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Two new methods for computing vibrational energy levels

CANADIAN JOURNAL OF CHEMISTRY 2015年第6期93卷 589-593页

作者： Carrington, Tucker, Jr. Queens Univ Dept Chem Kingston ON K7L 3N6 Canada

I review two new ideas for coping with the size of large product basis sets and large product grids when one computes vibrational energy levels. The first is based on a tensor reduction scheme. It exploits advantages of a sum-of-products potential. The key idea is to use a basis each of whose function is a sum of optimized products and to compress the number of terms in each basis function. When the potential does not have the sum-of-products form, it is usually necessary to use quadrature. The second idea uses a nondirect product grid that has structure and is therefore compatible with efficient matrix-vector products.

关键词： theory rovibrational spectroscopy iterative methods lanczos algorithm

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Decentralized Eigenvalue algorithms for Distributed Signal Detection in Wireless Networks

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2015年第2期63卷 427-440页

作者： Penna, Federico Stanczak, Slawomir Samsung Mobile Solut Lab MSL San Diego CA 92121 USA Heinrich Hertz Inst Nachrichtentech Berlin GmbH Fraunhofer Inst Telecommun D-10587 Berlin Germany

In this paper, we derive and analyze two algorithms-referred to as decentralized power method (DPM) and decentralized lanczos algorithm (DLA)-for distributed computation of one (the largest) or multiple eigenvalues of a sample covariance matrix over a wireless network. The proposed algorithms, based on sequential average consensus steps for computations of matrix-vector products and inner vector products, are first shown to be equivalent to their centralized counterparts in the case of exact distributed consensus. Then, closed-form expressions of the error introduced by nonideal consensus are derived for both algorithms. The error of the DPM is shown to vanish asymptotically under given conditions on the sequence of consensus errors. Finally, we consider applications to spectrum sensing in cognitive radio networks, and we show that virtually all eigenvalue-based tests proposed in the literature can be implemented in a distributed setting using either the DPM or the DLA. Simulation results are presented that validate the effectiveness of the proposed algorithms in conditions of practical interest (large-scale networks, small number of samples, and limited number of iterations).

关键词： Average consensus eigenvalue-based signal detection lanczos algorithm power method

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Comparison of Krylov subspace methods with preconditioning techniques for solving boundary value problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 1999年第11-12期38卷 197-206页

作者： Sundar, S Bhagavan, BK Indian Inst Technol Dept Math Kharagpur 721302 W Bengal India

In this paper, we made an attempt to establish the usefulness of lanczos solver with preconditioning technique over the preconditioned Conjugate Gradient (CG) solvers. We have presented here a detail comparative study with respect to convergence, speed as well as CPU-time, by considering appropriate boundary value problems. (C) 1999 Elsevier Science Ltd. All rights reserved.

关键词： Krylov subspace preconditioned conjugate gradient method lanczos algorithm preconditioned lanczos algorithm symmetric systems

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A reduction algorithm for reconstructing periodic Jacobi matrices in Minkowski spaces

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APPLIED MATHEMATICS AND COMPUTATION 2022年 419卷 126853-126853页

作者： Xu, Wei-Ru Bebiano, Natalia Chen, Guo-Liang Sichuan Normal Univ Laurent Math Ctr Sch Math Sci Chengdu 610066 Peoples R China Univ Coimbra Dept Math CMUC P-3001454 Coimbra Portugal East China Normal Univ Sch Math Sci Shanghai Key Lab PMMP Shanghai 200241 Peoples R China

The periodic Jacobi inverse eigenvalue problem concerns the reconstruction of a periodic Jacobi matrix from prescribed spectral data. In Minkowski spaces, with a given signature operator H = diag(1, 1, ..., 1, -1), the corresponding matrix is a periodic pseudo-Jacobi matrix. The inverse eigenvalue problem for such matrices consists in the reconstruction of pseudo-Jacobi matrices, with the same order and signature operator H. In this paper we solve this problem by applying Sylvester's identity and Householder transformation. The solution number and the corresponding reconstruction algorithm are here exhibited, and illustrative numerical examples are given. Comparing this approach with the known lanczos algorithm for reconstructing pseudo-Jacobi matrices, our method is shown to be more stable and effective. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Inverse eigenvalue problem Pseudo-Jacobi matrix Periodic pseudo-Jacobi matrix Sylvester's identity Householder transformation lanczos algorithm

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Methods for calculating vibrational energy levels

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CANADIAN JOURNAL OF CHEMISTRY 2004年第6期82卷 900-914页

作者： Carrington, T Univ Montreal Dept Chim Montreal PQ H3C 3J7 Canada

This article reviews new methods for computing vibrational energy levels of small polyatomic molecules. The principal impediment to the calculation of energy levels is the size of the required basis set. If one uses a product basis the Hamiltonian matrix for a four-atom molecule is too large to store in core memory. We discuss iterative methods that enable one to use a product basis to compute energy levels (and spectra) without storing a Hamiltonian matrix. Despite the advantages of iterative methods it is not possible, using product basis functions, to calculate vibrational spectra of molecules with more than four atoms. A very recent method combining contracted basis functions and the lanczos algorithm with which vibrational energy levels of methane have been computed is described. New ideas, based on exploiting preconditioning, for reducing the number of matrix-vector products required to converge energy levels of interest are also summarized.

