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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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Cited by 20 (2 self)
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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.
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.
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.
Complementary cycles of restarted GMRES
 Numerical Linear Algebra with Applications
"... Abstract. Restarted GMRES is one of the most popular methods for solving large nonsymmetric linear systems. It is generally thought that the information of previous GMRES cycles is lost at the time of a restart, so each cycle contributes to the global convergence individually. However, this is not t ..."
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Cited by 5 (0 self)
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Abstract. Restarted GMRES is one of the most popular methods for solving large nonsymmetric linear systems. It is generally thought that the information of previous GMRES cycles is lost at the time of a restart, so each cycle contributes to the global convergence individually. However, this is not the full story. In this paper, we shed light on the relationship between different GMRES cycles. It is shown that successive GMRES cycles can complement one another harmoniously. These groups of cycles, called complementary cycles, are defined and studied. Key words. Ritz values nonsymmetric linear systems, iterative methods, GMRES, restarting, harmonic
Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple righthand sides
, 2009
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Spectral deflation in Krylov solvers: A theory of coordinate space based methods
 ETNA
"... Abstract. For the iterative solution of large sparse linear systems we develop a theory for a family of augmented and deflated Krylov space solvers that are coordinate based in the sense that the given problem is transformed into one that is formulated in terms of the coordinates with respect to the ..."
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Cited by 4 (1 self)
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Abstract. For the iterative solution of large sparse linear systems we develop a theory for a family of augmented and deflated Krylov space solvers that are coordinate based in the sense that the given problem is transformed into one that is formulated in terms of the coordinates with respect to the augmented bases of the Krylov subspaces. Except for the augmentation, the basis is as usual generated by an Arnoldi or Lanczos process, but now with a deflated, singular matrix. The idea behind deflation is to explicitly annihilate certain eigenvalues of the system matrix, typically eigenvalues of small absolute value. The deflation of the matrix is based on an either orthogonal or oblique projection on a subspace that is complimentary to the deflated approximately invariant subspace. While an orthogonal projection allows us to find minimal residual norm solutions, the oblique projections, which we favor when the matrix is nonHermitian, allow us in the case of an exactly invariant subspace to correctly deflate both the right and the corresponding left (possibly generalized) eigenspaces of the matrix, so that convergence only depends on the nondeflated eigenspaces. The minimality of the residual is replaced by the minimality of a quasiresidual. Among the methods that we treat are primarily deflated versions of GMRES, MINRES, and QMR, but we also extend our approach to deflated, coordinate space based versions of other Krylov space methods including variants of CG and BICG. Numerical results will be published elsewhere.
IMPROVED SEED METHODS FOR SYMMETRIC POSITIVE DEFINITE LINEAR EQUATIONS WITH MULTIPLE RIGHTHAND
, 810
"... Abstract. We consider symmetric positive definite systems of linear equations with multiple righthand sides. The seed conjugate gradient method solves one righthand side with the conjugate gradient method and simultaneously projects over the Krylov subspace thus developed for the other righthand ..."
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Cited by 3 (1 self)
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Abstract. We consider symmetric positive definite systems of linear equations with multiple righthand sides. The seed conjugate gradient method solves one righthand side with the conjugate gradient method and simultaneously projects over the Krylov subspace thus developed for the other righthand sides. Then the next system is solved and used to seed the remaining ones. Rounding error in the conjugate gradient method limits how much the seeding can improve convergence. We propose three changes to the seed conjugate gradient method: only the first righthand side is used for seeding, this system is solved past convergence, and the roundoff error is controlled with some reorthogonalization. We will show that results are actually better with only one seeding, even in the case of related righthand sides. Controlling rounding error gives the potential for rapid convergence for the second and subsequent righthand sides.
RESTARTING THE NONSYMMETRIC LANCZOS ALGORITHM for Eigenvalues . . .
"... A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalus and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding nea ..."
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A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalus and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding nearbreakdown and discuss maintaining biorthogonality. A system of linear equations can be solved simultaneously with the eigenvalue computations. Deflation from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. The right and left eigenvectors generated while solving the linear equations can be used to help solve systems with multiple righthand sides.
unknown title
, 704
"... An iterative method to compute the sign function of a nonHermitian matrix and its application to the overlap Dirac operator at nonzero chemical potential ..."
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An iterative method to compute the sign function of a nonHermitian matrix and its application to the overlap Dirac operator at nonzero chemical potential
c ○ Copyright owned by the author(s) under the terms of the Creative Commons AttributionNonCommercialShareAlike Licence.
"... A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple righthand sides, is described. For the first righthand side, eigenvectors with small eigenvalues are computed while simultaneously sol ..."
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A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple righthand sides, is described. For the first righthand side, eigenvectors with small eigenvalues are computed while simultaneously solving the linear system. Two versions of this algorithm are given. The first is called LanDR and is based on conjugate gradient (CG) implementation of the Lanczos algorithm. This version will be optimal for the hermitian positive definite case. The second version is called MinResDR and is based on the minimum residual (MinRes) implementation of Lanczos algorithm. This version is optimal for indefinite hermitian systems where the CG algorithm is subject to instabilities. For additional righthand sides, we project over the calculated eigenvectors to speed up convergence. The algorithms used for subsequent righthand sides are called DCG and DMinRes respectively. After some introductory examples are given, we show tests for the case of Wilson fermions at kappa critical. A considerable speed up in the convergence is observed compared to unmodified CG and MinRes.