A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

Fang, Haw Ren; Saad, Yousef

doi:10.1137/110836535

Cited by 68 publications

(70 citation statements)

References 26 publications

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“…Since 0 ≤ p+q ≤ M , P 2 is a polynomial of A up to degree M , and can be expanded using a Chebyshev polynomial of the form (26).…”

Section: Linmentioning

confidence: 99%

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Randomized estimation of spectral densities of large matrices made accurate

Lin

2016

Numer. Math.

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Abstract. For a large Hermitian matrix A ∈ C N ×N , it is often the case that the only affordable operation is matrix-vector multiplication. In such case, randomized method is a powerful way to estimate the spectral density (or density of states) of A. However, randomized methods developed so far for estimating spectral densities only extract information from different random vectors independently, and the accuracy is therefore inherently limitedwhere Nv is the number of random vectors.In this paper we demonstrate that the "O(1/ √ Nv) barrier" can be overcome by taking advantage of the correlated information of random vectors when properly filtered by polynomials of A. Our method uses the fact that the estimation of the spectral density essentially requires the computation of the trace of a series of matrix functions that are numerically low rank. By repeatedly applying A to the same set of random vectors and taking different linear combination of the results, we can sweep through the entire spectrum of A by building such low rank decomposition at different parts of the spectrum. Under some assumptions, we demonstrate that a robust and efficient implementation of such spectrum sweeping method can compute the spectral density accurately with O(N 2 ) computational cost and O(N ) memory cost. Numerical results indicate that the new method can significantly outperform existing randomized methods in terms of accuracy. As an application, we demonstrate a way to accurately compute a trace of a smooth matrix function, by carefully balancing the smoothness of the integrand and the regularized density of states using a deconvolution procedure.

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“…Since 0 ≤ p+q ≤ M , P 2 is a polynomial of A up to degree M , and can be expanded using a Chebyshev polynomial of the form (26).…”

Section: Linmentioning

confidence: 99%

“…The spectrum sweeping method is not to be confused with another set of methods under the name of "spectrum slicing" methods [22,23,24,25,26,27]. The idea of the spectrum slicing methods is still to obtain a partial diagonalization of the matrix A.…”

Section: Introductionmentioning

confidence: 99%

Randomized estimation of spectral densities of large matrices made accurate

Lin

2016

Numer. Math.

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“…7 The dimensionality n and the number of nonzeros nnz of these matrices are listed in Table 9. Many of these matrices are produced by PARSEC [13], a real space density functional theory based code for electronic structure calculation in which the Hamiltonian is discretized by using finite difference.…”

Section: Comparison On Achieving Higher Accuraciesmentioning

confidence: 99%

“…8 Two recently developed solvers, the filtered Lanczos algorithm in [7] and the feast algorithm in [29], are not included because they are designed to compute all eigenvalues (and eigenvectors) within a given interval. A commonly accepted interface for computing k extreme eigenpairs does not yet exist for either solver, to the best of our knowledge.…”

mentioning

confidence: 99%

An Efficient Gauss--Newton Algorithm for Symmetric Low-Rank Product Matrix Approximations

Liu¹,

Wen²,

Zhang³

2015

SIAM J. Optim.

