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New results on the convergence of the conjugate gradient method

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2009年第3期16卷 223-236页

作者： Bouyouli, R. Meurant, G. Smoch, L. Sadok, H. Zone Univ Mivoix Univ Littoral Lab Math Pures & Appl F-62228 Calais France CEA DIF F-91680 Bruyeres Le Chatel France Univ Mohammed 5 Dept Math Fac Sci Rabat Morocco

This paper is concerned with proving theoretical results related to the convergence of the conjugate gradient (CG) method for solving positive definite symmetric linear systems. Considering the inverse of the projection of the inverse of the matrix, new relations for ratios of the A-norm of the error and the norm of the residual are provided, starting from some earlier results of Sadok (Numer algorithms 2005;40:201-216). The proofs of our results rely on the well-known correspondence between the CG method and the lanczos algorithm. Copyright (C) 2008 John Wiley & Sons, Ltd.

关键词： symmetric linear systems conjugate gradient algorithm lanczos algorithm Krylov subspace Ritz values

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algorithm for Solving Arbitrary Systems of Linear Equations

Algorithm for Solving Arbitrary Systems of Linear Equations

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9th WSEAS International Conference on Systems Theory and Scientific Computation

作者： Budanov, V. M. Moscow MV Lomonosov State Univ Inst Mech Moscow Russia

The paper presents the finite algorithm for solution of arbitrary system of linear equations. The algorithm is based on known approaches and concepts, especially Krylov subspace and lanczos basis. It is shown that the... 详细信息

ISBN: (纸本)9789604741090

关键词： System of linear equations lanczos algorithm

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Efficient Finite Element Electromagnetic Analysis for High-Frequency/High-Speed Circuits And Multiconductor Transmission Lines

Efficient Finite Element Electromagnetic Analysis for High-F...

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作者： Lee, Shih-hao University of Illinois – Urbana-Champaign

学位级别：doctor

This dissertation comprises the following four components. (1) Development of a robust and efficient 3-D finite element electromagnetic field solver with high-order vector elements for high-frequency and high-speed circuit simulations. The solver supports wave port and lumped port excitations as well as the incorporation of lumped networks and circuit models in a distributed finite element model. An adaptive multipoint model order reduction method is developed for fast broadband analysis. (2) Development of a fast and accurate multiconductor transmission line simulator and parameter extractor with improved model order reduction techniques. A methodology is further proposed for a combined quasi-TEM and full-wave transmission line analysis, which possesses their respective advantages and ensures full-wave accuracy from DC to very high frequencies. The transmission line analysis also takes into account the frequency dependence of dielectric materials. (3) Study of the low-frequency instability problem in the 3-D full-wave finite element simulation. The tree-cotree splitting is combined with several other techniques to improve the matrix conditioning and extend full-wave solutions down to very low frequencies for a more robust broadband characterization of high-speed digital circuits. (4) A combined domain decomposition–model order reduction (DD–MOR) method for efficient full-wave analysis of interconnections in multilayer printed circuit boards. The method not only brings a significant enhancement to computational efficiency while maintaining full-wave accuracy, but also provides great flexibility in the finite element mesh generation

关键词： electromagnetics microwave finite element method (FEM) nodal elements vector elements edge elements higher-order elements triangular elements tetrahedral elements finite element analysis full-wave analysis impedance boundary condition port boundary condition wave port lumped port deembedding thin-wire approximation thin wire modeling internal impedance lumped elements lumped circuits field-circuit simulation EM-circuit simulation stamping reduced-order modeling model order reduction (MOR) fast frequency sweep solution space projection (SSP) quasi-static analysis quasi-TEM analysis generalized eigenproblem lanczos algorithm modal analysis eigenanalysis frequency-dependent media anisotropic media Debye model multiconductor transmission lines transmission line parameters frequency-dependent losses conductor loss dielectric loss substrate loss characteristic impedance skin effect proximity effect parameter extraction resistance inductance capacitance conductance per-unit-length composite conductors waveguide filter coplanar waveguide (CPW) striplines microstrip lines high-frequency circuits RF circuits high-speed circuits tree-cotree splitting low-frequency breakdown low-frequency instability preconditioning bonding wire printed circuit board multilayer printed circuit board (PCB) via-holes signal integrity electromagnetic coupling interconnects domain decomposition Approximate Modal Interface (AMI) Approximate Modal Interface–Solution Space Projection (AMI–SSP) domain decomposition–model order reduction (DD–MOR) computer-aided design (CAD)

