CASINO manual - Theory of Condensed Matter

38 Analysis of the performance of CASINO on parallel computers
38.1 VMC in parallel
The VMC algorithm is perfectly parallel: no interprocessor communication is required during simulations.
Each processor carries out an independent random walk using a different random-number
sequence, and the results are averaged at the end of each block, so that running for a length of time
T on P processors generates the same amount of data as running for time P T on a single processor
(assuming the equilibration time to be negligible). VMC should therefore scale to an arbitrarily large
number of processors.
Note that, although the energy obtained by running for time T on P processors should be in statistical
agreement with that obtained by running for time P T on a single processor, the results will not be
exactly equal, because the random walks are different in the two cases.
38.2 Optimization in parallel
38.2.1 Standard variance minimization
The VMC stages of a variance-minimization calculation are perfectly parallel, as described above.
In the optimization stages, the configuration set is distributed evenly between the processors. The
master processor broadcasts the current set of optimizable parameters, then each processor calculates
the local energy of each of its configurations and reports the energies (and weights, if required) to
the master. The CPU time required to evaluate the local energies of the configuration set usually far
exceeds the time spent communicating (reporting one or two numbers per configuration to the master
and receiving a handful of parameter values at each iteration). In particular the time spent evaluating
the local energies increases with system size, whereas the time spent on interprocessor communication
is independent of system size. So the standard variance minimization method is essentially perfectly
parallel.
Note that the number of processor communications could easily be reduced further if each processor
were simply to report the sum of its local energies and the sum of the squares of the local energies to
the master.
38.2.2 Variance minimization for linear Jastrow parameters
The VMC stage of the optimization (including the construction and accumulation of the quartic
coefficients) is perfectly parallel. The optimization itself is carried out in serial on the master node.
However, this stage typically takes a fraction of a second, and is independent of system size. So the
varmin-linjas scheme is essentially perfectly parallel.
38.2.3 Energy minimization
The VMC stages of an energy minimization are perfectly parallel, as described above. For the matrix
algebra stages, the configurations are divided evenly between the processors, each of which separately
generates one section of the full matrices. The full matrices are then gathered on the master processor,
where the matrix algebra is done. The time taken to do the matrix algebra is usually insignificant
in comparison to the time taken in VMC and matrix generation. The time taken in interprocessor
communication is recorded and written out during energy minimization, and is typically at maximum
a few percent of the total time spent in an iteration (and often much less than one percent). Overall,
energy minimization is very nearly perfectly parallel.
38.3 DMC in parallel
38.3.1 Parallelization strategy
When performing DMC on a parallel machine, the population of configurations is usually distributed
evenly over the set of processors. The algorithm is not perfectly parallel because the populations on
207