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A gmres-Power algorithm for computing PageRank problems

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2018年 343卷 113-123页

作者： Gu, Chuanqing Jiang, Xianglong Shao, Chenchen Chen, Zhibing Shanghai Univ Dept Math Shanghai 200444 Peoples R China Shenzhen Univ Sch Math & Stat Shenzhen 518052 Peoples R China

The PageRank algorithm for determining the importance of Web pages has become a central technique in Web search. we propose a new method to speed up the convergence performance for computing PageRank when the damping factor is close to one, called as gmres-Power, which is based on a periodic combination of the power method with the gmres algorithm. The description and convergence analysis of the new algorithm are discussed in detail. Numerical results are reported to confirm the efficiency of the new algorithm. (C) 2018 Elsevier B.V. All rights reserved.

关键词： pageRank gmres algorithm Power method gmres-Power

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A NOTE ON INEXACT INNER PRODUCTS IN gmres

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2022年第3期43卷 1406-1422页

作者： Gratton, Serge Simon, Ehouarn Titley-peloquin, David Toint, Philippe L. INPT IRIT ENSEEIHT BP 7122 F-31071 Toulouse 7 France McGill Univ Dept Bioresource Engn Sainte Anne De Bellevue PQ H9X 3V9 Canada Univ Namur Namur Ctr Complex Syst naXys B-5000 Namur Belgium

We show to what extent the accuracy of the inner products computed in the gmres iterative solver can be reduced as the iterations proceed without affecting the convergence rate or final accuracy achieved by the iterates. We bound the loss of orthogonality in gmres with inexact inner products. We use this result to bound the ratio of the residual norm in inexact gmres to the residual norm in exact gmres and give a condition under which this ratio remains close to 1. We illustrate our results with examples in variable floating-point arithmetic.

关键词： Arnoldi algorithm gmres algorithm inexact inner products variable precision arithmetic

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Implementation of the spectral element discretization of the nonstationary velocity-vorticity-pressure Stokes formulation

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BOUNDARY VALUE PROBLEMS 2024年第1期2024卷 1-19页

作者： Chorfi, Nejmeddine Ouertani, Henda King Saud Univ Coll Sci Dept Math POB 2455 Riyadh Saudi Arabia King Saud Univ Coll Comp & Informat Sci Informat Technol Dept Riyadh 11451 Saudi Arabia

This paper deals with the resolution of the time-dependent velocity-vorticity-pressure formulation of Stokes problem in a two- and three-dimensional domain. The discretization is based on the spectral element method with respect to the space variable and Euler's implicit scheme with respect to the time variable. We implement the asymmetric linear system to solve the full spectral element discrete problem. Detailed numerical experiments lead to confirm the usefulness of the discretization.

关键词： Nonstationary Stokes problem Spectral element method Implicit Euler scheme gmres algorithm

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Jacobian-Free Poincare-Krylov Method to Determine the Stability of Periodic Orbits of Electric Power Systems

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IEEE TRANSACTIONS ON POWER SYSTEMS 2022年第1期37卷 429-442页

作者： Garcia, Norberto Luisa Romero, Maria Acha, Enrique Univ Michoacana Fac Elect Engn Morelia 58000 Michoacan Mexico Tampere Univ Fac Informat Technol & Commun Sci Tampere 33100 Finland

A first application of a time domain Poincare-Krylov approach for the study of the all-important stability of periodic solutions of nonlinear power networks is reported in this work. Whilst a Newton method solves the nonlinear algebraic equations resulting from the Poincare map method, an iterative Krylov-subspace method based on a gmres algorithm is applied for solving the Newton correction equations. More importantly, a QR factorization of the Hessenberg matrix involved in the gmres least square problem is implemented using Givens rotations in order to avoid the linear growth of the computational complexity in large-scale power networks. Further, the stability of periodic solutions is determined by computing the Floquet multipliers using Ritz values and the Hessenberg matrix. Numerical tests carried out on a modified three-phase version of the IEEE 118-node system with a hydro-turbine synchronous generator and a grid-tied power converter demonstrate that speedup factors up to 8 are attainable with the Krylov-Subspace approach with respect to the standard Poincare map method. An outstanding outcome of the comparative study is that the incorporation of QR factorization to the Hessenberg least square problem outperformed the classic periodic steady-state solvers, providing further computational savings up to 50%.

关键词： Power system stability Stability analysis Jacobian matrices Power system dynamics Numerical stability Steady-state Power system harmonics Stability periodic steady-state Poincare map method Newton method Krylov subspace gmres algorithm

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A data completion algorithm using an integral representation of the Steklov-Poincare operator

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2023年 422卷

作者： Abid, Chaima Abda, Amel Ben Fatma, Riadh Ben Boukari, Yosra Univ Tunis El Manar ENIT LAMSIN BP 37 Tunis 1002 Tunisia

In this paper, the Cauchy problem for the Helmholtz equation is investigated. The objective is to recover the missing data on some part of the boundary of a bounded domain from overspecified data on the remaining part of the boundary. We propose a preconditioned Krylov algorithm to solve this ill-posed problem, based on the represen-tation of the solution with surface integral equations and the Steklov-Poincare operator. We give a theoretical and numerical validation of the proposed method conducted in the 3D setting. We show the fast convergence of the proposed algorithm tested on various synthetic examples. The numerical results show a high precision of the reconstruction obtained for different levels of noisy data.(c) 2022 Elsevier B.V. All rights reserved.

