CASINO manual - Theory of Condensed Matter

different processors interact via the population-control mechanism. The population on each processor
fluctuates, and the cost of each time step is determined by the processor with the largest population.
It is therefore necessary to even up the distribution of configurations between processors from time to
time. Unfortunately, transferring configurations between processors can be costly, and is usually the
principal limitation on the scaling of the DMC method with the number of processors. (There is also
a small cost associated with the communication required to decide on a reference energy after each
time step, but we assume this to be negligible henceforth.)
[Note that with the release of casino 2.8 in Feb 2011, it was shown that the effective cost of configuration
transfers can be reduced to essentially zero by using fancy MPI tricks such as non-blocking
sends and receives, so the following discussion is largely academic, but we retain it for completeness.]
38.3.2 Behaviour of the population on each processor
Let T redist τ be the redistribution period, i.e., we redistribute the configuration population after every
T redist time steps τ. At any given time the population on a processor p must be increasing or decreasing
exponentially, because the mean energy e p of the configuration population on that processor is unlikely
to be exactly equal to the reference energy E T . We assume that e p −E T remains roughly constant over
the redistribution period, i.e., that the autocorrelation period is much longer than the redistribution
period. At the start of the redistribution period the population C p (1) on each processor is the same.
At the end of the redistribution period, the expected population on processor p is C p (T redist ) =
C p (1) exp[−(e p − E T )T redist τ]. Hence ¯C(T redist ) ≈ ¯C(1) exp[−(Ē − E T )T redist τ] + O(Tredist 2 τ 2 ), where
the bar denotes an average over the processors, and so the average growth or decay of the population
is the same as that of the entire population (which should be small, because E T is chosen so as to
ensure this).
38.3.3 Optimal redistribution period
Let A be the cost of propagating a single configuration over one time step. Let B be the cost of
transferring a single configuration between processors.
Let q be the processor with the largest number of configurations, i.e., with the lowest energy e q ≡
min{e p }. Both the cost of propagating configurations and the cost of transferring configurations are
determined by processor q. The expected number of configurations on processor q at the end of
the redistribution period (i.e., after T redist time steps) is max{C p (T redist )} ≈ ¯C(T redist ) + cT redist +
O(Tredist 2 ), where c = ¯C(1)(Ē − min{e p})τ. NB, 〈 ¯C(1)〉 = N C /P , where N C is the target population
and P is the number of processors, and 〈c〉 is a positive constant.
At the end of the redistribution period, cT redist configurations are to be transferred from processor q.
Hence the average cost of transferring configurations per time step is B〈c〉, which is independent of
T redist .
The average cost per time step of waiting for the processor q with the greatest number of configurations
to finish propagating all its excess configurations is
A〈c〉 [0 + 1 + . . . + (T redist − 1)]
= A〈c〉(T redist − 1)
. (459)
T redist 2
So the total average cost per time step in DMC is
T = AN C
P + A〈c〉(T redist − 1)
+ B〈c〉. (460)
2
Clearly the redistribution period should be chosen to be as small as possible to minimize T . Numerical
tests confirm that increasing the redistribution period only acts to slow down calculations. One should
therefore choose T redist = 1, i.e., redistribution should take place after every time step. We assume
this to be the case henceforth (the previously existing keyword redist period which allowed the user
to do otherwise has now anyway been deleted).
38.4 Scaling of the DMC algorithm with the number of processors
The cost A of propagating each configuration scales as N α . For typical systems, where extended
orbitals represented in a localized basis are used and the CPU time is dominated by the evaluation of
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