Numerical Methods for the QCD Overlap Operator: II. Optimal Krylov SubspaceMethods

Arnold, Guido; Cundy, Nigel; Eshof, Jasper van den; Frommer, Andreas; Krieg, Stefan; Lippert, Thomas; Schäfer, Katrin

doi:10.1007/3-540-28504-0_15

Cited by 24 publications

(48 citation statements)

References 9 publications

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“…The latter type of recurrence relations had previously been applied to iterative methods in [11,12]; see also Arnold et al [2] for a recent application to QCD computations. The derivation of our iterative methods for (1.2) differs from the derivation by Barth and Manteuffel [4] of their schemes in that we do not apply properties of orthogonal polynomials on the unit circle.…”

mentioning

confidence: 99%

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

Beckermann¹,

Reichel²

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. Matrices with a skew-symmetric part of low rank arise in many applications, including path following methods and integral equations. This paper explores the properties of the Arnoldi process when applied to such a matrix. We show that an orthogonal Krylov subspace basis can be generated with short recursion formulas and that the Hessenberg matrix generated by the Arnoldi process has a structure, which makes it possible to derive a progressive GMRES method. Eigenvalue computation is also considered.

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confidence: 99%

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

Beckermann¹,

Reichel²

2008

SIAM J. Matrix Anal. & Appl.

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“…12 Throughout these simulations, we took advantage of relaxation and preconditioning techniques developed in [46,47]. The accuracy of the preconditioner and the number of projected eigenvalues were optimised for our HMC program, which gave a gain factor of ∼ 30.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The transmission update, given by algorithm 2 and using equation (46) can be shown to be area conserving. For simplicity, it is assumed that d 1 is a function of the gauge field at the crossing only and has no additional dependence on Π, and all the other d k s are functions of the gauge field of the crossing and (η, Π) only.…”

Section: A21 Transmissionmentioning

confidence: 99%

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Numerical methods for the QCD overlap operator IV: Hybrid Monte Carlo

Cundy

Krieg

Arnold

et al. 2009

Computer Physics Communications

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The computational costs of calculating the matrix sign function of the overlap operator together with fundamental numerical problems related to the discontinuity of the sign function in the kernel eigenvalues are the major obstacles towards simulations with dynamical overlap fermions using the Hybrid Monte Carlo algorithm. In a previous paper of the present series we introduced optimal numerical approximation of the sign function and have developed highly advanced preconditioning and relaxation techniques which speed up the the inversion of the overlap operator by nearly an order of magnitude.In this fourth paper of the series we construct an HMC algorithm for overlap fermions. We approximate the matrix sign function using the Zolotarev rational approximation, treating the smallest eigenvalues of the Wilson operator exactly within the fermionic force. Based on this we derive the fermionic force for the overlap operator. We explicitly solve the problem of the Dirac delta-function terms arising through zero crossings of eigenvalues of the Wilson operator. The main advantage of this scheme is that its energy violations scale better than O(∆τ 2 ) and thus are comparable with the violations of the standard leapfrog algorithm over the course of a trajectory. We explicitly prove that our algorithm satisfies reversibility and area conservation. We present test results from our algorithm on 4 4 , 6 4 , and 8 4 lattices.

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“…For example, the Wilson-Dirac formulation [19,10] and overlap fermion [36,1] computations both lead to such problems. Deflation of eigenvalues is used for QCD problems, for example in [11,14,13,37,10,33,64,26].…”

Section: Example From Qcdmentioning

confidence: 99%

Deflated and Restarted Symmetric Lanczos Methods for Eigenvalues and Linear Equations with Multiple Right-Hand Sides

Abdel-Rehim¹,

Morgan²,

Nicely³

et al. 2010

SIAM J. Sci. Comput.

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Abstract.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 right-hand sides. Experiments are given with large matrices from quantum chromodynamics that have many right-hand sides.Key words. linear equations, deflation, eigenvalues, Lanczos, conjugate gradient, QCD, multiple right-hand sides, symmetric, Hermitian AMS subject classifications. 65F10, 65F15, 15A06, 15A181. Introduction. We consider a large matrix A that is either real symmetric or complex Hermitian. We are interested in solving the system of equations Ax = b, possibly with multiple right-hand sides, and in solving the associated eigenvalue problem. Both eigenvalues and eigenvectors are desired. Symmetric and Hermitian problems can take advantage of fast algorithms such as the conjugate gradient method (CG) [22,52] for linear equations and the related Lanczos algorithm [25,46] for eigenvalues. However, regular CG can be improved upon for the case of multiple right-hand sides, and Lanczos may have storage and accuracy issues while computing eigenvectors. We give new methods for these problems.An approach is presented called Lanczos with deflated restarting or Lan-DR. It simultaneously solves the linear equations and computes the eigenvalues and eigenvectors. A restarted Krylov subspace approach is used for the linear equations, but it also saves approximate eigenvectors at the restart and uses them in the subspace for the next cycle. The restarting of the Lanczos algorithm does not slow down the convergence as it normally would, because once the approximate eigenvectors are accurate enough, they essentially remove or deflate the associated eigenvalues from the problem. The eigenvalue portion of Lan-DR has already been presented in [68], but as mentioned, we add on the solution of linear equations. Also, some reorthogonalization is necessary to control roundoff error. We give some new approaches for this reorthogonalization. We also give a Minres/harmonic version of the algorithm.

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Numerical Methods for the QCD Overlap Operator: II. Optimal Krylov SubspaceMethods

Cited by 24 publications

References 9 publications

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

Numerical methods for the QCD overlap operator IV: Hybrid Monte Carlo

Deflated and Restarted Symmetric Lanczos Methods for Eigenvalues and Linear Equations with Multiple Right-Hand Sides

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