Changes to docs to highlight new stuff in MLD internals.
Salvatore Filippone
9 years ago
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<H1><A NAME="SECTION00090000000000000000"></A><A NAME="sec:adding"></A>
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<H2><A NAME="SECTION00089000000000000000"></A><A NAME="sec:auxil"></A>
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<BR>
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<BR>
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Adding new smoother and solver objects to MLD2P4
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Auxiliary Methods
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</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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<P>
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Various functionalities are implemented as additional methods of the
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Developers can add completely new smoother and/or solver classes
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preconditioner object.
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derived from the base objects in the library (see Remark 2 in Section <A HREF="node20.html#sec:precset">6.2</A>),
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</BIG></BIG></BIG>
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without recompiling the library itself.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">To do so, it is necessary first to select the base type to be extended.
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In our experience, it is quite likely that the new application needs
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only the definition of a ``solver'' object, which is almost
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always acting only on the local part of the distributed matrix.
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The parallel actions required to connect the various solver objects
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are most often already provided by the block-Jacobi or the additive
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Schwarz smoothers. To define a new solver, the developer will then
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have to define its components and methods, perhaps taking one of the
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predefined solvers as a starting point, if possible.
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</FONT></FONT></FONT>
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<P>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Once the new smoother/solver class has been developed, to use it in
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the context of the multilevel preconditioners it is necessary to:
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<BR><HR>
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</FONT></FONT></FONT>
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<!--Table of Child-Links-->
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<UL>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI>declare in the application program a variable of the new type;
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</LI>
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<LI>pass that variable as the argument to the <code>set</code> routine as in the
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following:
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<DIV ALIGN="CENTER">
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<code>call p%set(smoother,info [,ilev,ilmax,pos])</code>
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<BR><code>call p%set(solver,info [,ilev,ilmax,pos])</code>
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</DIV>
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<UL CLASS="ChildLinks">
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</LI>
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<LI><A NAME="tex2html428"
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<LI>link the code implementing the various methods into the application executable.
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HREF="node28.html">Method: dump</A>
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</LI>
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<LI><A NAME="tex2html429"
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</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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HREF="node29.html">Method: clone</A>
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The new solver object is then dynamically included in the
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<LI><A NAME="tex2html430"
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preconditioner structure, and acts as a <I>mold</I> to which the
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HREF="node30.html">Method: sizeof</A>
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preconditioner will conform, even though the MLD2P4 library has not
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<LI><A NAME="tex2html431"
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been modified to account for this new development.
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HREF="node31.html">Method: allocate_wrk</A>
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</FONT></FONT></FONT>
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<LI><A NAME="tex2html432"
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<P>
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HREF="node32.html">Method: free_wrk</A>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">It is possible to define new values for the keyword <code>WHAT</code> in the
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<code>set</code> routine; if the library code does not recognize a keyword,
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it passes it down the composition hierarchy (levels containing
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<BR><HR>
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smoothers containing in turn solvers), so that it can be eventually caught by
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the new solver.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">An example is provided in the source code distribution under the
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folder <code>tests/newslv</code>. In this example we are implementing a new
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incomplete factorization variant (which is simply the ILU(0)
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factorization under a new name). Because of the specifics of this case, it is
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possible to reuse the basic structure of the ILU solver, with its
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L/D/U components and the methods needed to apply the solver; only a
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few methods, such as the description and most importantly the build,
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need to be ovverridden (rewritten).
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The interfaces for the calls shown above are defined using
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</FONT></FONT></FONT>
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<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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<code>smoother</code> </FONT></FONT></FONT></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>class(mld_x_base_smoother_type)</code> </FONT></FONT></FONT></TD>
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</TR>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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</FONT></FONT></FONT></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The user-defined new smoother to be employed in the
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preconditioner.</FONT></FONT></FONT></TD>
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</TR>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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<code>solver</code> </FONT></FONT></FONT></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>class(mld_x_base_solver_type)</code> </FONT></FONT></FONT></TD>
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</TR>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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</FONT></FONT></FONT></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The user-defined new solver to be employed in the
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preconditioner.
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</FONT></FONT></FONT></TD>
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</TR>
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</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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The other arguments are defined in the way described in
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Sec. <A HREF="node20.html#sec:precset">6.2</A>. As an example, in the <code>tests/newslv</code>
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code we define a new object of type <code>mld_d_tlu_solver_type</code>, and
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we pass it as follows:
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</FONT></FONT></FONT><PRE>
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! sparse matrix and preconditioner
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type(psb_dspmat_type) :: a
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type(mld_dprec_type) :: prec
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type(mld_d_tlu_solver_type) :: tlusv
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......
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!
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! prepare the preconditioner: an ML with defaults, but with TLU solver at
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! intermediate levels. All other parameters are at default values.
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!
