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Out-of-core SVD performance for document indexing

APPLIED NUMERICAL MATHEMATICS 2007年第11-12期57卷 1230-1239页

作者： Martin, Dian I. Martin, John C. Berry, Michael W. Browne, Murray Univ Tennessee Dept Comp Sci Knoxville TN 37996 USA Small Bear Tech Consulting Thorn Hill TN 37881 USA

The following study documents a formal evaluation of the performance tradeoffs and scalability for computing the sparse matrix singular value decomposition (SVD) as part of the Latent Semantic Analysis (LSA) of a given document collection with an out-of-core process. Most software packages capable of computing the SVD do all of their processing in-core, which involves keeping all vectors for the computation in memory. This limits the size of document collections that can be processed. The goal of the study was specifically to evaluate software capable of performing the SVD calculations out-of-core, minimizing memory usage by keeping only a small set of work vectors in memory at a time. Performance measures of interest for this study included the time of execution, both in CPU time and wall clock time, as well the memory and disk usage for computing the SVD. (c) 2007 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： lanczos algorithm latent semantic analysis out-of-core singular value decomposition

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An Approach Inspired by Quantum Mechanics for the Modeling of Large Power Systems

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IEEE TRANSACTIONS ON POWER SYSTEMS 2024年第1期39卷 1360-1369页

作者： Guichard, Pierrick Retiere, Nicolas Mayou, Didier Univ Grenoble Alpes Inst NEEL CNRS F-38042 Grenoble France Univ Grenoble Alpes CNRS Grenoble INP G2Elab F-38000 Grenoble France

The ever increasing complexity of power systems mandates a further improvement of the numerical methods used for their simulation. We propose a new approach that is inspired by an exact analogy with quantum mechanics, but requires no specific knowledge of quantum physics. This approach allows in particular to use local methods commonly applied in quantum mechanics, such as lanczos' algorithm. It is applied to computing the load flow and extended to simple dynamic studies where the nodes are described by swing equations. The first set of results are for an ideal 2D-lattice grid to illustrate the method and its main features. Then a contingency analysis by the proposed approach is performed for various realistic power systems, including a model of the European network. This demonstrates the interest of the new approach and its efficiency when compared with state-of-the-art methods, especially for large power system.

关键词： Load flow local approach lanczos algorithm Hamiltonian formalism contingency analysis

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DIFFERENTIATING THE METHOD OF CONJUGATE GRADIENTS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2014年第1期35卷 110-126页

作者： Gratton, Serge Titley-Peloquin, David Toint, Philippe Ilunga, Jean Tshimanga IRIT CERFACS F-31057 Toulouse France IRIT ENSEEIHT F-31071 Toulouse France Univ Namur B-5000 Namur Belgium

The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations Ax = b, where A is an element of R-n (x) (n) is symmetric positive definite. Let x(k) denote the kth iterate of CG. This is a nonlinear differentiable function of b. In this paper we obtain expressions for J(k), the Jacobian matrix of xk with respect to b. We use these expressions to obtain bounds on parallel to J(k)parallel to(2), the spectral norm condition number of x(k), and discuss algorithms to compute or estimate J(k)v and J(k)(T)v for a given vector v.

关键词： conjugate gradients algorithm lanczos algorithm perturbation analysis linearization automatic differentiation

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Modeling of first-order photobleaching kinetics using Krylov subspace spectral methods

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2018年第6期75卷 2153-2172页

作者： Sheikholeslami, Somayyeh Lambers, James V. Univ Southern Mississippi Dept Phys & Astron Hattiesburg MS 39406 USA Univ Southern Mississippi Dept Math Hattiesburg MS 39406 USA

We solve the first order 2-D reaction-diffusion equations which describe binding-diffusion kinetics using the photobleaching scanning profile of a confocal laser scanning microscope, approximated by a Gaussian laser profile. We show how to solve the first-order photobleaching kinetics partial differential equations (PDEs) using a time-stepping method known as a Krylov subspace spectral (KSS) method. KSS methods are explicit methods for solving time-dependent variable-coefficient partial differential equations. They approximate Fourier coefficients of the solution using Gaussian quadrature rules in the spectral domain. In this paper, we show how a KSS method can be used to obtain not only an approximate numerical solution, but also an approximate analytical solution when using initial conditions that come from pre-bleach steady states and also general initial conditions, to facilitate asymptotic analysis. Analytical and numerical results are presented. It is observed that although KSS methods are explicit, it is possible to use a time step that is far greater than what the CFL condition would indicate. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： lanczos algorithm Spectral methods Gaussian quadrature FRAP Photobleaching kinetics equation

