The Experts below are selected from a list of 8994 Experts worldwide ranked by ideXlab platform
Ivan V Oseledets - One of the best experts on this subject based on the ideXlab platform.
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calculating vibrational spectra of molecules using tensor train decomposition
Journal of Chemical Physics, 2016Co-Authors: Maxim Rakhuba, Ivan V OseledetsAbstract:We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (locally optimal block preconditioned conjugate gradient method, Inverse Iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH3CN on a laptop in one hour using only 100 MB of memory to represent all computed eigenfunctions.
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calculating vibrational spectra of molecules using tensor train decomposition
arXiv: Numerical Analysis, 2016Co-Authors: Maxim Rakhuba, Ivan V OseledetsAbstract:We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (LOBPCG, Inverse Iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH$_3$CN on a laptop in one hour using only $100$ MB of memory to represent all computed eigenfunctions.
Alastair Spence - One of the best experts on this subject based on the ideXlab platform.
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numerical computation of the complex eigenvalues of a matrix by solving a square system of equations
Journal of Natural Sciences Research, 2015Co-Authors: Richard O Akinola, Alastair SpenceAbstract:It is well known that if the largest or smallest eigenvalue of a matrix has been computed by some numerical algorithms and one is interested in computing the corresponding eigenvector, one method that is known to give such good approximations to the eigenvector is Inverse Iteration with a shift. For complex eigenpairs, instead of using Ruhe’s normalization, we show that the natural two norm normalization for the matrix pencil, yields an underdetermined system of equation and by adding an extra equation, the augmented system becomes square which can be solved by LU factorization at a cheaper rate and quadratic convergence is guaranteed. While the underdetermined system of equations can be solved using QR factorization as shown in an earlier work by the same authors, converting it to a square system of equations has the added advantage that besides using LU factorization, it can be solved by several approaches including iterative methods. We show both theoretically and numerically that both algorithms are equivalent in the absence of roundoff errors.
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lyapunov Inverse Iteration for identifying hopf bifurcations in models of incompressible flow
SIAM Journal on Scientific Computing, 2012Co-Authors: Howard C Elman, Karl Meerbergen, Alastair SpenceAbstract:The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [K. Meerbergen and A. Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1982--1999] avoids this computation, instead performing an Inverse Iteration for a certain set of real eigenvalues that requires the solution of a large-scale Lyapunov equation at each Iteration. In this study, we refine the Lyapunov Inverse Iteration method to make it more robust and efficient, and we examine its performance on challenging test problems arising from fluid dynamics. Various implementation issues are discussed, including the use of inexact inner Iterations and the impact of the choice of iterative solution for the Lyapunov equations, and the effect of eigenvalue distribution on performance. Numerical experiments demonstrate the robustness of the algorithm.
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Inverse Iteration for Purely Imaginary Eigenvalues with Application to the Detection of Hopf Bifurcations in Large-Scale Problems
SIAM Journal on Matrix Analysis and Applications, 2010Co-Authors: Karl Meerbergen, Alastair SpenceAbstract:The detection of a Hopf bifurcation in a large-scale dynamical system that depends on a physical parameter often consists of computing the right-most eigenvalues of a sequence of large sparse eigenvalue problems. Guckenheimer, Gueron, and Harris-Warrick [SIAM J. Numer. Anal., 34 (1997), pp. 1-21] proposed a method that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenvalues. This method utilizes a certain sum of Kronecker products and involves the solution of matrices of squared dimension, which is impractical for large-scale applications. However, if good starting guesses are available for the parameter and the purely imaginary eigenvalue at the Hopf point, then efficient algorithms are available. In this paper, we propose a method for obtaining such good starting guesses, based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem (possibly arising after a linearization process). The problem is formulated as an inexact Inverse Iteration method that requires the solution of a sequence of Lyapunov equations with low rank right-hand sides. It is this last fact that makes the method feasible for large systems. The power of our method is tested on four numerical examples.
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a tuned preconditioner for inexact Inverse Iteration applied to hermitian eigenvalue problems
Ima Journal of Numerical Analysis, 2007Co-Authors: Melina A Freitag, Alastair SpenceAbstract:In this paper, we consider the computation of an eigenvalue and the corresponding eigenvector of a large sparse Hermitian positive-definite matrix using inexact Inverse Iteration with a fixed shift. For such problems, the large sparse linear systems arising at each Iteration are often solved approximately by means of symmetrically preconditioned MINRES. We consider preconditioners based on the incomplete Cholesky factorization and derive a new tuned Cholesky preconditioner which shows considerable improvement over the standard preconditioner. This improvement is analysed using the convergence theory for MINRES. We also compare the spectral properties of the tuned preconditioned matrix with those of the standard preconditioned matrix. In particular, we provide both a perturbation result and an interlacing result, and these results show that the spectral properties of the tuned preconditioner are similar to those of the standard preconditioner. For Rayleigh quotient shifts, comparison is also made with a technique introduced by Simoncini & Elden (2002, BIT, 42, 159-182) which involves changing the right-hand side of the Inverse Iteration step. Several numerical examples are given to illustrate the theory described in the paper.
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convergence of inexact Inverse Iteration with application to preconditioned iterative solves
Bit Numerical Mathematics, 2007Co-Authors: Melina A Freitag, Alastair SpenceAbstract:In this paper we study inexact Inverse Iteration for solving the generalised eigenvalue problem A x=λM x. We show that inexact Inverse Iteration is a modified Newton method and hence obtain convergence rates for various versions of inexact Inverse Iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We also show how modifying the right hand side in Inverse Iteration still provides a convergent method, but the rate of convergence will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to the usual preconditioner, but which has advantages both for the standard form of Inverse Iteration and for the version with a modified right hand side. Numerical examples are given to illustrate the theoretical results.