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call prec%init('ML', info)
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call prec%hierarchy_build(a,desc_a,info)
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nlv = prec%get_nlevs()
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call prec%set(tlusv, info,ilev=1,ilmax=max(1,nlv-1))
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call prec%smoothers_build(a,desc_a,info)
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<H2><A NAME="SECTION000120000000000000000">
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|
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Bibliography</A>
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|
||||||
</H2><DL COMPACT><DD>
|
|
||||||
|
|
||||||
|
<H3><A NAME="SECTION00089300000000000000">
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||||||
|
Method: sizeof</A>
|
||||||
|
</H3><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
<P>
|
<P>
|
||||||
<P></P><DT><A NAME="MUMPS">1</A>
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
<DD>
|
<DIV ALIGN="CENTER"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><code>sz = p%sizeof()</code>
|
||||||
P. R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. L'Excellent, C. Weisbecker,
|
<BR></BIG></BIG></BIG></DIV><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
<EM>Improving multifrontal methods by means of block low-rank representations</EM>,
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|
||||||
SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474.
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|
||||||
See also <TT>http://mumps.enseeiht.fr</TT>.<P></P><DT><A NAME="BREZINA_VANEK">2</A>
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|
||||||
<DD>
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|
||||||
M. Brezina, P. Vanek,
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|
||||||
<EM>A Black-Box Iterative Solver Based on a Two-Level Schwarz Method</EM>,
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|
||||||
Computing, 63, 1999, 233-263.<P></P><DT><A NAME="Briggs2000">3</A>
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|
||||||
<DD>
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|
||||||
W. L. Briggs, V. E. Henson, S. F. McCormick,
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|
||||||
<EM>A Multigrid Tutorial, Second Edition</EM>,
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|
||||||
SIAM, 2000.<P></P><DT><A NAME="para_04">4</A>
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|
||||||
<DD>
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|
||||||
A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone,
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|
||||||
<EM>Extending PSBLAS to Build Parallel Schwarz Preconditioners</EM>,
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|
||||||
in J. Dongarra, K. Madsen, J. Wasniewski, editors,
|
|
||||||
Proceedings of PARA 04 Workshop on State of the Art
|
|
||||||
in Scientific Computing, Lecture Notes in Computer Science,
|
|
||||||
Springer, 2005, 593-602.<P></P><DT><A NAME="aaecc_07">5</A>
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|
||||||
<DD>
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|
||||||
A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone,
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|
||||||
<EM>2LEV-D2P4: a package of high-performance preconditioners
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|
||||||
for scientific and engineering applications</EM>,
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|
||||||
Applicable Algebra in Engineering, Communications and Computing,
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|
||||||
18 (3) 2007, 223-239.<P></P><DT><A NAME="CAI_SARKIS">6</A>
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|
||||||
<DD>
|
|
||||||
X. C. Cai, M. Sarkis,
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|
||||||
<EM>A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems</EM>,
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|
||||||
SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797.<P></P><DT><A NAME="apnum_07">7</A>
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|
||||||
<DD>
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|
||||||
P. D'Ambra, S. Filippone, D. di Serafino,
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|
||||||
<EM>On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners</EM>,
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|
||||||
Applied Numerical Mathematics, Elsevier Science,
|
|
||||||
57 (11-12), 2007, 1181-1196.<P></P><DT><A NAME="MLD2P4_TOMS">8</A>
|
|
||||||
<DD>
|
|
||||||
P. D'Ambra, D. di Serafino, S. Filippone,
|
|
||||||
<I>MLD2P4: a Package of Parallel Multilevel
|
|
||||||
Algebraic Domain Decomposition Preconditioners
|
|
||||||
in Fortran 95</I>, ACM Trans. Math. Softw., 37(3), 2010, art. 30.<P></P><DT><A NAME="UMFPACK">9</A>
|
|
||||||
<DD>
|
|
||||||
T. A. Davis,
|
|
||||||
<EM>Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal
|
|
||||||
Method with a Column Pre-ordering Strategy</EM>,
|
|
||||||
ACM Transactions on Mathematical Software, 30, 2004, 196-199.
|
|
||||||
(See also <TT>http://www.cise.ufl.edu/~davis/</TT>)<P></P><DT><A NAME="SUPERLU">10</A>
|
|
||||||
<DD>
|
|
||||||
J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu,
|
|
||||||
<EM>A supernodal approach to sparse partial pivoting</EM>,
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|
||||||
SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755.<P></P><DT><A NAME="blas3">11</A>
|
|
||||||
<DD>
|
|
||||||
J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling,
|
|
||||||
<I>A set of Level 3 Basic Linear Algebra Subprograms</I>,
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|
||||||
ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17.<P></P><DT><A NAME="blas2">12</A>
|
|
||||||
<DD>
|
|
||||||
J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson,
|
|
||||||
<I>An extended set of FORTRAN Basic Linear Algebra Subprograms</I>,
|
|
||||||
ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17.<P></P><DT><A NAME="PSBLASGUIDE">13</A>
|
|
||||||
<DD>
|
|
||||||
S. Filippone, A. Buttari,
|
|
||||||
<EM>PSBLAS 3.5.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library</EM>, 2012,
|
|
||||||
available from <TT>https://github.com/sfilippone/psblas3/tree/master/docs</TT>.<P></P><DT><A NAME="PSBLAS3">14</A>
|
|
||||||
<DD>
|
|
||||||
S. Filippone, A. Buttari,
|
|
||||||
<EM>Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003</EM>.
