A seismic-wave finite-difference code is run on a cluster of many-core nodes.

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Scientific Computing for Physicists, Ch Many processors: speed-up, decomposition and the cost of communication

Problem

A seismic-wave finite-difference code is run on a cluster of many-core nodes. ((a)) Profiling shows a fraction

f=0.90

of the one-process run time parallelises perfectly; the rest is serial set-up and output. Using Amdahl's law, compute the speed-up

S(16)

on 16 processes and the ceiling

S(\infty)

. ((b)) The same code is run instead in weak-scaling mode, holding the work per process fixed, with

f'=0.98

of the run time on 32 processes spent in the part that shares out. Compute Gustafson's scaled speed-up

S(32)

on 32 processes. ((c)) The domain is decomposed into slabs, exchanging one halo row with each of two neighbours every step; one halo message carries

32768\,\mathrm{B}

. A ping-pong test gives latency

\alpha_{\rm lat}=2.0\,\mu\mathrm{s}

and bandwidth

\beta_{\rm net}=1.0\times10^{10}\,\mathrm{B\,s^{-1}}

. Find the time spent in the two halo messages each step, and express it as a percentage of a slab update measured at

4.20\,\mathrm{ms}

. ((d)) The rows

i=0,\dots,N-1

of a triangular pair loop (

i<j

) are split into

P=8

equal contiguous ranges, one per process. The first of

P

equal ranges holds a fraction

(2P-1)/P^2

of all the

i<j

pairs. Find that fraction for

P=8

and the resulting bound on the speed-up from this decomposition.

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