Numerical methods for the QCDd overlap operator. I. Sign-function and error bounds

By J. V. D. Eshof, A. Frommer, T. Lippert, K. Schilling, Henk A. van der Vorst. 2002

177 citations recorded by the source corpus.

Abstract The numerical and computational aspects of the overlap formalism in lattice quantum chromodynamics are extremely demanding due to a matrix–vector product that involves the sign function of the Hermitian Wilson matrix. In this paper we investigate several methods to compute the product of the matrix sign-function with a vector, in particular Lanczos based methods and partial fraction expansion methods. Our goal is two-fold: we give realistic comparisons between known methods together with novel approaches and we present error bounds which allow to guarantee a given accuracy when terminating the Lanczos method and the multishift-CG solver, applied within the partial fraction expansion methods.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/bbffaf5792dd1f59701f67dacad9806616faf3bb

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In Nuclear and High Energy Physics

Citations in the corpus

A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions (2013, methodology): “Functions of this type also arise in the context of computation of Neumann-to-Dirichlet and Dirichlet-to-Neumann maps [16, 3], the solution of systems of stochastic differential equations [2], and in quantum chromodynamics [22].”
A restarted Lanczos approximation to functions of a symmetric matrix (2010, methodology): “It was found in [ 23 ] that the use of f(t) = sign(t) in (1.1) leads to rather erratic convergence.”
Light Hadron Masses from Lattice QCD (2012, background): “While it is possible to do this with polynomial (Giusti et al., 2003) or rational (Edwards et al., 1999; van den Eshof et al., 2002; Neuberger, 1998a) approximations, it typically requires at least O(100) applications of the kernel Wilson operator to perform one matrix-vector multiplication with…”
Iterative Krylov Methods for Large Linear Systems (2003, result): “Our preliminary experiments with this approach have been encouraging, for more details see [190].”