Oleksandr Kyriienko - One of the best experts on this subject based on the ideXlab platform.
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quantum Inverse Iteration algorithm for programmable quantum simulators
npj Quantum Information, 2020Co-Authors: Oleksandr KyriienkoAbstract:We propose a quantum Inverse Iteration algorithm, which can be used to estimate ground state properties of a programmable quantum device. The method relies on the Inverse power Iteration technique, ...
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quantum Inverse Iteration algorithm for programmable quantum simulators
arXiv: Quantum Physics, 2019Co-Authors: Oleksandr KyriienkoAbstract:We propose a quantum Inverse Iteration algorithm which can be used to estimate the ground state properties of a programmable quantum device. The method relies on the Inverse power Iteration technique, where the sequential application of the Hamiltonian Inverse to an initial state prepares an approximate groundstate. To apply the Inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.
Maxim Rakhuba - One of the best experts on this subject based on the ideXlab platform.
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calculating vibrational spectra of molecules using tensor train decomposition
Journal of Chemical Physics, 2016Co-Authors: Maxim Rakhuba, Ivan V OseledetsAbstract:We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (locally optimal block preconditioned conjugate gradient method, Inverse Iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH3CN on a laptop in one hour using only 100 MB of memory to represent all computed eigenfunctions.
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calculating vibrational spectra of molecules using tensor train decomposition
arXiv: Numerical Analysis, 2016Co-Authors: Maxim Rakhuba, Ivan V OseledetsAbstract:We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (LOBPCG, Inverse Iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH$_3$CN on a laptop in one hour using only $100$ MB of memory to represent all computed eigenfunctions.
Melina A Freitag - One of the best experts on this subject based on the ideXlab platform.
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gmres convergence bounds for eigenvalue problems
Computational methods in applied mathematics, 2018Co-Authors: Melina A Freitag, Patrick Kurschner, Jennifer PestanaAbstract:The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right hand side. Within the solution of eigenvalue problems via Inverse Iteration or subspace Iteration, the right hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g. tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace Iteration with either a tuned or polynomial preconditioner should be used in practice.
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gmres convergence bounds for eigenvalue problems
arXiv: Numerical Analysis, 2016Co-Authors: Melina A Freitag, Patrick Kurschner, Jennifer PestanaAbstract:The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right hand side. Within the solution of eigenvalue problems via Inverse Iteration or subspace Iteration, the right hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds give rise to adapted preconditioners applied to the eigenvalue problems, e.g. tuned and polynomial preconditioners. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace Iteration with either a tuned or polynomial preconditioner should be used in practice.
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a tuned preconditioner for inexact Inverse Iteration applied to hermitian eigenvalue problems
Ima Journal of Numerical Analysis, 2007Co-Authors: Melina A Freitag, Alastair SpenceAbstract:In this paper, we consider the computation of an eigenvalue and the corresponding eigenvector of a large sparse Hermitian positive-definite matrix using inexact Inverse Iteration with a fixed shift. For such problems, the large sparse linear systems arising at each Iteration are often solved approximately by means of symmetrically preconditioned MINRES. We consider preconditioners based on the incomplete Cholesky factorization and derive a new tuned Cholesky preconditioner which shows considerable improvement over the standard preconditioner. This improvement is analysed using the convergence theory for MINRES. We also compare the spectral properties of the tuned preconditioned matrix with those of the standard preconditioned matrix. In particular, we provide both a perturbation result and an interlacing result, and these results show that the spectral properties of the tuned preconditioner are similar to those of the standard preconditioner. For Rayleigh quotient shifts, comparison is also made with a technique introduced by Simoncini & Elden (2002, BIT, 42, 159-182) which involves changing the right-hand side of the Inverse Iteration step. Several numerical examples are given to illustrate the theory described in the paper.
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convergence of inexact Inverse Iteration with application to preconditioned iterative solves
Bit Numerical Mathematics, 2007Co-Authors: Melina A Freitag, Alastair SpenceAbstract:In this paper we study inexact Inverse Iteration for solving the generalised eigenvalue problem A x=λM x. We show that inexact Inverse Iteration is a modified Newton method and hence obtain convergence rates for various versions of inexact Inverse Iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We also show how modifying the right hand side in Inverse Iteration still provides a convergent method, but the rate of convergence will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to the usual preconditioner, but which has advantages both for the standard form of Inverse Iteration and for the version with a modified right hand side. Numerical examples are given to illustrate the theoretical results.
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convergence theory for inexact Inverse Iteration applied to the generalised nonsymmetric eigenproblem
Electronic Transactions on Numerical Analysis, 2007Co-Authors: Melina A Freitag, Alastair SpenceAbstract:In this paper we consider the computation of a finite eigenvalue and corresponding right eigenvector of a large sparse generalised eigenproblem Ax = �Mx using inexact Inverse Iteration. Our convergence theory is quite general and requires few assumptions on A and M. In particular, there is no need for M to be symmetric posi- tive definite or even nonsingular. The theory includes both fixed and variable shift strategies, and the bounds obtained are improvements on those currently in the literature. In addition, the analysis developed here is used to provide a convergence theory for a version of inexact simplified Jacobi-Davidson. Several numerical examples are presented to illustrate the theory: including applications in nuclear reactor stability, with M singular and nonsymmetric, the linearised Navier-Stokes equations and the bounded finline dielectric waveguide.
