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Document DESC%CNV method and MOLD arguments to CDASB and PREC%BUILD

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Salvatore Filippone
2017-12-14 17:09:59 +00:00
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@@ -1,10 +1,6 @@
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<!--Converted with LaTeX2HTML 2012 (1.2)
original version by: Nikos Drakos, CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
<HTML>
<HEAD>
<TITLE>Basic Nomenclature</TITLE>
@@ -13,7 +9,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -25,30 +21,31 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -64,24 +61,24 @@ PDE.
<P>
Each point of the discretization mesh will have (at least) one
associated equation/variable, and therefore one index. We say that
point <IMG
point <SPAN CLASS="MATH"><IMG
WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$i$"> <EM>depends</EM> on point <IMG
ALT="$i$"></SPAN> <EM>depends</EM> on point <SPAN CLASS="MATH"><IMG
WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$j$"> if the equation for a
variable associated with <IMG
ALT="$j$"></SPAN> if the equation for a
variable associated with <SPAN CLASS="MATH"><IMG
WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$i$"> contains a term in <IMG
ALT="$i$"></SPAN> contains a term in <SPAN CLASS="MATH"><IMG
WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$j$">, or equivalently
if <IMG
ALT="$j$"></SPAN>, or equivalently
if <SPAN CLASS="MATH"><IMG
WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
ALT="$a_{ij} \ne0$">.
ALT="$a_{ij} \ne0$"></SPAN>.
After the partition of the discretization mesh into <EM>sub-domains</EM>
assigned to the parallel processes,
we classify the points of a given sub-domain as following.
@@ -112,7 +109,7 @@ 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<A NAME="tex2html3"
HREF="footnode.html#foot210"><SUP>2</SUP></A>; therefore
HREF="footnode.html#foot210"><SUP><SPAN CLASS="arabic">2</SPAN></SUP></A>; therefore
the cardinality of the boundary points set denotes the amount of data
sent to other domains.
</DD>
@@ -126,43 +123,43 @@ 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&nbsp;[<A
HREF="node132.html#2007c">4</A>,<A
HREF="node132.html#2007d">3</A>].
HREF="node133.html#2007c">4</A>,<A
HREF="node133.html#2007d">3</A>].
<P>
We denote the sets of internal, boundary and halo points for a given
subdomain by <IMG
subdomain by <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img7.png"
ALT="$\cal I$">, <IMG
ALT="$\cal I$"></SPAN>, <SPAN CLASS="MATH"><IMG
WIDTH="15" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
ALT="$\cal B$"> and <IMG
ALT="$\cal B$"></SPAN> and <SPAN CLASS="MATH"><IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\cal H$">.
ALT="$\cal H$"></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 <IMG
one subdomain to a process. If each process <SPAN CLASS="MATH"><IMG
WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$i$"> owns one
ALT="$i$"></SPAN> owns one
subdomain, the number of rows in the local sparse matrix is
<!-- MATH
$|{\cal I}_i| + |{\cal B}_i|$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img10.png"
ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert$">, and the number of local columns
ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert$"></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 <!-- MATH
$|{\cal I}_i| + |{\cal B}_i| +|{\cal H}_i|$
-->
<IMG
<SPAN CLASS="MATH"><IMG
WIDTH="118" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert +\vert{\cal H}_i\vert$">.
ALT="$\vert{\cal I}_i\vert + \vert{\cal B}_i\vert +\vert{\cal H}_i\vert$"></SPAN>.
<P>
@@ -191,33 +188,34 @@ Point classfication.</CAPTION>
<P>
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''
explicitly note that &ldquo;Halo&rdquo; points are also often called &ldquo;ghost&rdquo;
points in the literature.
<P>
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