On the Neuberger overlap operator

“…To check that our results are not biased by the choice of the ǫ matrix expansion, we have repeated the computation of the low-lying spectrum on some of the configurations with the Lanczos algorithm proposed in [50]. The results obtained by both methods are compatible within errors.…”

Section: Lattice Implementationmentioning

confidence: 93%

Ultraviolet filtering of lattice configurations and applications to Monte Carlo dynamics

Pérez

González-Arroyo

Sastre

2011

J. High Energ. Phys.

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We present a detailed study of a filtering method based upon Dirac quasi-zeromodes in the adjoint representation. The procedure induces no distortions on configurations which are solutions of the euclidean classical equations of motion. On the other hand, it is very effective in reducing the short-wavelength stochastic noise present in Monte-Carlo generated configurations. After testing the performance of the method in various situations, we apply it successfully to study the effect of Monte-Carlo dynamics on topological structures like instantons.

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“…The convergence is measured using the norm of the residual error. For the overlap matrix-vector multiplication is used the double pass Lanczos algorithm (without small eigenspace projection of H W ) as described in [9]. Together with the algorithms described in the previous sections Fig.…”

Section: Comparison Of Methodsmentioning

confidence: 99%

The two-grid algorithm confronts a shifted unitary orthogonal method

Boriçi

2005

Nuclear Physics B - Proceedings Supplements

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In this paper I describe a new optimal Krylov subspace solver for shifted unitary matrices called the Shifted Unitary Orthogonal Method (SUOM). This algorithm is used as a benchmark against any improvement like the two-grid algorithm. I use the latter to show that the overlap operator can be inverted by successive inversions of the truncated overlap operator. This strategy results in large gains compared to SUOM.It is well-known that overlap fermions [1] lead to much more expensive computations than standard fermions, i.e. Wilson or Kogut-Sussking fermions. This is obvious since for every application of the overlap operator an extra linear system solving is needed. For the time being, it seems that to get chiral symmetry at finite lattice spacing one should wait for a Petaflops computer being built.However, algorithmic research is far from exhausted. In this paper I give an example that this is the case if one uses the two-grid algorithm [2]. Before I do this, I introduce briefly an optimal Krylov subspace solver for shifted unitary matrices. SUOM: A NEW OPTIMAL KRYLOV SOLVERConsider the task of solving the linear system:where V = γ 5 sign(H W ) is a unitary matrix, 1l the identity matrix, H W the Hermitian Wilson operator, c 1 = (1 + m q )/2, c 2 = (1 − m q )/2 and m q the bare fermion mass. The overlap operator D is non-Hermitian. For such operators GMRES (Generalised Minimal Residual) and FOM (Full Orthogonalisation Method) are known to be the fastest. It is shown that when the norm-minimising process of GMRES is converging rapidly, the residual norms in the corresponding Galerkin process of FOM exhibit similar behaviour [3]. But they are based on long recurrences and thus require to store a large number of vectors of the size of matrix columns. However, exploiting the fact that the overlap operator is a shifted unitary matrix one can construct a GM-RES type algorithm with short recurrences [4]. Similarly, a short recurrences algorithm can be obtained from FOM. The method is based on an observation of Rutishauser [5] that for upper Hessenberg unitary matrices one can write H = LU −1 , where L and U are lower and upper bidiagonal matrices. Applying this decomposition for the Arnoldi iteration:one obtains an algorithm which constructs Arnoldi vectors Q k by short recurrences [6]:(1.3)Projecting the linear system (1.1) onto the Krylov subspace one gets:k )y k = e 1 (1.4) which can be equivalently written as:Note that the matrix on the left hand side is tridiagonal. It can be shown that one can solve 1

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On the Neuberger overlap operator

Abstract: We compute Neuberger's overlap operator by the Lanczos algorithm applied to the Wilson-Dirac operator. Locality of the operator for quenched QCD data and its eigenvalue spectrum in an instanton background are studied.Comment: Revised version: 16 pages, 5 figures, algorithm adde

Cited by 48 publications

References 14 publications

Truncated overlap fermions

Truncated overlap fermions

Ultraviolet filtering of lattice configurations and applications to Monte Carlo dynamics

The two-grid algorithm confronts a shifted unitary orthogonal method

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