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Adaptive Rational Interpolation: Arnoldi and lanczos-like Equations

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EUROPEAN JOURNAL OF CONTROL 2008年第4期14卷 342-354页

作者： Frangos, Michalis Jaimoukha, Imad M. Univ London Imperial Coll Sci Technol & Med Dept Elect & Elect Engn London SW7 2AZ England

The Arnoldi and lanczos algorithms, which belong to the class of Krylov subspace methods, are increasingly used for model reduction of large-scale systems. The standard versions of the algorithms tend to create reduced order models that poorly approximate low frequency dynamics. Rational Arnoldi and lanczos algorithms produce reduced models that approximate dynamics at various frequencies. This paper tackles the issue of developing simple Arnoldi and lanczos equations,for the rational case. This allows a simple error analysis to be carried out for both algorithms and permits the development of computationally efficient model reduction algorithms, where the frequencies at which the dynamics are to be matched can be updated adaptively.

关键词： Model reduction Krylov subspaces Arnoldi algorithm lanczos algorithm rational interpolation adaptive interpolation

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Computation of large invariant subspaces using polynomial filtered lanczos iterations with applications in density functional theory

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2008年第1期30卷 397-418页

作者： Bekas, C. Kokiopoulou, E. Saad, Yousef Univ Minnesota Dept Comp Sci & Engn Minneapolis MN 55455 USA

The most expensive part of all electronic structure calculations based on density functional theory lies in the computation of an invariant subspace associated with some of the smallest eigenvalues of a discretized Hamiltonian operator. The dimension of this subspace typically depends on the total number of valence electrons in the system, and can easily reach hundreds or even thousands when large systems with many atoms are considered. At the same time, the discretization of Hamiltonians associated with large systems yields very large matrices, whether with planewave or real-space discretizations. The combination of these two factors results in one of the most significant bottlenecks in computational materials science. In this paper we show how to efficiently compute a large invariant subspace associated with the smallest eigenvalues of a symmetric/Hermitian matrix using polynomially filtered lanczos iterations. The proposed method does not try to extract individual eigenvalues and eigenvectors. Instead, it constructs an orthogonal basis of the invariant subspace by combining two main ingredients. The first is a filtering technique to dampen the undesirable contribution of the largest eigenvalues at each matrix-vector product in the lanczos algorithm. This technique employs a well-selected low pass filter polynomial, obtained via a conjugate residual-type algorithm in polynomial space. The second ingredient is the lanczos algorithm with partial reorthogonalization. Experiments are reported to illustrate the efficiency of the proposed scheme compared to state-of-the-art implicitly restarted techniques.

关键词： polynomial filtering conjugate residual lanczos algorithm density functional theory

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A renormalization scheme for calculating the spectrum of a vibronic system occurring in molecules or impurities in insulators

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PHYSICA B-CONDENSED MATTER 2008年第18期403卷 3093-3096页

作者： Baher, S. Mehrjoo, S. Lorestan Univ Dept Phys Khorramabad *** Iran

An iteration scheme which makes use of a numerical renormalization group approach is used to calculate the spectrum of vibronic levels. This spectrum resulted from dynamic effects occurring in certain molecules or impurities in insulators. The Hamiltonian of these systems is expressed in the matrix form, using products of suitable electron-phonon states as a basis. In applying this method to multimode electron-phonon systems, phonon modes are coupled in a chain-like fashion. Then a finite chain calculation in terms of Hubbard X-operators is explored by setting up the vibronic Hamiltonian. Calculations are based on lanczos algorithm, in which only the nearest neighbor matrix elements along the chain need to be taken into account. The iterative scheme is then applied to a two-level electronic system coupled to phonons. A single-particle Green's function corresponding to a two-level system is applied to calculate the spectral density of states, which, coupled to single mode is carried out. The strength of lines in density of states is affected by the coupling constant as well as the temperature dependence of some measurable quantities. (C) 2008 Elsevier B.V. All rights reserved.

