1,146 research outputs found
Multigrid Methods in Lattice Field Computations
The multigrid methodology is reviewed. By integrating numerical processes at
all scales of a problem, it seeks to perform various computational tasks at a
cost that rises as slowly as possible as a function of , the number of
degrees of freedom in the problem. Current and potential benefits for lattice
field computations are outlined. They include: solution of Dirac
equations; just operations in updating the solution (upon any local
change of data, including the gauge field); similar efficiency in gauge fixing
and updating; operations in updating the inverse matrix and in
calculating the change in the logarithm of its determinant; operations
per producing each independent configuration in statistical simulations
(eliminating CSD), and, more important, effectively just operations per
each independent measurement (eliminating the volume factor as well). These
potential capabilities have been demonstrated on simple model problems.
Extensions to real life are explored.Comment: 4
Renormalization group maps for Ising models in lattice gas variables
Real space renormalization group maps, e.g., the majority rule
transformation, map Ising type models to Ising type models on a coarser
lattice. We show that each coefficient of the renormalized Hamiltonian in the
lattice gas variables depends on only a finite number of values of the
renormalized Hamiltonian. We introduce a method which computes the values of
the renormalized Hamiltonian with high accuracy and so computes the
coefficients in the lattice gas variables with high accuracy. For the critical
nearest neighbor Ising model on the square lattice with the majority rule
transformation, we compute over 1,000 different coefficients in the lattice gas
variable representation of the renormalized Hamiltonian and study the decay of
these coefficients. We find that they decay exponentially in some sense but
with a slow decay rate. We also show that the coefficients in the spin
variables are sensitive to the truncation method used to compute them.Comment: 22 pages, 9 color postscript figures; minor revisions in version
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