Arnoldi iteration

Résumé

In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices. The Arnoldi method belongs to a class of linear algebra algorithms that give a partial result after a small number of iterations, in contrast to so-called direct methods which must complete to give any useful results (see for example, Householder transformation). The partial result in this case being the first few vectors of the basis the algorithm is building. When applied to Hermitian matrices it reduces to the Lanczos algorithm. The Arnoldi iteration was invented by W. E. Arnoldi in 1951. Krylov subspaces and the power iteration An intuitive method for finding the largest (in absolute value) eigenv

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https://en.wikipedia.org/wiki/Arnoldi_iteration

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Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Daniel Kressner

Novel memory-efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree d. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor d. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so-called quadratic Arnoldi method and two-level orthogonal Arnoldi procedure methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift-and-invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree 30 arising from the interpolation of nonlinear eigenvalue problems, which stem from boundary element discretizations of PDE eigenvalue problems. Copyright (C) 2013 John Wiley & Sons, Ltd.

Wiley-Blackwell2014

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Eigensolution of large conservative gyroscopic systems by a subspace forward iteration method

Thomas Gmür

1989

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PowerSGD: Practical Low-Rank Gradient Compression for Distributed Optimization

Martin Jaggi, Thijs Vogels

We study lossy gradient compression methods to alleviate the communication bottleneck in data-parallel distributed optimization. Despite the significant attention received, current compression schemes either do not scale well, or fail to achieve the target test accuracy. We propose a new low-rank gradient compressor based on power iteration that can i) compress gradients rapidly, ii) efficiently aggregate the compressed gradients using all-reduce, and iii) achieve test performance on par with SGD. The proposed algorithm is the only method evaluated that achieves consistent wall-clock speedups when benchmarked against regular SGD using highly optimized off-the-shelf tools for distributed communication. We demonstrate reduced training times for convolutional networks as well as LSTMs on common datasets. Our code is available at https://github.com/epfml/powersgd.

NEURAL INFORMATION PROCESSING SYSTEMS (NIPS)2019

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