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Excitonic couplings and electronic coherence in bridged naphthalene dimers

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PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA 1999年第23期96卷 13003-13008页

作者： Tretiak, S Zhang, WM Chernyak, V Mukamel, S Univ Rochester Dept Chem Rochester NY 14627 USA Univ Rochester Rochester Theory Ctr Opt Sci & Engn Rochester NY 14627 USA

The electronic excitations of naphthalene and a family of bridged naphthalene dimers are calculated and analyzed by using the Collective Electronic Oscillator method combined with the oblique lanczos algorithm. All experimentally observed trends in absorption profiles and radiative lifetimes are reproduced. Each electronic excitation is linked to the corresponding real-space transition density matrix, which represents the motions of electrons and holes created in the molecule by photon absorption. Two-dimensional plots of these matrices help visualize the degree of exciton localization and explain the dependence of the electronic interaction between chromophores on their separation.

关键词： excitonic couplings electronic coherence bridged naphthalene dimers Collective Electronic Oscillator method oblique lanczos algorithm photon absorption twodimensional plots

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LINEAR PREDICTION AND SINGLE-CHANNEL RECORDING

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JOURNAL OF NEUROSCIENCE METHODS 1995年第1-2期60卷 69-78页

作者： CARTER, AA OSWALD, RE CORNELL UNIV COLL VET MEDDEPT PHARMACOLITHACANY 14853

The measurement of individual single-channel events arising from the gating of ion channels provides a detailed data set from which the kinetic mechanism of a channel can be deduced. In many cases, the pattern of dwells in the open and closed states is very complex, and the kinetic mechanism and parameters are not easily determined. Assuming a Markov model for channel kinetics, the probability density function for open and closed time dwells should consist of a sum of decaying exponentials. One method of approaching the kinetic analysis of such a system is to determine the number of exponentials and the corresponding parameters which comprise the open and closed dwell time distributions. These can then be compared to the relaxations predicted from the kinetic model to determine, where possible, the kinetic constants. We report here the use of a linear technique, linear prediction/singular value decomposition, to determine the number of exponentials and the exponential parameters. Using simulated distributions and comparing with standard maximum-likelihood analysis, the singular value decomposition techniques provide advantages in some situations and are a useful adjunct to other single-channel analysis techniques.

关键词： SINGULAR VALUE DECOMPOSITION lanczos algorithm MAXIMUM-LIKELIHOOD ANALYSIS PATCH-CLAMP RECORDING

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Quantum state complexity meets many-body scars

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JOURNAL OF PHYSICS-CONDENSED MATTER 2024年第15期36卷 155601-155601页

作者： Nandy, Sourav Mukherjee, Bhaskar Bhattacharyya, Arpan Banerjee, Aritra Jozef Stefan Inst Ljubljana SI-1000 Slovenia UCL Dept Phys & Astron Gower St London WC1E 6BT England Indian Inst Technol Gandhinagar Gandhinagar 382355 Gujarat India Birla Inst Technol & Sci Pilani Campus Jhunjhunu 333031 Rajasthan India Okinawa Inst Sci & Technol 1919-1 Tancha Onna Son Okinawa 9040495 Japan

Scar eigenstates in a many-body system refers to a small subset of non-thermal finite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realised by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie algebra. We demonstrate why a careful usage of the forward scattering approximation (FSA), instead of any other method, is required to extract the most appropriate set of lanczos coefficients in this case as the consequence of this approximate symmetry. Only such a method leads to a well defined notion of a closed Krylov subspace and consequently, that of spread complexity. We show this using three separate initial states, namely |Z2⟩,|Z3⟩ and the vacuum state, due to the disparate classes of scar states hosted by these sectors. We also discuss systematic methods of remedying the imperfections in the FSA setup stemming from these approximate symmetries.

