15.6 Comparison of algorithms for scalar field theory 509
Updateφkusing(15.103)with time steps(15.105);
FFT:φk→φn;
END LangStep.We have used ‘FFT’ for the forward transform (from real space to reciprocal space)
and ‘FFT’ for the backward transform. The random forcesRkcan be generated
in two ways. The simplest way is to generate a set of random forcesRnon the
real space grid, and then Fourier-transform this set to the reciprocal grid. A more
efficient way is to generate the forces directly on the Fourier grid. The forcesRk
satisfy the following requirements:Rk=R∗−k, as a result of theRnbeing real; and
the variance satisfies〈|Rk|^2 〉=〈R^2 n〉=1. Thus, fork=−k(modulo 2π/L), the
real and imaginary part of the random forceRkboth have width 1/
√
- Ifk=−k
(modulo 2π/L) then the random force has a real part only, which should be drawn
from a distribution with width 1.
15.6 Comparison of algorithms for scalar field theory
In the previous sections we have described seven different methods for simulating
the scalar field theory on a lattice. We now present a comparison of the performance
of the different methods (Table 15.3). We have takenm = 0.1 andg =0.01
as the bare parameters on a 16×16 lattice. The simulations were carried out
on a standard workstation. The results should not be taken too seriously because
different platforms and different, more efficient codings could give different results.
Moreover, some methods can be parallelised more efficiently than others, which
is important when doing large-scale calculations (seeChapter 16). Finally, no real
effort has been put into optimising the programs (except for standard optimisation at
compile time), so the results should be interpreted as trends rather than as rigorous
comparisons.
We give the CPU time needed for one simulation step and the correlation time,
measured in simulation steps. The error in the run time is typically a few per cent,
and that in the correlation time is typically between 5 and 10 per cent. For each
method we include results for an 8×8anda16×16 lattice to show how the CPU time
per step and the correlation time scale with the lattice size. All programs give the
correct results for the renormalised mass and coupling constant, which have been
presented before. The number of MC or MD steps in these simulations varied from
30 000 to 100 000, depending on the method used. In the Andersen method, we used
100 steps between momentum refreshing forh=0.05 and 50 steps forh=0.1.
The time steps used in the hybrid algorithm were chosen such as to stabilise the
acceptance rate at 70%. In this algorithm, 10 steps were used between the updates.
The time steph=0.2 given for the Fourier-accelerated Langevin method is in fact