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<head><title>Basic Nomenclature</title>
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<h4 class="subsectionHead"><span class="titlemark">2.1 </span> <a
id="x6-40002.1"></a>Basic Nomenclature</h4>
<!--l. 158--><p class="noindent" >Our computational model implies that the data allocation on the parallel distributed
memory machine is guided by the structure of the physical model, and specifically by
the discretization mesh of the PDE.
<!--l. 163--><p class="indent" > Each point of the discretization mesh will have (at least) one associated
equation/variable, and therefore one index. We say that point <span
class="cmmi-10">i </span><span
class="cmti-10">depends </span>on point <span
class="cmmi-10">j </span>if
the equation for a variable associated with <span
class="cmmi-10">i </span>contains a term in <span
class="cmmi-10">j</span>, or equivalently if
<span
class="cmmi-10">a</span><sub><span
class="cmmi-7">ij</span></sub><span
class="cmmi-10">&#x2260;</span>0. After the partition of the discretization mesh into <span
class="cmti-10">sub-domains </span>assigned
to the parallel processes, we classify the points of a given sub-domain as
following.
<dl class="description"><dt class="description">
<span
class="cmbx-10">Internal.</span> </dt><dd
class="description">An internal point of a given domain <span
class="cmti-10">depends </span>only on points of the
same domain. If all points of a domain are assigned to one process, then
a computational step (e.g., a matrix-vector product) of the equations
associated with the internal points requires no data items from other
domains and no communications.
</dd><dt class="description">
<span
class="cmbx-10">Boundary.</span> </dt><dd
class="description">A point of a given domain is a boundary point if it <span
class="cmti-10">depends </span>on
points belonging to other domains.
</dd><dt class="description">
<span
class="cmbx-10">Halo.</span> </dt><dd
class="description">A halo point for a given domain is a point belonging to another domain
such that there is a boundary point which <span
class="cmti-10">depends </span>on it. Whenever performing
a computational step, such as a matrix-vector product, the values associated
with halo points are requested from other domains. A boundary point of a
given domain is usually a halo point for some other domain<span class="footnote-mark"><a
href="userhtml7.html#fn2x0"><sup class="textsuperscript">2</sup></a></span><a
id="x6-4001f2"></a> ;
therefore the cardinality of the boundary points set denotes the amount
of data sent to other domains.
</dd><dt class="description">
<span
class="cmbx-10">Overlap.</span> </dt><dd
class="description">An overlap point is a boundary point assigned to multiple domains.
Any operation that involves an overlap point has to be replicated for each
assignment.</dd></dl>
<!--l. 202--><p class="noindent" >Overlap points do not usually exist in the basic data distributions; however they are a
feature of Domain Decomposition Schwarz preconditioners which are the subject of
related research work&#x00A0;<span class="cite">[<a
href="userhtmlli2.html#X2007c">4</a>,&#x00A0;<a
href="userhtmlli2.html#X2007d">3</a>]</span>.
<!--l. 207--><p class="indent" > We denote the sets of internal, boundary and halo points for a given subdomain
by <span
class="cmsy-10"><img
src="cmsy10-49.png" alt="I" class="10x-x-49" /></span>, <span
class="cmsy-10"><img
src="cmsy10-42.png" alt="B" class="10x-x-42" /> </span>and <span
class="cmsy-10"><img
src="cmsy10-48.png" alt="H" class="10x-x-48" /></span>. Each subdomain is assigned to one process; each process usually owns
one subdomain, although the user may choose to assign more than one subdomain to
a process. If each process <span
class="cmmi-10">i </span>owns one subdomain, the number of rows in
the local sparse matrix is <span
class="cmsy-10">|<img
src="cmsy10-49.png" alt="I" class="10x-x-49" /></span><sub><span
class="cmmi-7">i</span></sub><span
class="cmsy-10">| </span>+ <span
class="cmsy-10">|<img
src="cmsy10-42.png" alt="B" class="10x-x-42" /></span><sub><span
class="cmmi-7">i</span></sub><span
class="cmsy-10">|</span>, and the number of local columns (i.e.
those for which there exists at least one non-zero entry in the local rows) is
<span
class="cmsy-10">|<img
src="cmsy10-49.png" alt="I" class="10x-x-49" /></span><sub><span
class="cmmi-7">i</span></sub><span
class="cmsy-10">| </span>+ <span
class="cmsy-10">|<img
src="cmsy10-42.png" alt="B" class="10x-x-42" /></span><sub><span
class="cmmi-7">i</span></sub><span
class="cmsy-10">| </span>+ <span
class="cmsy-10">|<img
src="cmsy10-48.png" alt="H" class="10x-x-48" /></span><sub><span
class="cmmi-7">i</span></sub><span
class="cmsy-10">|</span>.
<!--l. 217--><p class="indent" > <hr class="figure"><div class="figure"
>
<a
id="x6-40022"></a>
<div class="center"
>
<!--l. 218--><p class="noindent" >
<!--l. 221--><p class="noindent" ><img
src="points.png" alt="PIC"
></div>
<br /> <div class="caption"
><span class="id">Figure&#x00A0;2: </span><span
class="content">Point classfication.</span></div><!--tex4ht:label?: x6-40022 -->
<!--l. 227--><p class="indent" > </div><hr class="endfigure">
<!--l. 229--><p class="indent" > This classification of mesh points guides the naming scheme that we adopted in
the library internals and in the data structures. We explicitly note that &#8220;Halo&#8221; points
are also often called &#8220;ghost&#8221; points in the literature.
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