, enabling to monitor the computational effort while ensuring the final accuracy of each individual solution. Numerical experiments are reported to illustrate the numerical and computational features of both the new block Krylov solvers and the new search space block expansion polices.

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A New Projected Variant of the Deflated Block Conjugate Gradient Method

2019

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A New Shifted Block GMRES Method with Inexact Breakdowns for Solving Multi-Shifted and Multiple Right-Hand Sides Linear Systems

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Cited by 9 publications

References 37 publications

K$ω$ -- Open-source library for the shifted Krylov subspace method of the form $(zI-H)x=b$

K$ω$ -- Open-source library for the shifted Krylov subspace method of the form $(zI-H)x=b$

A Block Minimum Residual Norm Subspace Solver with Partial Convergence Management for Sequences of Linear Systems

A New Projected Variant of the Deflated Block Conjugate Gradient Method

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