A Parallel Davidson-Type Algorithm for Several Eigenvalues

Borges, Leonardo; Oliveira, Suely

doi:10.1006/jcph.1998.6003

Cited by 13 publications

(5 citation statements)

References 29 publications

(49 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Every new preconditioned solver for finding an extreme eigenpair should be compared with the "ideal" algorithm in terms of the speed of convergence, costs of every iteration, and memory requirements. As the number of publications on different preconditioned eigenvalue solvers and their applications, e.g., recent papers [36,43,44,24,2,34,33], keeps growing rapidly, a need for such benchmarking becomes evident.…”

Section: Conclusion Let Us Formulate Here the Main Points Of The Papermentioning

confidence: 99%

Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method

Knyazev

2001

SIAM J. Sci. Comput.

660

588

View full text Add to dashboard Cite

We describe new algorithms of the locally optimal block preconditioned conjugate gradient (LOBPCG) method for symmetric eigenvalue problems, based on a local optimization of a three-term recurrence, and suggest several other new methods. To be able to compare numerically different methods in the class, with different preconditioners, we propose a common system of model tests, using random preconditioners and initial guesses. As the "ideal" control algorithm, we advocate the standard preconditioned conjugate gradient method for finding an eigenvector as an element of the null-space of the corresponding homogeneous system of linear equations under the assumption that the eigenvalue is known. We recommend that every new preconditioned eigensolver be compared with this "ideal" algorithm on our model test problems in terms of the speed of convergence, costs of every iteration, and memory requirements. We provide such comparison for our LOBPCG method. Numerical results establish that our algorithm is practically as efficient as the "ideal" algorithm when the same preconditioner is used in both methods. We also show numerically that the LOBPCG method provides approximations to first eigenpairs of about the same quality as those by the much more expensive global optimization method on the same generalized block Krylov subspace. We propose a new version of block Davidson's method as a generalization of the LOBPCG method. Finally, direct numerical comparisons with the Jacobi-Davidson method show that our method is more robust and converges almost two times faster.

show abstract

Section: Conclusion Let Us Formulate Here the Main Points Of The Papermentioning

confidence: 99%

Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method

Knyazev

2001

SIAM J. Sci. Comput.

660

588

View full text Add to dashboard Cite

show abstract

“…General surveys of these techniques can be found in [8,15,25,28]. In our work we introduce Davidson-type algorithms [4,7,24] as a basis for graph partitioning algorithms. Furthermore, we use graph theory to incorporate information about the problem into the Davidson eigensolver.…”

Section: The Davidson's Eigensolvermentioning

confidence: 99%

“…Relax ri2 times the system Ljtj = yj using Gauss-Seidel. 4. Return to as an approximate solution to Loto = ro- Figure 1 illustrates our multilevel preconditioner for a graph which has been coarsened to four levels.…”

Section: Algorithm 2 -Multilevel Preconditioner For Laplacian Systemmentioning

confidence: 99%

A Graph Based Davidson Algorithm for the Graph Partitioning Problem

Holzrichter

Oliveira

1999

Int. J. Found. Comput. Sci.

Self Cite

View full text Add to dashboard Cite

The problem of partitioning a graph such that the number of edges incident to vertices in different partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing fill-in in matrix factorizations and loadbalancing for parallel algorithms. Spectral graph partitioning algorithms partition a graph using the eigenvector associated with the second smallest eigenvalue of a matrix called the graph Laplacian. The focus of this paper is the use graph theory to compute this eigenvector more quickly. Keywords: Graph partitioning algorithms, spectral algorithms, Fiedler vector, eigenvalue solvers. Int. J. Found. Comput. Sci. 1999.10:225-246. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only. X 1 A 1 * 1 J j " 3n \, • %-Int. J. Found. Comput. Sci. 1999.10:225-246. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only.

show abstract

“…The computation of the smallest eigenvalues of large sparse symmetric matrices is of considerable interest in quantum chemistry and quantum physics applications, and there are other types of methods that are commonly used. In particular, there are procedures [6,11,13,21,22,24] based on Davidson's method [9] and procedures [3,8,7,15,16,23] based on the Lanczos method. Of particular interest are the procedures described in [24], where adaptive Chebyshev filtering is used in conjunction with Davidson's method and [3] where Chebyshev filtering is used in conjunction with a Lanczos procedure.…”

Section: Introductionmentioning

confidence: 99%

A Rayleigh–Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices

Anderson¹

2010

Journal of Computational Physics

View full text Add to dashboard Cite

A Parallel Davidson-Type Algorithm for Several Eigenvalues

Cited by 13 publications

References 29 publications

Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method

Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method

A Graph Based Davidson Algorithm for the Graph Partitioning Problem

A Rayleigh–Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices

Contact Info

Product

Resources

About