关键词： Data completion algorithm Cauchy problem gmres algorithm Boundary integral equations Steklov-Poincar? operator

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VARYING THE S IN YOUR S-STEP gmres

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2017年 47卷 206-230页

作者： Imberti, David Erhel, Jocelyne INRIA Campus Univ Beaulieu F-35042 Rennes France

Krylov subspace methods are commonly used iterative methods for solving large sparse linear systems. However, they suffer from communication bottlenecks on parallel computers. Therefore, s-step methods have been developed, where the Krylov subspace is built block by block so that s matrix-vector multiplications can be done before orthonormalizing the block. Then Communication-Avoiding algorithms can be used for both kernels. This paper introduces a new variation on the s-step gmres method in order to reduce the number of iterations necessary to ensure convergence with a small overhead in the number of communications. Namely, we develop an s-step gmres algorithm, where the block size is variable and increases gradually. Our numerical experiments show a good agreement with our analysis of condition numbers and demonstrate the efficiency of our variable s-step approach.

关键词： Communication-Avoiding s-step Krylov subspace method gmres algorithm variable s-step

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Numerical solution of 2 x 2 block linear systems by block Gram-Schmidt methods

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2017年第8期94卷 1562-1573页

作者： Okulicka, Felicja Smoktunowicz, Alicja Warsaw Univ Technol Fac Math & Informat Sci Koszykowa 75 PL-00662 Warsaw Poland

In this paper we propose and analyse some algorithms for solving block linear systems which are based upon the block Gram-Schmidt method. In particular, we prove that the algorithm BCGS2 (Reorthogonalized Block Classical Gram-Schmidt) using Householder Q-R decomposition implemented in floating-point arithmetic is backward stable, under the mild assumptions. Numerical tests were done in MATLAB to illustrate our theoretical results. A particular emphasis is on symmetric saddle-point problems, which arise in many important practical applications. We compare the results with the generalized minimal residual (gmres) algorithm.

关键词： Block Q-R factorization Householder transformation condition number numerical stability saddle-point problem gmres algorithm 15A12 15A23 15A60 65H10

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Implementation of Some Iterative Matrix Solvers in Simulation of the Wave Propagation Problem using the Time-harmonic Finite Difference algorithm 17

Implementation of Some Iterative Matrix Solvers in Simulatio...

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17th International Conference Computational Problems of Electrical Engineering (CPEE)

作者： Butrylo, Boguslaw Choroszucho, Agnieszka Bialystok Tech Univ Fac Elect Engn Bialystok Poland

ISBN: (纸本)9781509028009

In this paper, the formulation of two-dimensional finite difference frequency domain (FDFD) for heterogeneous and lossy materials is presented. But the main attention is focus on the efficient formulation of the matrix solver. Contemporary, the numerical models are consisted of a large-scale sparse complex matrix, therefore the specific formulation of the solver enable to improve the properties of the proposed algorithm. Two iterative subroutines for the complex equation are presented and validated, i.e. the BiConjugate Gradient Stabilized method (BiCGStab) and the Generalized Minimum Residual method with restarts (gmres(m)). The stability and performance of these algorithms are tested using model based on typical building construction including concrete column.

关键词： finite difference frequency scheme heterogeneous materials iterative large-scale solver gmres algorithm BiCGStab algorithm

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Solution of skin-effect problems by means of the hybrid SDBCI method

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COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING 2014年第6期33卷 1935-1949页

作者： Aiello, Giovanni Alfonzetti, Salvatore Salerno, Nunzio Univ Catania Dipartimento Ingn Elettr Elettron & Informat Catania Italy

Purpose - The purpose of this paper is to present a modified version of the hybrid Finite Element Method-Dirichlet Boundary Condition Iteration method for the solution of open-boundary skin effect problems. Design/methodology/approach - The modification consists of overlapping the truncation and the integration boundaries of the standard method, so that the integral equation becomes singular as in the well-known Finite Element Method-Boundary Element Method (FEM-BEM) method. The new method is called FEM-SDBCI. Assuming an unknown Dirichlet condition on the truncation boundary, the global algebraic system is constituted by the sparse FEM equations and by the dense integral equations, in which singularities arise. Analytical formulas are provided to compute these singular integrals. The global system is solved by means of a Generalized Minimal Residual iterative procedure. Findings - The proposed method leads to slightly less accurate numerical results than FEM-BEM, but the latter requires much more computing time. Practical implications - Then FEM-SDBCI appears more appropriate than FEM-BEM for applications which require a shorter computing time, for example in the stochastic optimization of electromagnetic devices. Originality/value - Note that FEM-SDBCI assumes a Dirichlet condition on the truncation boundary, whereas FEM-BEM assumes a Neumann one.

关键词： Finite element methods Integral equations gmres algorithm Hybrid technique Skin-effect

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Imposing symmetry in augmented linear systems

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2014年第6期21卷 781-795页

作者： Szularz, M. Univ Ulster Sch Comp & Informat Engn Coleraine BT52 1SA Londonderry North Ireland

This paper discusses the methods of imposing symmetry in the augmented system formulation (ASF) for least-squares (LS) problems. A particular emphasis is on upper Hessenberg problems, where the challenge lies in leaving all zero-by-definition elements of the LS matrix unperturbed. Analytical solutions for optimal perturbation matrices are given, including upper Hessenberg matrices. Finally, the upper Hessenberg LS problems represented by unsymmetric ASF that indicate a normwise backward stability of the problem (which is not the case in general) are identified. It is observed that such problems normally arise from Arnoldi factorization (for example, in the generalized minimal residual (gmres) algorithm). The problem is illustrated with a number of practical (arising in the gmres algorithm) and some purpose-built' examples. Copyright (c) 2014 John Wiley & Sons, Ltd.

关键词： gmres algorithm least-squares problem Hessenberg matrix Arnoldi method augmented system formulation generalized Kiebasiski-Schwetlick lemma sensitivity of QR factorization quadratic programming with box constraints

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