关键词： vibrational energy levels kinetic energy operators lanczos algorithm contracted basis functions preconditioning

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A Krylov subspace projection method for simultaneous solution of Helmholtz problems at multiple frequencies

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2003年第41-42期192卷 4609-4640页

作者： Wagner, MM Pinsky, PM Oberai, AA Malhotra, M Stanford Univ Div Mech & Computat Stanford CA 94305 USA Boston Univ Dept Aerosp & Mech Engn Boston MA 02215 USA Sun Microsyst Inc Palo Alto CA USA

A Krylov subspace projection method which provides simultaneous solutions of the Helmholtz equation at multiple frequencies in one solution step is presented. The projector is obtained with an unsymmetric block lanczos algorithm applied to a transfer function derived from a finite element discretization. This approach is equivalent to a matrix valued Pade approximation of the transfer function. The proposed method is an extension of the formulation presented in [J. Comput. Acoust. 8 (2000) 223] to unsymmetric systems and allows the treatment of a much wider range of practical problems, including near-field and fluid-structure interaction computations. (C) 2003 Elsevier B.V. All rights reserved.

关键词： multiple-frequency solution Krylov subspace lanczos algorithm Pade approximation Helmholtz equation Dirichlet-to-Neumann map acoustics

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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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A deflated version of the conjugate gradient algorithm

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2000年第5期21卷 1909-1926页

作者： Saad, Y Yeung, M Erhel, J Guyomarc'h, F Univ Minnesota Dept Comp Sci & Engn Minneapolis MN 55455 USA Univ Calif Irvine Dept Math Irvine CA 92697 USA Inst Natl Rech Informat & Automat F-35042 Rennes France

We present a deflated version of the conjugate gradient algorithm for solving linear systems. The new algorithm can be useful in cases when a small number of eigenvalues of the iteration matrix are very close to the origin. It can also be useful when solving linear systems with multiple right-hand sides, since the eigenvalue information gathered from solving one linear system can be recycled for solving the next systems and then updated.

关键词： conjugate gradient deflation multiple right-hand sides lanczos algorithm

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Direct calculation of long time correlation functions using an optical potential

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1998年第5期68卷 317-328页

作者： Alacid, M Leforestier, C Univ Montpellier 2 UMR 5636 Lab Struct & Dynam Syst Mol & Solides F-34095 Montpellier 05 France

We present an L-2 method aimed at directly computing autocorrelation functions [Phi(0)/Phi(f)] for systems displaying long time recurrences. By making use of a lanczos scheme, as previously proposed by Wyatt [Chem. Phys. Lett. 121, 301 (1985)], the method avoids explicit time propagation of the wavefunction. The problem associated with spurious recurrences, due to the finite size of the L-2-box, is solved in terms of an optical potential located in the asymptotic region. The resulting complex representation of the Hamiltonian operator is handled by a complex symmetric lanczos scheme, which retains the same basic advantages as its real version. The method is illustrated on the ozone photodissociation process which displays a very detailed recurrence structure over a long time period. It is shown that such a direct calculation of the correlation function is about one order of magnitude faster than an actual wavepacket propagation. The accuracy of the method is assessed by comparison to calculations performed without any optical potential but using a very large box size along the dissociation coordinate. (C) 1998 John Wiley & Sons, Inc.

关键词： autocorrelation function optical potential lanczos algorithm ozone photodissociation hyperspherical coordinates

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FastSKAT: Sequence kernel association tests for very large sets of markers

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GENETIC EPIDEMIOLOGY 2018年第6期42卷 516-527页

作者： Lumley, Thomas Brody, Jennifer Peloso, Gina Morrison, Alanna Rice, Kenneth Univ Auckland Dept Stat Auckland New Zealand Univ Washington Cardiovasc Hlth Res Unit Seattle WA 98195 USA Boston Univ Dept Biostat Boston MA 02215 USA Univ Texas Hlth Sci Ctr Houston Houston TX 77030 USA Univ Washington Dept Biostat F-600 Hlth Sci BldgCampus Mail Stop 357232 Seattle WA 98195 USA

The sequence kernel association test (SKAT) is widely used to test for associations between a phenotype and a set of genetic variants that are usually rare. Evaluating tail probabilities or quantiles of the null distribution for SKAT requires computing the eigenvalues of a matrix related to the genotype covariance between markers. Extracting the full set of eigenvalues of this matrix (an nxn matrix, for n subjects) has computational complexity proportional to n(3). As SKAT is often used when n>104, this step becomes a major bottleneck in its use in practice. We therefore propose fastSKAT, a new computationally inexpensive but accurate approximations to the tail probabilities, in which the k largest eigenvalues of a weighted genotype covariance matrix or the largest singular values of a weighted genotype matrix are extracted, and a single term based on the Satterthwaite approximation is used for the remaining eigenvalues. While the method is not particularly sensitive to the choice of k, we also describe how to choose its value, and show how fastSKAT can automatically alert users to the rare cases where the choice may affect results. As well as providing faster implementation of SKAT, the new method also enables entirely new applications of SKAT that were not possible before;we give examples grouping variants by topologically associating domains, and comparing chromosome-wide association by class of histone marker.

关键词： convolution genetic association lanczos algorithm randomized trace estimator stochastic singular value decomposition

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