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<H2><A NAME="SECTION00031000000000000000">
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Basic Nomenclature</A>
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</H2>
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<P>
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Our computational model implies that the data allocation on the
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parallel distributed memory machine is guided by the structure of the
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physical model, and specifically by the discretization mesh of the
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PDE.
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<P>
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Each point of the discretization mesh will have (at least) one
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associated equation/variable, and therefore one index. We say that
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point <SPAN CLASS="MATH"><IMG
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$i$"></SPAN> <EM>depends</EM> on point <SPAN CLASS="MATH"><IMG
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WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$j$"></SPAN> if the equation for a
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variable associated with <SPAN CLASS="MATH"><IMG
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$i$"></SPAN> contains a term in <SPAN CLASS="MATH"><IMG
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WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$j$"></SPAN>, or equivalently
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if <SPAN CLASS="MATH"><IMG
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WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$a_{ij} \ne0$"></SPAN>.
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After the partition of the discretization mesh into <EM>sub-domains</EM>
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assigned to the parallel processes,
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we classify the points of a given sub-domain as following.
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<DL>
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<DT><STRONG>Internal.</STRONG></DT>
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<DD>An internal point of
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a given domain <EM>depends</EM> only on points of the
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same domain.
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If all points of a domain are assigned to one
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process, then a computational step (e.g., a
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matrix-vector product) of the
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equations associated with the internal points requires no data
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items from other domains and no communications.
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<P>
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</DD>
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<DT><STRONG>Boundary.</STRONG></DT>
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<DD>A point of
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a given domain is a boundary point if it <EM>depends</EM> on points
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belonging to other domains.
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<P>
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</DD>
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<DT><STRONG>Halo.</STRONG></DT>
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<DD>A halo point for a given domain is a point belonging to
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another domain such that there is a boundary point which <EM>depends</EM>
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on it. Whenever performing a computational step, such as a
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matrix-vector product, the values associated with halo points are
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requested from other domains. A boundary point of a given
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domain is usually a halo point for some other domain<A NAME="tex2html3"
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HREF="footnode.html#foot210"><SUP><SPAN CLASS="arabic">2</SPAN></SUP></A>; therefore
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the cardinality of the boundary points set denotes the amount of data
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sent to other domains.
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</DD>
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<DT><STRONG>Overlap.</STRONG></DT>
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<DD>An overlap point is a boundary point assigned to
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multiple domains. Any operation that involves an overlap point
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has to be replicated for each assignment.
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</DD>
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</DL>
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Overlap points do not usually exist in the basic data
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distributions; however they are a feature of Domain Decomposition
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Schwarz preconditioners which are the subject of related research
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work [<A
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HREF="node133.html#2007c">4</A>,<A
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HREF="node133.html#2007d">3</A>].
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<P>
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We denote the sets of internal, boundary and halo points for a given
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subdomain by <SPAN CLASS="MATH"><IMG
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WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$\cal I$"></SPAN>, <SPAN CLASS="MATH"><IMG
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WIDTH="15" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img8.png"
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ALT="$\cal B$"></SPAN> and <SPAN CLASS="MATH"><IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\cal H$"></SPAN>.
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Each subdomain is assigned to one process; each process usually
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owns one subdomain, although the user may choose to assign more than
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one subdomain to a process. If each process <SPAN CLASS="MATH"><IMG
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$i$"></SPAN> owns one
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subdomain, the number of rows in the local sparse matrix is
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<!-- MATH
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$|{\cal I}_i| + |{\cal B}_i|$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.png"
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ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert$"></SPAN>, and the number of local columns
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(i.e. those for which there exists at least one non-zero entry in the
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local rows) is <!-- MATH
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$|{\cal I}_i| + |{\cal B}_i| +|{\cal H}_i|$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="118" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.png"
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ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert +\vert{\cal H}_i\vert$"></SPAN>.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig:points"></A><A NAME="274"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
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Point classfication.</CAPTION>
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<TR><TD>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="318" HEIGHT="285" ALIGN="BOTTOM" BORDER="0"
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SRC="img13.png"
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ALT="\includegraphics[scale=0.65]{figures/points.eps}">
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<!-- MATH
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$\rotatebox{-90}{\includegraphics[scale=0.65]{figures/points}}$
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-->
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<IMG
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WIDTH="131" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
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SRC="img12.png"
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ALT="\rotatebox{-90}{\includegraphics[scale=0.65]{figures/points}}">
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</DIV></TD></TR>
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</TABLE>
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</DIV>
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<P>
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This classification of mesh points guides the naming scheme that we
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adopted in the library internals and in the data structures. We
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explicitly note that “Halo” points are also often called “ghost”
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points in the literature.
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<P>
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