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23
An ArnoldiSchur Algorithm for Large Eigenproblems
, 2000
"... Sorensen's iteratively restarted Arnoldi algorithm is one of the most successful and flexible methods for finding a few eigenpairs of a large matrix. However, the need to preserve structure of the Arnoldi decomposition, on which the algorithm is based, restricts the range of transformations ..."
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Cited by 64 (2 self)
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Sorensen's iteratively restarted Arnoldi algorithm is one of the most successful and flexible methods for finding a few eigenpairs of a large matrix. However, the need to preserve structure of the Arnoldi decomposition, on which the algorithm is based, restricts the range of transformations that can be performed on it. In consequence, it is difficult to deflate converged Ritz vectors from the decomposition. Moreover, the potential forward instability of the implicit QR algorithm can cause unwanted Ritz vectors to persist in the computation. In this paper we introduce a generalized Arnoldi decomposition that solves both problems in a natural and efficient manner.
GMRES with deflated restarting
 SIAM J. Sci. Comput
"... Abstract. A modification is given of the GMRES iterative method for nonsymmetric systems of linear equations. The new method deflates eigenvalues using Wu and Simon’s thick restarting approach. It has the efficiency of implicit restarting, but is simpler and does not have the same numerical concerns ..."
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Cited by 60 (8 self)
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Abstract. A modification is given of the GMRES iterative method for nonsymmetric systems of linear equations. The new method deflates eigenvalues using Wu and Simon’s thick restarting approach. It has the efficiency of implicit restarting, but is simpler and does not have the same numerical concerns. The deflation of small eigenvalues can greatly improve the convergence of restarted GMRES. Also, it is demonstrated that using harmonic Ritz vectors is important, because then the whole subspace is a Krylov subspace that contains certain important smaller subspaces.
NEARLY OPTIMAL PRECONDITIONED METHODS FOR HERMITIAN EIGENPROBLEMS UNDER LIMITED MEMORY. PART II: SEEKING MANY EIGENVALUES
, 2006
"... In a recent companion paper, we proposed two methods, GD+k and JDQMR, as nearly optimal methods for finding one eigenpair of a real symmetric matrix. In this paper, we seek nearly optimal methods for a large number, nev, of eigenpairs, that work with a search space whose size is O(1), independent ..."
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Cited by 25 (7 self)
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In a recent companion paper, we proposed two methods, GD+k and JDQMR, as nearly optimal methods for finding one eigenpair of a real symmetric matrix. In this paper, we seek nearly optimal methods for a large number, nev, of eigenpairs, that work with a search space whose size is O(1), independent from nev. The motivation is twofold: avoid the additional O(nevN) storage, and the O(nev 2 N) iteration costs. First, we provide an analysis of the oblique projectors required in the JacobiDavidson method, and we identify ways to avoid them during the inner iterations, either completely, or partially. Second, we develop a comprehensive set of performance models for GD+k, JacobiDavidson type methods, and ARPACK. Based both on theoretical arguments and on our models we argue that any eigenmethod with O(1) basis size, preconditioned or not, will be superseded asymptotically by Lanczos type methods that use O(nev) vectors in the basis. However, this may not happen until nev> O(1000). Third, we perform an extensive set of experiments with our methods and against other stateoftheart software that validate our models, and confirm our GD+k and JDQMR methods as nearly optimal within the class of O(1) basis size methods.
Deflated iterative methods for linear equations with multiple righthand sides
, 2004
"... Abstract. A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple righthand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems ar ..."
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Cited by 20 (7 self)
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Abstract. A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple righthand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems are solved by combining restarted GMRES with a projection over the previously determined eigenvectors. This approach offers an alternative to block methods, and it can also be combined with a block method. It is useful when there are a limited number of small eigenvalues that slow the convergence. An example is given showing significant improvement for a problem from quantum chromodynamics. The second and subsequent righthand sides are solved much quicker than without the deflation. This new approach is relatively simple to implement and is very efficient compared to other deflation methods.
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.
Deflated GMRES for systems with multiple shifts and multiple righthand sides
, 2007
"... Abstract. We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple righthand sides. First, for a single righthand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including latti ..."
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Cited by 15 (1 self)
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Abstract. We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple righthand sides. First, for a single righthand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and nonHermitian. Some Krylov iterative methods such as GMRES and BiCGStab have been used to solve multiply shifted systems for about the cost of solving just one system. Restarted GMRES can be improved by deflating eigenvalues for matrices that have a few small eigenvalues. We show that a particular deflated method, GMRESDR, can be applied to multiply shifted systems. In quantum chromodynamics, it is common to have multiple righthand sides with multiple shifts for each righthand side. We develop a method that efficiently solves the multiple righthand sides by using a deflated version of GMRES and yet keeps costs for all of the multiply shifted systems close to those for one shift. An example is given showing this can be extremely effective with a quantum chromodynamics matrix.
Multiple explicitly restarted Arnoldi method for solving large eigenproblems
 SIAM Journal on scientific computing SJSC, Volume 27, Number
, 2005
"... In this paper we propose a new approach for calculating some eigenpairs of large sparse nonHermitian matrices. This method, called Multiple Explicitly Restarted Arnoldi (MERAM), is particularly well suited for environments that combine different parallel programming paradigms. This technique is bas ..."
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Cited by 8 (5 self)
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In this paper we propose a new approach for calculating some eigenpairs of large sparse nonHermitian matrices. This method, called Multiple Explicitly Restarted Arnoldi (MERAM), is particularly well suited for environments that combine different parallel programming paradigms. This technique is based on a multiple use of Explicitly Restarted Arnoldi method and improves its convergence. This technique is implemented and tested on a distributed enviromnment consisting of two interconnected parallel machines. MERAM technique is compared to Explicitly Restarted Arnoldi (ERAM) method, and one can notice that the accelaration of convergence is improved effectively. In some cases, more than one twofold improvement can be seen in MERAM results. We also implemented MERAM on a cluster of workstations. According to our experiments, MERAM converges better than Explicitly Restarted Block Arnoldi method and, for some matrices, more quickly than PARPACK package which implements Implicitly Restarted Arnoldi Method. Key words. Large eigenproblem, Arnoldi method, explicit restarting, parallel programming, asynchronous communication, heterogeneous environment.
DEFLATED AND RESTARTED SYMMETRIC LANCZOS METHODS FOR EIGENVALUES AND LINEAR EQUATIONS WITH MULTIPLE Righthand Sides
, 2008
"... A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equat ..."
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Cited by 8 (5 self)
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A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is necessary to control roundoff error, and several approaches are discussed. The eigenvectors generated while solving the linear equations can be used to help solve systems with multiple righthand sides. Experiments are given with large matrices from quantum chromodynamics that have many righthand sides.