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<head><title>Basic Nomenclature</title>
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</head><body
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<!--l. 155--><div class="crosslinks"><p class="noindent">[<a
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href="userhtmlsu4.html" >next</a>] [<a
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<h4 class="subsectionHead"><span class="titlemark">2.1 </span> <a
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id="x6-40002.1"></a>Basic Nomenclature</h4>
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<!--l. 158--><p class="noindent" >Our computational model implies that the data allocation on the parallel distributed
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memory machine is guided by the structure of the physical model, and specifically by
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the discretization mesh of the PDE.
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<!--l. 163--><p class="indent" > Each point of the discretization mesh will have (at least) one associated
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equation/variable, and therefore one index. We say that point <span
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class="cmmi-10">i </span><span
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class="cmti-10">depends </span>on point <span
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class="cmmi-10">j </span>if
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the equation for a variable associated with <span
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class="cmmi-10">i </span>contains a term in <span
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class="cmmi-10">j</span>, or equivalently if
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<span
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class="cmmi-10">a</span><sub><span
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class="cmmi-7">ij</span></sub><span
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class="cmmi-10">≠</span>0. After the partition of the discretization mesh into <span
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class="cmti-10">sub-domains </span>assigned
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to the parallel processes, we classify the points of a given sub-domain as
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following.
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<dl class="description"><dt class="description">
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<span
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class="cmbx-10">Internal.</span> </dt><dd
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class="description">An internal point of a given domain <span
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class="cmti-10">depends </span>only on points of the
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same domain. If all points of a domain are assigned to one process, then
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a computational step (e.g., a matrix-vector product) of the equations
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associated with the internal points requires no data items from other
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domains and no communications.
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</dd><dt class="description">
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<span
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class="cmbx-10">Boundary.</span> </dt><dd
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class="description">A point of a given domain is a boundary point if it <span
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class="cmti-10">depends </span>on
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points belonging to other domains.
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</dd><dt class="description">
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<span
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class="cmbx-10">Halo.</span> </dt><dd
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class="description">A halo point for a given domain is a point belonging to another domain
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such that there is a boundary point which <span
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class="cmti-10">depends </span>on it. Whenever performing
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a computational step, such as a matrix-vector product, the values associated
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with halo points are requested from other domains. A boundary point of a
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given domain is usually a halo point for some other domain<span class="footnote-mark"><a
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href="userhtml7.html#fn2x0"><sup class="textsuperscript">2</sup></a></span><a
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id="x6-4001f2"></a> ;
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therefore the cardinality of the boundary points set denotes the amount
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of data sent to other domains.
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</dd><dt class="description">
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<span
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class="cmbx-10">Overlap.</span> </dt><dd
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class="description">An overlap point is a boundary point assigned to multiple domains.
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Any operation that involves an overlap point has to be replicated for each
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assignment.</dd></dl>
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<!--l. 202--><p class="noindent" >Overlap points do not usually exist in the basic data distributions; however they are a
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feature of Domain Decomposition Schwarz preconditioners which are the subject of
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related research work <span class="cite">[<a
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href="userhtmlli2.html#X2007c">4</a>, <a
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href="userhtmlli2.html#X2007d">3</a>]</span>.
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<!--l. 207--><p class="indent" > We denote the sets of internal, boundary and halo points for a given subdomain
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by <span
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class="cmsy-10"><img
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src="cmsy10-49.png" alt="I" class="10x-x-49" /></span>, <span
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class="cmsy-10"><img
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src="cmsy10-42.png" alt="B" class="10x-x-42" /> </span>and <span
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class="cmsy-10"><img
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src="cmsy10-48.png" alt="H" class="10x-x-48" /></span>. Each subdomain is assigned to one process; each process usually owns
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one subdomain, although the user may choose to assign more than one subdomain to
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a process. If each process <span
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class="cmmi-10">i </span>owns one subdomain, the number of rows in
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the local sparse matrix is <span
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class="cmsy-10">|<img
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src="cmsy10-49.png" alt="I" class="10x-x-49" /></span><sub><span
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class="cmmi-7">i</span></sub><span
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class="cmsy-10">| </span>+ <span
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class="cmsy-10">|<img
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src="cmsy10-42.png" alt="B" class="10x-x-42" /></span><sub><span
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class="cmmi-7">i</span></sub><span
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class="cmsy-10">|</span>, and the number of local columns (i.e.
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those for which there exists at least one non-zero entry in the local rows) is
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<span
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class="cmsy-10">|<img
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src="cmsy10-49.png" alt="I" class="10x-x-49" /></span><sub><span
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class="cmmi-7">i</span></sub><span
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class="cmsy-10">| </span>+ <span
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class="cmsy-10">|<img
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src="cmsy10-42.png" alt="B" class="10x-x-42" /></span><sub><span
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class="cmmi-7">i</span></sub><span
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class="cmsy-10">| </span>+ <span
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class="cmsy-10">|<img
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src="cmsy10-48.png" alt="H" class="10x-x-48" /></span><sub><span
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class="cmmi-7">i</span></sub><span
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class="cmsy-10">|</span>.
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<!--l. 217--><p class="indent" > <hr class="figure"><div class="figure"
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>
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<a
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id="x6-40022"></a>
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<div class="center"
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>
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<!--l. 218--><p class="noindent" >
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<!--l. 221--><p class="noindent" ><img
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src="points.png" alt="PIC"
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></div>
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<br /> <div class="caption"
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><span class="id">Figure 2: </span><span
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class="content">Point classfication.</span></div><!--tex4ht:label?: x6-40022 -->
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<!--l. 227--><p class="indent" > </div><hr class="endfigure">
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<!--l. 229--><p class="indent" > This classification of mesh points guides the naming scheme that we adopted in
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the library internals and in the data structures. We explicitly note that “Halo” points
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are also often called “ghost” points in the literature.
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<!--l. 236--><div class="crosslinks"><p class="noindent">[<a
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href="userhtmlsu4.html" >next</a>] [<a
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href="userhtmlsu1.html" >front</a>] [<a
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href="userhtmlse2.html#userhtmlsu2.html" >up</a>] </p></div>
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<!--l. 236--><p class="indent" > <a
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id="tailuserhtmlsu1.html"></a>
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</body></html>
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