关键词： quantum dynamics quantum many-body scars Rydberg atoms Krylov complexity lanczos algorithm quantum complexity

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Estimating the trace of matrix functions with application to complex networks

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NUMERICAL algorithmS 2023年第1期92卷 503-522页

作者： Fuentes, Rafael Diaz Donatelli, Marco Fenu, Caterina Mantica, Giorgio Univ Cagliari Dept Math & Comp Sci Via Osped 72 I-09124 Cagliari Italy Univ Insubria Dipartimento Sci & Alta Tecnol Via Valleggio 11 I-22100 Como Italy Univ Insubria Ctr Complex & Nonlinear Syst Via Valleggio 11 I-22100 Como Italy Ist Nazl Fis Nucl Sez Milano Milan Italy INdAM GNFM Florence Italy

The approximation of trace(f(Omega)), where f is a function of a symmetric matrix Omega, can be challenging when Omega is exceedingly large. In such a case even the partial lanczos decomposition of Omega is computationally demanding and the stochastic method investigated by Bai et al. (J. Comput. AppL Math. 74:71-89, 1996) is preferred. Moreover, in the last years, a partial global lanczos method has been shown to reduce CPU time with respect to partial lanczos decomposition. In this paper we review these techniques, treating them under the unifying theory of measure theory and Gaussian integration. This allows generalizing the stochastic approach, proposing a block version that collects a set of random vectors in a rectangular matrix, in a similar fashion to the partial global lanczos method. We show that the results of this technique converge quickly to the same approximation provided by Bai et al. (J. Comput. Appl. Math. 74:71-89, 1996), while the block approach can leverage the same computational advantages as the partial global lanczos. Numerical results for the computation of the Von Neumann entropy of complex networks prove the robustness and efficiency of the proposed block stochastic method.

关键词： Gauss quadrature lanczos algorithm Monte Carlo method Trace computation Network analysis

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REML estimation with intrinsic Matern dependence in the spatial linear mixed model

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ELECTRONIC JOURNAL OF STATISTICS 2016年第2期10卷 2856-2893页

作者： Dutta, Somak Mondal, Debashis Iowa State Univ Ames IA 50011 USA Oregon State Univ Corvallis OR 97331 USA

We present a new matrix-free residual maximum likelihood (REML) analysis for irregularly spaced spatial data, where observations usually represent average values over very small regions that are interpreted as points. The REML analysis is obtained after embedding the sampling locations in a fine scale rectangular lattice, treating unobserved sites as missing data. The spatial random fields considered here are based on fractional Laplacian differencing on the lattice and they are unique in approximating continuum intrinsic Matern dependence. Here, using the h-likelihood method, we derive REML estimating equations that allow for singular precision matrices, estimation of covariate effects, prediction of unobserved spatial effects and REML estimation of precision parameters as a solution to an explicit gamma non-linear model. Furthermore, we devise a sophisticated computational algorithm that enables us to achieve scalable matrix-free statistical computations. In particular, these matrix-free computations include the use of (1) the two-dimensional discrete cosine transformation that arises in the spectral decomposition of the precision matrix of our spatial random fields and that allows fast matrix-free matrix-vector multiplication, (2) a matrix-free pre-conditioned lanczos algorithm that solves non-sparse matrix equations with linear constraints, (3) a matrix-free Hutchinson's trace estimator that stochastically approximates the trace of a matrix, (4) a robust trust region method that always finds a local maximum of the non-concave residual log-likelihood function and (5) some preliminary computations of the log REML likelihood function based on Taylor series approximation. Using computer experiments, we provide further understanding on not just the number and values but also the basins of attraction of the local and global maxima of the REML function. This understanding significantly simplifies the problem of finding global maxima. We further demonstrate through comp

关键词： Arsenic contamination discrete cosine transform fractional differencing Hutchinson's trace approximations incomplete Cholesky lanczos algorithm long range dependence power variogram matrix-free computations trust region method

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The Anderson model of localization:: A challenge for modern eigenvalue methods

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 1999年第6期20卷 2089-2102页

作者： Elsner, U Mehrmann, V Milde, F Römer, RA Schreiber, M Tech Univ Fak Math D-09107 Chemnitz Germany Tech Univ Inst Phys D-09107 Chemnitz Germany

We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large, sparse, real, symmetric, and indefinite matrices of the Anderson model of localization. We compare the lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the implicitly restarted Arnoldi method coupled with polynomial and several shift-and-invert convergence accelerators as well as with a sparse hybrid tridiagonalization method. We demonstrate that for our problem the lanczos implementation is faster and more memory efficient than the other approaches. This seemingly innocuous problem presents a major challenge for all modern eigenvalue algorithms.

关键词： eigenvalue eigenvector problems lanczos algorithm Arnoldi algorithm Anderson model of localization shift-and-invert polynomial convergence accelerators

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