|
|
||||||
ACM Transactions on on Mathematical Software, 38 (4), 2012, art. 23.<P></P><DT><A NAME="psblas_00">15</A>
|
|
||||||
<DD>
|
|
||||||
S. Filippone, M. Colajanni,
|
|
||||||
<EM>PSBLAS: A Library for Parallel Linear Algebra
|
|
||||||
Computation on Sparse Matrices</EM>,
|
|
||||||
ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550.<P></P><DT><A NAME="MPI2">16</A>
|
|
||||||
<DD>
|
|
||||||
W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir,
|
|
||||||
<EM>MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions</EM>,
|
|
||||||
MIT Press, 1998.<P></P><DT><A NAME="blas1">17</A>
|
|
||||||
<DD>
|
|
||||||
C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh,
|
|
||||||
<I>Basic Linear Algebra Subprograms for FORTRAN usage</I>,
|
|
||||||
ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323.<P></P><DT><A NAME="SUPERLUDIST">18</A>
|
|
||||||
<DD>
|
|
||||||
X. S. Li, J. W. Demmel,
|
|
||||||
<EM>SuperLU_DIST: A Scalable Distributed-memory
|
|
||||||
Sparse Direct Solver for Unsymmetric Linear Systems</EM>,
|
|
||||||
ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140.<P></P><DT><A NAME="Notay2008">19</A>
|
|
||||||
<DD>
|
|
||||||
Y. Notay, P. S. Vassilevski,
|
|
||||||
<EM>Recursive Krylov-based multigrid cycles</EM>,
|
|
||||||
Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487. <P></P><DT><A NAME="Saad_book">20</A>
|
|
||||||
<DD>
|
|
||||||
Y. Saad,
|
|
||||||
<EM>Iterative methods for sparse linear systems</EM>, 2nd edition, SIAM, 2003.<P></P><DT><A NAME="dd2_96">21</A>
|
|
||||||
<DD>
|
|
||||||
B. Smith, P. Bjorstad, W. Gropp,
|
|
||||||
<EM>Domain Decomposition: Parallel Multilevel Methods for Elliptic
|
|
||||||
Partial Differential Equations</EM>,
|
|
||||||
Cambridge University Press, 1996.<P></P><DT><A NAME="MPI1">22</A>
|
|
||||||
<DD>
|
|
||||||
M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra,
|
|
||||||
<EM>MPI: The Complete Reference. Volume 1 - The MPI Core</EM>, second edition,
|
|
||||||
MIT Press, 1998.<P></P><DT><A NAME="Stuben_01">23</A>
|
|
||||||
<DD>
|
|
||||||
K. Stüben,
|
|
||||||
<EM>An Introduction to Algebraic Multigrid</EM>,
|
|
||||||
in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid,
|
|
||||||
Academic Press, 2001.<P></P><DT><A NAME="TUMINARO_TONG">24</A>
|
|
||||||
<DD>
|
|
||||||
R. S. Tuminaro, C. Tong,
|
|
||||||
<EM>Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines</EM>, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.<P></P><DT><A NAME="VANEK_MANDEL_BREZINA">25</A>
|
|
||||||
<DD>
|
|
||||||
P. Vanek, J. Mandel, M. Brezina,
|
|
||||||
<EM>Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems</EM>,
|
|
||||||
Computing, 56 (3) 1996, 179-196.
|
|
||||||
|
|
||||||
<P>
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<P>
|
||||||
</DL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
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Return memory footprint in bytes.
|
||||||
|
</BIG></BIG></BIG>
|
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|
||||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><BR><HR>
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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HREF="node27.html">Auxiliary Methods</A>
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<H1><A NAME="SECTION000130000000000000000">
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<H3><A NAME="SECTION00089400000000000000">
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About this document ...</A>
|
Method: allocate_wrk</A>
|
||||||
</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
</H3><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
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</FONT></FONT></FONT><P>
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|
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This document was generated using the
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||||||
<A HREF="http://www.latex2html.org/"><STRONG>LaTeX</STRONG>2<tt>HTML</tt></A> translator Version 2012 (1.2)
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|
||||||
<P>
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<P>
|
||||||
Copyright © 1993, 1994, 1995, 1996,
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
<A HREF="http://cbl.leeds.ac.uk/nikos/personal.html">Nikos Drakos</A>,
|
<DIV ALIGN="CENTER"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><code>call p%allocate_wrk(info[, vmold])</code>
|
||||||
Computer Based Learning Unit, University of Leeds.