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We derive and study a Gauss-Newton method for computing a symmetric lowrank product XX T , where X ∈ R n×k for k < n, that is the closest to a given symmetric matrix A ∈ R n×n in Frobenius norm. When A = B T B (or BB T ), this problem essentially reduces to finding a truncated singular value decomposition of B. Our Gauss-Newton method, which has a particularly simple form, shares the same order of iteration-complexity as a gradient method when k n, but can be significantly faster on a wide range of problems. In this paper, we prove global convergence and a Q-linear convergence rate for this algorithm and perform numerical experiments on various test problems, including those from recently active areas of matrix completion and robust principal component analysis. Numerical results show that the proposed algorithm is capable of providing considerable speed advantages over Krylov subspace methods on suitable application problems where high-accuracy solutions are not required. Moreover, the algorithm possesses a higher degree of concurrency than Krylov subspace methods, thus offering better scalability on modern multi-/manycore computers. Introduction.Low-rank matrix approximation techniques, such as dominant eigenvalue or singular value decompositions (SVDs), appear in a wide variety of scientific and engineering applications. For example, principal component analyses (PCA) in statistics use them to compute a few directions of maximal variance. Many data dimensionality reduction methods in machine learning use them to perform low-dimensional embedding of high-dimensional data. More recently, identifying dominant eigenvalue or SVDs of a sequence of closely related matrices has become an indispensable algorithmic component for many first-order optimization methods for various convex optimization problems, such as semidefinite programming, low-rank matrix completion, robust principal component analysis, sparse principal component analysis, sparse inverse covariance matrix estimation, and nearest correlation matrix estimation (see [9] for a more detailed summary). More often than not, the computational cost of doing low-rank approximations forms a major bottleneck in the overall efficiency of solution processes.

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“…When interior eigenvalues are to be computed, spectral transformations are typically employed to move the desired eigenvalues to the ends of the eigenspectrum. Transformations include shift-and-invert operators and various polynomial filters; see, e.g., [1,14]. The computational overhead that is added to the recursive expansion by incorporating iterative eigenvalue methods depends on several factors such as matrix sparsity and hardware and software details.…”

Section: B155mentioning

confidence: 99%

Interior Eigenvalues from Density Matrix Expansions in Quantum Mechanical Molecular Dynamics

Rubensson¹,

Niklasson²

2014

SIAM J. Sci. Comput.

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An accelerated polynomial expansion scheme to construct the density matrix in quantum mechanical molecular dynamics simulations is proposed. The scheme is based on recursive density matrix expansions, e.g., [A. M. N. Niklasson, Phys. Rev. B, 66 (2002), 155115], which are accelerated by a scale-and-fold technique [E. H. Rubensson, J. Chem. Theory Comput., 7 (2011), pp. 1233-1236. The acceleration scheme requires interior eigenvalue estimates, which may be expensive and cumbersome to come by. Here we show how such eigenvalue estimates can be extracted from the recursive expansion by a simple and robust procedure at a negligible computational cost. Our method is illustrated with density functional tight-binding Born-Oppenheimer molecular dynamics simulations, where the computational effort is dominated by the density matrix construction. In our analysis we identify two different phases of the recursive polynomial expansion, the conditioning and purification phases, and we show that the acceleration represents an improvement of the conditioning phase, which typically gives a significant reduction of the computational cost. Introduction.With the fast growth of computational processing power, atomistic simulations based on calculations of the electronic structure have become a powerful approach to the study of a broad range of problems in materials science, chemistry, and biology [26,33,64]. Nevertheless, the computational cost associated with electronic structure calculations normally limits applications to fairly small systems. In particular, the cubic, O(N 3 ), scaling of the computational cost as a function of the number of atoms, N , for the regular solution of the quantum mechanical eigenvalue problem is considered to be a most limiting factor. A number of different electronic structure technologies have therefore been developed that circumvent this bottleneck with a computational effort that scales only linearly with the system size [6,18]. The reduction in the cost is typically achieved by utilizing sparse matrix algebra in an iterative construction of the density matrix, which avoids the full regular solution of the quantum mechanical eigenvalue problem. The matrix sparsity arises in localized atomic basis set representations due to the short-range character of the electronic wavefunctions for nonmetallic materials [3,27,28]. With linear scaling

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A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

Cited by 68 publications

References 26 publications

Randomized estimation of spectral densities of large matrices made accurate

Randomized estimation of spectral densities of large matrices made accurate

An Efficient Gauss--Newton Algorithm for Symmetric Low-Rank Product Matrix Approximations

Interior Eigenvalues from Density Matrix Expansions in Quantum Mechanical Molecular Dynamics

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