<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <html > <head><title>Basic Nomenclature</title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> <meta name="generator" content="TeX4ht (https://tug.org/tex4ht/)"> <meta name="originator" content="TeX4ht (https://tug.org/tex4ht/)"> <!-- html,3 --> <meta name="src" content="userhtml.tex"> <link rel="stylesheet" type="text/css" href="userhtml.css"> </head><body > <!--l. 155--><div class="crosslinks"><p class="noindent">[<a href="userhtmlsu4.html" >next</a>] [<a href="#tailuserhtmlsu1.html">tail</a>] [<a href="userhtmlse2.html#userhtmlsu2.html" >up</a>] </p></div> <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">≠</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 <span class="cite">[<a href="userhtmlli2.html#X2007c">4</a>, <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="x6-4002r2"></a> <div class="center" > <!--l. 218--><p class="noindent" > <!--l. 221--><p class="noindent" ><img src="points.png" alt="PIC" width="46" height="46" ></div> <br /> <div class="caption" ><span class="id">Figure 2: </span><span class="content">Point classfication.</span></div><!--tex4ht:label?: x6-4002r2 --> <!--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 “Halo” points are also often called “ghost” points in the literature. <!--l. 236--><div class="crosslinks"><p class="noindent">[<a href="userhtmlsu4.html" >next</a>] [<a href="userhtmlsu1.html" >front</a>] [<a href="userhtmlse2.html#userhtmlsu2.html" >up</a>] </p></div> <!--l. 236--><p class="indent" > <a id="tailuserhtmlsu1.html"></a> </body></html>