A diffusion code's array-form update was characterised in Chapter 1 as moving and…

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

Problem

A diffusion code's array-form update was characterised in Chapter 1 as moving

152\,\mathrm{B}

and doing 4 floating-point operations per cell. On a node of 28 cores the one-core triad bandwidth is

16\,\mathrm{GB\,s^{-1}}

and the all-core triad bandwidth is

224\,\mathrm{GB\,s^{-1}}

. ((a)) Assume the stencil streams at the one-core triad bandwidth on one core, and that each added thread adds one core's bandwidth until the node's limit is reached. Predict the thread count at which this stencil stops speeding up on the node. ((b)) A colleague times a true serial (single-process, unthreaded) build at

T_{\rm serial}=8.4\,\mathrm{s}

, but reports speed-up using the parallel code run on one process,

T_{\rm par,1}=9.6\,\mathrm{s}

, against the same parallel code on 8 threads,

T_{\rm par,8}=1.3\,\mathrm{s}

, each timed by Chapter 8's rules (warm-up runs, repeats, median and interquartile range). State the colleague's reported speed-up and the correct speed-up by the rule that a scaling study is reported against the best serial code, and say which is right. ((c)) A grid of

N_{\rm g}=1024

cells per side has a summed field total whose terms are bounded by

5.0\,\mathrm{K}

in magnitude. Using Chapter 2's summation bound

(n-1)(\epsilon_{\rm mach}/2)\sum_i|x_i|

, with

n=N_{\rm g}^2

, find the tolerance, twice that bound, within which a parallel and a serial sum of the field must agree, and state whether a measured difference of

2\times10^{-10}\,\mathrm{K}

passes it.

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