arXiv subjects

Walter Wilcox

Publications and source records attributed to Walter Wilcox.

At least 55 records · Page 3Linked to original sources

arXiv2007

Deflated GMRES for Systems with Multiple Shifts and Multiple Right-Hand Sides

We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple right-hand sides. First, for a single right-hand 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 non-Hermitian. 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, GMRES-DR, can be applied to multiply shifted systems. In quantum chromodynamics, it is common to have multiple right-hand sides with multiple shifts for each right-hand side. We develop a method that efficiently solves the multiple right-hand 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.

math-ph↗

arXiv2005

Disconnected Loops with Twisted Mass Lattice QCD

We give a general introduction and discussion of the issues involved in using the twisted mass formulation of lattice fermions in the context of disconnected loop calculations, including a short orientation on the present experimental situation for nucleon strange quark form factors. A prototype calculation of the disconnected part of the nucleon scalar form factor is described.

hep-lat↗

arXiv2004

Electric Polarizability of Neutral Hadrons from Lattice QCD

By simulating a uniform electric field on a lattice and measuring the change in the rest mass, we calculate the electric polarizability of neutral mesons and baryons using the methods of quenched lattice QCD. Specifically, we measure the electric polarizability coefficient from the quadratic response to the electric field for 10 particles: the vector mesons $ρ^0$ and $K^{*0}$; the octet baryons n, $Σ^0$, $Λ_{o}^{0}$, $Λ_{s}^{0}$, and $Ξ^0$; and the decouplet baryons $Δ^0$, $Σ^{*0}$, and $Ξ^{*0}$. Independent calculations using two fermion actions were done for consistency and comparison purposes. One calculation uses Wilson fermions with a lattice spacing of $a=0.10 $fm. The other uses tadpole improved Lüsher-Weiss gauge fields and clover quark action with a lattice spacing $a=0.17 $fm. Our results for neutron electric polarizability are compared to experiment.

hep-lat↗

arXiv2004

Deflated Iterative Methods for Linear Equations with Multiple Right-Hand Sides

A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple right-hand 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 an iterative method with a projection over the previously determined eigenvectors. Restarted GMRES is considered for the iterative method as well as non-restarted methods such as BiCGSTAB. These methods offer an alternative to block methods, and they can also be combined with a block approach. An example is given showing significant improvement for a problem from quantum chromodynamics.

math-ph↗

arXiv2003

Lattice Results on the Connected Neutron Charge Radius

We describe a calculation using quenched lattice QCD of the connected part of the neutron electric form factor for momentum transfers in the range $0.3 {\rm GeV^2} \stackrel{<}{\sim} -q^2 \stackrel{<}{\sim} 1.0 {\rm GeV^2}$. We extract the implied charge radius using a Galster parameterization and consider various ways of extrapolating the neutron charge radius value to the chiral limit. We find that the measured charge radii may be reconciled to experiment by standard phenomenology and lowest or next to lowest order contributions from chiral perturbation theory.

hep-lat↗

arXiv2003

Deflation of Eigenvalues for Iterative Methods in Lattice QCD

Work on generalizing the deflated, restarted GMRES algorithm, useful in lattice studies using stochastic noise methods, is reported. We first show how the multi-mass extension of deflated GMRES can be implemented. We then give a deflated GMRES method that can be used on multiple right-hand sides of $Ax=b$ in an efficient manner. We also discuss and give numerical results on the possibilty of combining deflated GMRES for the first right hand side with a deflated BiCGStab algorithm for the subsequent right hand sides.

hep-lat↗

arXiv2002

Electric Polarizability of Hadrons

The electric polarizability of a hadron allows an external electric field to shift the hadron mass. We try to calculate the electric polarizability for several hadrons from their quadratic response to the field at a=0.17fm using an improved gauge field and the clover quark action. Results are compared to experiment where available.

hep-lat↗

arXiv2002

Continuum Moment Equations on the Lattice

An analysis is given as to why one can not directly evaluate continuum moment equations, i.e., equations involving powers of the position variable times charge, current, or energy/momentum operators, on the lattice. I examine two cases: a three point function evaluation of the nucleon magnetic moment and a four point function (charge overlap) evaluation of the pseudoscalar charge radius.

hep-lat↗

arXiv2001

Heatbath Noise Methods in Lattice QCD

In a recent paper, de Forcrand has pointed out that matrix inversions using Gaussian noise need not be iterated to full convergence, but instead may be solved approximately and treated as a heatbath. Gaussian noise however is not optimal for minimizing variance. It shown here how his algorithm may be generalized to a mixture of Gaussian and Z(N) noise, resulting in a smaller effective variance for some operators.

hep-lat↗

arXiv2001

Deflation of Eigenvalues for GMRES in Lattice QCD

Versions of GMRES with deflation of eigenvalues are applied to lattice QCD problems. Approximate eigenvectors corresponding to the smallest eigenvalues are generated at the same time that linear equations are solved. The eigenvectors improve convergence for the linear equations, and they help solve other right-hand sides.

