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<h2 id="Contents">Contents<a class="anchor-link" href="#Contents"></a></h2><p>In this notebook, we will learn</p>
<ul>
<li>How to paralleize the Jacobi method</li>
<li>How to parallelize the Jacobi method</li>
<li>How the data partition can impact the performance of a distributed algorithm</li>
<li>How to use latency hiding to improve parallel performance</li>
</ul>
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<li>We need to get remote entries from 2 neighbors (2 messages per iteration)</li>
<li>We need to communicate 1 entry per message</li>
<li>Thus, communication complexity is $O(1)$</li>
<li>Communication/computation ration is $O(P/N)$, making the algorithm potentially scalable if $P&lt;&lt;N$.</li>
<li>Communication/computation ratio is $O(P/N)$, making the algorithm potentially scalable if $P&lt;&lt;N$.</li>
</ul>
</div>
</div>
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<span class="k">end</span>
</pre></div>
<ul>
<li>The outer loop cannot be parallelized (like in the 1d case).</li>
<li>The outer loop cannot be parallelized (like in the 1D case).</li>
<li>The two inner loops are trivially parallel</li>
</ul>
</div>
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<h3 id="Parallelization-strategies">Parallelization strategies<a class="anchor-link" href="#Parallelization-strategies"></a></h3><p>In 2d one has more flexibility in order to distribute the data over the processes. We consider these three alternatives:</p>
<h3 id="Parallelization-strategies">Parallelization strategies<a class="anchor-link" href="#Parallelization-strategies"></a></h3><p>In 2D, one has more flexibility in order to distribute the data over the processes. We consider these three alternatives:</p>
<ul>
<li>1D block row partition (each worker handles a subset of consecutive rows and all columns)</li>
<li>2D block partition (each worker handles a subset of consecutive rows and columns)</li>
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</thead>
<tbody>
<tr>
<td>1d block</td>
<td>1D block</td>
<td>2</td>
<td>O(N)</td>
<td>N²/P</td>
<td>O(P/N)</td>
</tr>
<tr>
<td>2d block</td>
<td>2D block</td>
<td>4</td>
<td>O(N/√P)</td>
<td>N²/P</td>
<td>O(√P/N)</td>
</tr>
<tr>
<td>2d cyclic</td>
<td>2D cyclic</td>
<td>4</td>
<td>O(N²/P)</td>
<td>N²/P</td>
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<h3 id="Which-partition-is-the-best-one?">Which partition is the best one?<a class="anchor-link" href="#Which-partition-is-the-best-one?"></a></h3><ul>
<li>Both 1d and 2d block partitions are potentially scalable if $P&lt;&lt;N$</li>
<li>The 2d block partition has the lowest communication complexity</li>
<li>The 1d block partition requires to send less messages (It can be useful if the fixed cost of sending a message is high)</li>
<li>Both 1D and 2D block partitions are potentially scalable if $P&lt;&lt;N$</li>
<li>The 2D block partition has the lowest communication complexity</li>
<li>The 1D block partition requires to send less messages (It can be useful if the fixed cost of sending a message is high)</li>
<li>The best strategy for a given problem size will thus depend on the machine.</li>
<li>Cyclic partitions are impractical for this application (but they are useful in others)</li>
</ul>