关键词： Hubbard X-operators Green's function density of states lanczos algorithm two-level system

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On large-scale diagonalization techniques for the Anderson model of localization

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SIAM REVIEW 2008年第1期50卷 91-112页

作者： Schenk, Olaf Bollhoefer, Matthias Roemer, Rudolf A. Univ Basel Dept Comp Sci CH-4056 Basel Switzerland Tech Univ Carolo Wilhelmina Braunschweig Inst Computat Math D-38106 Braunschweig Germany Univ Warwick Dept Phys Coventry CV4 7AL W Midlands England Univ Warwick Ctr Comp Sci Coventry CV4 7AL W Midlands England

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques ill the implicitly restarted lanczos method and in the Jacobi-Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

关键词： Anderson model of localization large-scale eigenvalue problem lanczos algorithm Jacobi-Davidson algorithm Cullum-Willoughby implementation symmetric indefinite matrix multilevel preconditioning maximum weighted matching

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Use of Non-uniform Grids for Reduced Spatial Sampling in FDTD Simulation

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FREQUENZ 2008年第7-8期62卷 156-159页

作者： Ramachandran, Aravind Cangellaris, Andreas C. Univ Illinois Dept Elect & Comp Engn Urbana IL 61801 USA

An algorithm for the generation of non-uniform finite-difference grids for the analysis of electrically long wave propagating structures is presented. The methodology is based on the use of a Pade-Chebyshev approximation of the input impedance of the discrete model of the electrically-long dimension of the structure, generated using an equidistant grid, and the subsequent use of the lanczos algorithm to cast the approximation in terms of a reduced-order model that is interpreted in terms of a non-uniform grid of reduced spatial sampling. A numerical study is used to demonstrate that structures with non-uniform material properties can be modeled using a non-uniform grid constructed for the simpler case of a uniform structure.

关键词： FDTD transmission line Pade-Chebyshev approximation Gaussian spectral rule lanczos algorithm

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Gauss Quadrature Rule Based on Parameter Estimation

Gauss Quadrature Rule Based on Parameter Estimation

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7th International Symposium on Instrumentation and Control Technology

作者： Xie Kai Yang Shulin Beijing Inst Graph Commun Coll Informat & Mech Engn Beijing 102600 Peoples R China

ISBN: (纸本)9780819473639

In this paper, we consider the estimation of the unknown parameter for the problem of reconstructing a high resolution image from multiple under-sampled, shifted, degraded frames. L-curve criterion is used to estimate regularization parameters. However, the computational of the L-curve is quiet costly for large-size problems. The paper proposes an efficient approximate technique based on Gauss quadrature rule. The technique translates some matrix computation into Gauss quadrature with singular decomposition. It can reduce the computational complexity of the L-curve.

关键词： Gauss quadrature lanczos algorithm L-curve

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A REFINED UNSYMMETRIC lanczos EIGENSOLVER FOR COMPUTING ACCURATE EIGENTRIPLETS OF A REAL UNSYMMETRIC MATRIX

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2007年 28卷 95-113页

作者： Tremblay, Jean Christophe Carrington, Tucker, Jr. Univ Montreal Dept Chim Montreal PQ H3C 3J7 Canada

For most unsymmetric matrices it is difficult to compute many accurate eigenvalues using the primitive form of the unsymmetric lanczos algorithm (ULA). In this paper we propose a modification of the ULA. It is related to ideas used in [J. Chem. Phys. 122 (2005), 244107 (11 pages)] to compute resonance lifetimes. Using the refined ULA we suggest, the calculation of accurate extremal and interior eigenvalues is feasible. The refinement is simple: approximate right and left eigenvectors computed using the ULA are used to form a small projected matrix whose eigenvalues and eigenvectors are easily computed. There is no re-biorthogonalization of the lanczos vectors and no need to store large numbers of vectors in memory. The method can therefore be used to compute eigenvalues of very large matrices. The idea is tested on several matrices.

关键词： Eigenproblem unsymmetric matrices lanczos algorithm

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