hep-lat↗

arXiv2000

Disconnected Electromagnetic Form Factors

Preliminary results of a calculation of disconnected nucleon electromagnetic factors factors on the lattice are presented. The implementation of the numerical subtraction scheme is outlined. A comparison of results for electric and magnetic disconnected form factors on two lattice sizes with those of the Kentucky group is presented. Unlike previous results, the results found in this calculation are consistent with zero in these sectors.

hep-lat↗

arXiv1999

Noise Methods for Flavor Singlet Quantities

A discussion of methods for reducing the noise variance of flavor singlet quantities ("disconnected diagrams") in lattice QCD is given. After an introduction, the possible advantage of partitioning the Wilson fermion matrix into disjoint spaces is discussed and a numerical comparison of the variance for three possible partitioning schemes is carried out. The measurement efficiency of lattice operators is examined and shown to be strongly influenced by the Dirac and color partitioning choices. Next, the numerical effects of an automated subtraction algorithm on the noise variance of various disconnected loop matrix elements are examined. It is found that there is a dramatic reduction in the variance of the Wilson point-split electromagnetic currents and that this reduction persists at small quark mass.

hep-lat↗

arXiv1999

Finite Volume Effects in Self Coupled Geometries

By integrating the pressure equation at the surface of a self coupled curvilinear boundary, one may obtain asymptotic estimates of energy shifts, which is especially useful in lattice QCD studies of nonrelativistic bound states. Energy shift expressions are found for periodic (antiperiodic) boundary conditions on antipodal points, which require Neumann (Dirichlet) boundary conditions for even parity states and Dirichlet (Neumann) boundary conditions for odd parity states. It is found that averaging over periodic and antiperiodic boundary conditions is an effective way of removing the asymptotic energy shifts from the boundary. Asymptotic energy shifts from boxes with self coupled walls are also considered and shown to be effectively antipodal. The energy shift equations are illustrated by the solution of the bounded harmonic oscillator and hydrogen atoms.

hep-lat↗

arXiv1999

Perturbative Subtraction Methods

The effects of an automated, tenth order in $κ$ subtraction scheme on the noise variance of various Wilson QCD disconnected matrix elements are examined. It is found that there is a dramatic reduction in the variance of the lattice point-split electromagnetic currents and that this reduction persists at small quark mass.

hep-lat↗

arXiv1998

Lyapunov Exponents for the Intermittent Transition to Chaos

The dependence of the Lyapunov exponent on the closeness parameter, $ε$, in tangent bifurcation systems is investigated. We study and illustrate two averaging procedures for defining Lyapunov exponents in such systems. First, we develop theoretical expressions for an isolated tangency channel in which the Lyapunov exponent is defined on single channel passes. Numerical simulations were done to compare theory to measurement across a range of $ε$ values. Next, as an illustration of defining the Lyapunov exponent on many channel passes, a simulation of the intermittent transition in the logistic map is described. The modified theory for the channels is explained and a simple model for the gate entrance rates is constructed. An important correction due to the discrete nature of the iterative flow is identified and incorporated in an improved model. Realistic fits to the data were made for the Lyapunov exponents from the logistic gate and from the full simulation. A number of additional corrections which could improve the treatment of the gates are identified and briefly discussed.

chao-dyn↗

arXiv1998

Lattice Investigation of the DMO Sum Rule

An evaluation of charged pion polarizability using correlation functions measured on a 16^{3}\times 24 lattice in the context of the Das, Mathur, Okubo (DMO) sum rule is carried out. The calculation is limited to the so-called intrinsic part of the polarizability. This contribution, equivalent to a Euclidean time integral over vector and axial vector momentum-differentiated propagators, is evaluated in both a continuous and discrete sense. In the continuous case, the time behavior of the correlation functions is fit to the continuum quark model; the discrete case is handled by a straightforward application of Simpson's rule for integration after subtracting the pion contributions. A comparison of the implied vector meson and pion decay constants with phenomenological values is carried out. Results for the intrinsic polarizability are extrapolated across four quark mass values to the chiral limit. An extensive discussion of the lattice systematics in this calculation is given.

hep-lat↗

arXiv1997

Disconnected Loop Noise Methods in Lattice QCD

A comparison of the noise variance between algorithms for calculating disconnected loop signals in lattice QCD is carried out. The methods considered are the Z(N) noise method and the Volume method. We find that the noise variance is strongly influenced by the Dirac structure of the operator.

hep-lat↗

arXiv1996

Precise Neutron Electric Form Factor from Lattice QCD

A new calculation of the neutron electric form factor is described. Preliminary results for the connected part on a $20^{3}\times 32$ lattice are presented. The methods for calculating the disconnected part are briefly described.

hep-lat↗