Parallel Processing: A KISS Approach - University of North Dakota

The algorithm we will use to simulate the heat transfer is a simple 3 element
neighborhood averaging, where each array element becomes the average of itself
and its 2 neighbors. Equation (1) depicts the algorithm:
M
1
1
'
i
= ∑ M
( i + x)
3 x=−1
To facilitate the simulation of the constant heat and constant cold sources we will
extend array “M” to 14 elements with the outer elements having fixed values. The
constant heat source is colored red and the constant cold source is colored blue:
Heat
Cold
The extended version of “M” has array indices of 0 to 13 (C/C++) and since the
two end points are points of constant value, this extension al lows us to directly
implem ent equation (1) by allowing “i” to range from 1 to 12. We no longer have
to deal with the (original problem’s) end points separately.
Before we partition this into a parallel problem, note that the algorithm is
iterative. A single pass of the neighborhood averaging procedure will not produce
correct results. We must apply the neighborhood averaging procedure repeatedly
until convergence occurs (until no significant changes occur from iteration to
iteration).
We will arbitrarily partition the problem space into 3 non-overlapping equal sized
segments and assign each segment to 1 of 3 nodes on the cluster:
Node 1 Node 2 Node 3
Heat
Cold
Immediately a problem arises. The calculation involves a neighborhood average,
yet the array elements (shown in gray) along the segment edges do not have all of
the required neighbors as these “missing” neighbors have been assigned to other
nodes. In order to retain usage of our equation and insure identical results to the
sequential version, the necessity for “halo” elements (shown in green) arises:
Node 1 Node 2 Node 3
Heat
Cold
We now have a problem that is uniform across the 3 nodes, in that all local arrays
are of the same size (1x6) and whose outer elements have fixed values (at least
with regards to the “local” neighborhood averaging procedure). If this problem
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