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25
Computing and deflating eigenvalues while solving multiple right hand side linear systems with an application to quantum chromodynamics
, 2008
"... Abstract. We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the t ..."
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Abstract. We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm capitalizes on the iteration vectors produced by CG to update only a small window of about ten vectors that approximate the eigenvectors. While this window is restarted in a locally optimal way, the CG algorithm for the linear system is unaffected. Yet, in all our experiments, this small window converges to the required eigenvectors at a rate identical to unrestarted Lanczos. After the solution of the linear system, eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems. In this case, eigenvectors identified in earlier systems can be used to deflate, and thus accelerate, the convergence of subsequent systems. We have used this algorithm with excellent results in lattice QCD applications, where hundreds of right hand sides may be needed. Specifically, about 70 eigenvectors are obtained to full accuracy after solving 24 right hand sides. Deflating these from the large number of subsequent right hand sides removes the dreaded critical slowdown, where the conditioning of the matrix increases as the quark mass reaches a critical value. Our experiments show almost a constant number of iterations for our method, regardless of quark mass, and speedups of 8 over original CG for light quark masses.
PRIMME: PReconditioned Iterative Multimethod Eigensolver: METHODS AND SOFTWARE DESCRIPTION
, 2006
"... This paper describes the PRIMME software package for the solving large, sparse Hermitian and real symmetric eigenvalue problems. The difficulty and importance of these problems have increased over the years, necessitating the use of preconditioning and near optimally converging iterative methods. O ..."
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Cited by 15 (6 self)
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This paper describes the PRIMME software package for the solving large, sparse Hermitian and real symmetric eigenvalue problems. The difficulty and importance of these problems have increased over the years, necessitating the use of preconditioning and near optimally converging iterative methods. On the other hand, the complexity of tuning or even using such methods has kept them outside the reach of many users. Responding to this problem, our goal was to develop a general purpose software that requires minimal or no tuning, yet it provides the best possible robustness and efficiency. PRIMME is a comprehensive package that brings stateoftheart methods from “bleeding edge ” to production, with a flexible, yet highly usable interface. We review the theory that gives rise to the near optimal methods GD+k and JDQMR, and present the various algorithms that constitute the basis of PRIMME. We also describe the software implementation, interface, and provide some sample experimental results.
JADAMILU: a software code for computing selected eigenvalues of large sparse symmetric matrices
, 2007
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Adaptive Projection Subspace Dimension for the ThickRestart Lanczos Method
"... The ThickRestart Lanczos (TRLan) method is an effective method for solving largescale Hermitian eigenvalue problems. The performance of the method strongly depends on the dimension of the projection subspace used at each restart. In this article, we propose an objective function to quantify the ef ..."
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Cited by 9 (3 self)
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The ThickRestart Lanczos (TRLan) method is an effective method for solving largescale Hermitian eigenvalue problems. The performance of the method strongly depends on the dimension of the projection subspace used at each restart. In this article, we propose an objective function to quantify the effectiveness of the selection of subspace dimension, and then introduce an adaptive scheme to dynamically select the dimension to optimize the performance. We have developed an opensource software package a–TRLan to include this adaptive scheme in the TRLan method. When applied to calculate the electronic structure of quantum dots, a–TRLan runs up to 2.3x faster than a stateoftheart preconditioned conjugate gradient eigensolver.
Solving Largescale Eigenvalue Problems in SciDAC Applications
"... Abstract. Largescale eigenvalue problems arise in a number of DOE applications. This paper provides an overview of the recent development of eigenvalue computation in the context of two SciDAC applications. We emphasize the importance of Krylov subspace methods, and point out its limitations. We di ..."
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Abstract. Largescale eigenvalue problems arise in a number of DOE applications. This paper provides an overview of the recent development of eigenvalue computation in the context of two SciDAC applications. We emphasize the importance of Krylov subspace methods, and point out its limitations. We discuss the value of alternative approaches that are more amenable to the use of preconditioners, and report the progress on using the multilevel algebraic substructuring techniques to speed up eigenvalue calculation. In addition to methods for linear eigenvalue problems, we also examine new approaches to solving two types of nonlinear eigenvalue problems arising from SciDAC applications. 1.
Controlling inner iterations in the Jacobi–Davidson method
, 2006
"... The Jacobi–Davidson method is an eigenvalue solver which uses the iterative (and in general inaccurate) solution of inner linear systems to progress, in an outer iteration, towards a particular solution of the eigenproblem. In this paper we prove a relation between the residual norm of the inner lin ..."
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Cited by 7 (0 self)
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The Jacobi–Davidson method is an eigenvalue solver which uses the iterative (and in general inaccurate) solution of inner linear systems to progress, in an outer iteration, towards a particular solution of the eigenproblem. In this paper we prove a relation between the residual norm of the inner linear system and the residual norm of the eigenvalue problem. We show that the latter may be estimated inexpensively during the inner iterations. On this basis, we propose a stopping strategy for the inner iterations to maximally exploit the strengths of the method. These results extend previous results obtained for the special case of Hermitian eigenproblems with the conjugate gradient or the symmetric QMR method as inner solver. The present analysis applies to both standard and generalized eigenproblems, does not require symmetry, and is compatible with most iterative methods for the inner systems. It is also easily extended to other type of inner–outer eigenvalue solvers, as inexact inverse iteration or inexact Rayleigh quotient iteration. The effectiveness of our approach is illustrated by a few numerical experiments, including the comparison of a standard Jacobi–Davidson code with the same code enhanced by our stopping strategy.
Block KrylovSchur method for large symmetric eigenvalue problems, tech
, 2004
"... Abstract. Stewart’s recent KrylovSchur algorithm offers two advantages over Sorensen’s implicitly restarted Arnoldi (IRA) algorithm. The first is ease of deflation of converged Ritz vectors, the second is the avoidance of the potential forward instability of the QR algorithm. In this paper we devel ..."
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Cited by 5 (1 self)
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Abstract. Stewart’s recent KrylovSchur algorithm offers two advantages over Sorensen’s implicitly restarted Arnoldi (IRA) algorithm. The first is ease of deflation of converged Ritz vectors, the second is the avoidance of the potential forward instability of the QR algorithm. In this paper we develop a block version of the KrylovSchur algorithm for symmetric eigenproblems. Details of this block algorithm are discussed, including how to handle the rank deficient cases and how to use different block sizes. Numerical results on the efficiency of the block KrylovSchur method are reported.
Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple righthand sides
, 2009
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TracePenalty Minimization for Largescale Eigenspace Computation
, 2013
"... The RayleighRitz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively highdimensional eigenspaces of large sparse matrices. Although operations involved in RR steps can be parallelized to a certain level, their parallel scalability, which is limited b ..."
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Cited by 3 (2 self)
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The RayleighRitz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively highdimensional eigenspaces of large sparse matrices. Although operations involved in RR steps can be parallelized to a certain level, their parallel scalability, which is limited by some inherent sequential steps, is lower than dense matrixmatrix multiplications. The primary motivation of this paper is to develop a methodology that reduces the use of the RR procedure in exchange for matrixmatrix multiplications. We propose an unconstrained penalty model and establish its equivalence to the eigenvalue problem. This model enables us to deploy gradienttype algorithms that makes heavy use of dense matrixmatrix multiplications. Although the proposed algorithm does not necessarily reduce the total number of arithmetic operations, it leverages highly optimized operations on modern high performance computers to achieve parallel scalability. Numerical results based on a preliminary implementation, parallelized using OpenMP, show that our approach is promising.