Walter Wilcox

Deflated GMRES for systems with multiple shifts and multiple right-hand sides

Abstract 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.

The Nucleon's strange electromagnetic and scalar matrix elements

Quenched lattice QCD simulations and quenched chiral perturbation theory are used together for this study of strangeness in the nucleon. Dependences of the matrix elements on strange quark mass, valence quark mass and momentum transfer are discussed in both the lattice and chiral frameworks. The combined results of this study are in good agreement with existing experimental data and predictions are made for upcoming experiments. Possible future refinements of the theoretical method are suggested.

The pion form factor in lattice QCD

The pion electric form factor is computed for several values of momentum-transfer on a quenched 103 × 20 lattice with SU(3) colour at β = 5.9. With small errors and with pionic energies consistent with continuum dispersion, the data is compatible with, although consistently above, the monopole form suggested by vector dominance.

PI NN AND PSEUDOSCALAR FORM FACTORS FROM LATTICE QCD

The {pi}{ital NN} form factor {ital g}{sub {pi}{ital NN}}({ital q}{sup 2}) is obtained from a quenched lattice QCD calculation of the pseudoscalar form factor {ital g}{sub {ital P}}({ital q}{sup 2}) of the proton with pion pole dominance. We find that {ital g}{sub {pi}{ital NN}}({ital q}{sup 2}) is well fitted with a monopole form which agrees with the Goldberger-Treiman relation. The monopole mass is determined to be 0.75{plus_minus}0.14 GeV, which shows that {ital g}{sub {pi}{ital NN}}({ital q}{sup 2}) is rather soft. The extrapolated {pi}{ital N} coupling constant {ital g}{sub {pi}{ital NN}}=12.7{plus_minus}2.4 is quite consistent with the phenomenological values. We also compare {ital g}{sub {pi}{ital NN}}({ital q}{sup 2}) with the axial form factor {ital g}{sub {ital A}}({ital q}{sup 2}) to check pion dominance in the induced pseudoscalar form factor {ital h}{sub {ital A}}({ital q}{sup 2}) vis-a-vis the chiral Ward identity.

Chiral limit of nucleon lattice electromagnetic form factors

We calculate electric and magnetic form factors of protons and neutrons in quenched Monte Carlo lattice QCD on a 16 3 × 24 lattice at β=6.0 using Wilson fermions. We employ a method which characterizes one of the nucleon fields as a fixed zero-momentum secondary source. Extrapolating the overall data set to the chiral limit, we find acceptable fits for either dipole or monopole forms and extract proton and neutron magnetic moments, the magnitude of which are 10 to 15% low compared to experiment. In the extrapolation of the dipole fit of the form factors, we find that the dipole-to-nucleon mass ratio is about 7% low compared to experiment

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

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.

Magnetic moments of vector, axial, and tensor mesons in lattice QCD

We present a calculation of magnetic moments for selected spin-1 mesons using the techniques of lattice QCD. This is carried out by introducing a progressively small static magnetic field on the lattice and measuring the linear response of a hadrons mass shift. The calculations are done on 24{sup 4} quenched lattices using standard Wilson actions, with {beta}=6.0 and pion mass down to 500 MeV. The results are compared to those from the form factor method.

Magnetic polarizability of hadrons from lattice QCD in the background field method

We present a calculation of hadron magnetic polarizability using the techniques of lattice QCD. This is carried out by introducing a uniform external magnetic field on the lattice and measuring the quadratic part of a hadrons mass shift. The calculation is performed on a 24{sup 4} lattice with standard Wilson actions at beta=6.0 (spacing a=0.1 fm) and pion mass down to about 500 MeV. Results are obtained for 30 particles covering the entire baryon octet (n, p, {sigma}{sup 0}, {sigma}{sup -}, {sigma}{sup +}, {xi}{sup -}, {xi}{sup 0}, {lambda}) and decuplet ({delta}{sup 0}, {delta}{sup -}, {delta}{sup +}, {delta}{sup ++}, {sigma}*{sup 0}, {sigma}*{sup -}, {sigma}*{sup +}, {xi}*{sup 0}, {xi}*{sup -}, {omega}{sup -}), plus selected mesons ({pi}{sup 0}, {pi}{sup +}, {pi}{sup -}, K{sup 0}, K{sup +}, K{sup -}, {rho}{sup 0}, {rho}{sup +}, {rho}{sup -}, K*{sup 0}, K*{sup +}, K*{sup -}). The results are compared with available values from experiments and other theoretical calculations.

A study of hadron electric polarizability in quenched lattice QCD

Abstract The dependence of neutral hadron masses on external static electric fields is studied in quenched lattice QCD using the staggered fermion scheme. Electric polarizabilities are extracted for the pseudoscalar meson and the baryon at four quark masses. The vector meson polarizability cannot be reliably obtained from the present set of lattice data. The calculated neutron polarizability is in very good agreement with a recent experimental result. Our result for the pseudoscalar meson is about one quarter of that of the neutron.

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 {rho}{sup 0} and K{sup *0}; the octet baryons n, {Sigma}{sup 0}, {Lambda}{sub o}{sup 0}, {Lambda}{sub s}{sup 0}, and {Xi}{sup 0}; and the decouplet baryons {Delta}{sup 0}, {Sigma}{sup 0}, and {Xi}{sup 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 Luesher-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.