|
<BR></BIG></BIG></BIG></DIV><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
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||||||
<BR>
|
|
||||||
Copyright © 1997, 1998, 1999,
|
|
||||||
<A HREF="http://www.maths.mq.edu.au/~ross/">Ross Moore</A>,
|
|
||||||
Mathematics Department, Macquarie University, Sydney.
|
|
||||||
<P>
|
<P>
|
||||||
The command line arguments were: <BR>
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
<STRONG>latex2html</STRONG> <TT>-local_icons -noaddress -dir ../../html userhtml.tex</TT>
|
Allocate internal work vectors. Each application of the preconditioner
|
||||||
|
uses a number of work vectors which are allocated internally as
|
||||||
|
necessary; therefore allocation and deallocation of memory occurs
|
||||||
|
multiple times during the execution of a Krylov method. In most cases
|
||||||
|
this strategy is perfectly acceptable, but
|
||||||
|
on some platforms, most notably GPUs, memory allocation is
|
||||||
|
a slow operation, and the default behaviour would lead to a
|
||||||
|
slowdown. This method allows to trade space for time by preallocating
|
||||||
|
the internal workspace outside of the invocation of a Krylov
|
||||||
|
method. When using GPUs or other specialized devices, the <code>vmold</code>
|
||||||
|
argument is also necessary to ensure the internal work vectors are of
|
||||||
|
the appropriate dynamic type to exploit the accelerator hardware; when
|
||||||
|
allocation occurs internally this is taken care of based on the dynamic
|
||||||
|
type of the <code>x</code> argument to the <code>apply</code> method.
|
||||||
|
</BIG></BIG></BIG>
|
||||||
<P>
|
<P>
|
||||||
The translation was initiated by Salvatore Filippone on 2017-10-23<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><B>Arguments</B></BIG> </BIG></BIG></BIG>
|
||||||
<BR><HR>
|
<P></P>
|
||||||
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
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|
||||||
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG><TABLE CELLPADDING=3>
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||||||
|
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
<code>info</code> </BIG></BIG></BIG></TD>
|
||||||
|
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> <code>integer, intent(out)</code>.</BIG></BIG></BIG></TD>
|
||||||
|
</TR>
|
||||||
|
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
</BIG></BIG></BIG></TD>
|
||||||
|
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> Error code. If no error, 0 is returned. See Section <A HREF="node34.html#sec:errors">8</A> for details.</BIG></BIG></BIG></TD>
|
||||||
|
</TR>
|
||||||
|
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
<code>vmold</code> </BIG></BIG></BIG></TD>
|
||||||
|
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> <code>class(psb_</code><SPAN CLASS="textit">x</SPAN><code>_base_vect_type), intent(in), optional</code>. </BIG></BIG></BIG></TD>
|
||||||
|
</TR>
|
||||||
|
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
</BIG></BIG></BIG></TD>
|
||||||
|
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> The desired dynamic type for internal vector
|
||||||
|
components; this allows e.g. running on GPUs. </BIG></BIG></BIG></TD>
|
||||||
|
</TR>
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HREF="node33.html">Adding new smoother and</A>
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HREF="node27.html">Auxiliary Methods</A>
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Method: free_wrk</A>
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<DIV ALIGN="CENTER"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><code>call p%free_wrk(info)</code>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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Deallocate internal work vectors.
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"><B>Arguments</B></BIG> </BIG></BIG></BIG>
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<code>info</code> </BIG></BIG></BIG></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> <code>integer, intent(out)</code>.</BIG></BIG></BIG></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> Error code. If no error, 0 is returned. See Section <A HREF="node34.html#sec:errors">8</A> for details.</BIG></BIG></BIG></TD>
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HREF="node32.html">Method: free_wrk</A>
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<H1><A NAME="SECTION00090000000000000000"></A><A NAME="sec:adding"></A>
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<BR>
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||||||
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Adding new smoother and solver objects to MLD2P4
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</H1><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
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<P>
|
||||||
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Developers can add completely new smoother and/or solver classes
|
||||||
|
derived from the base objects in the library (see Remark 2 in Section <A HREF="node20.html#sec:precset">6.2</A>),
|
||||||
|
without recompiling the library itself.
|
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</BIG></BIG></BIG>
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<P>
|
||||||
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">To do so, it is necessary first to select the base type to be extended.
|
||||||
|
In our experience, it is quite likely that the new application needs
|
||||||
|
only the definition of a “solver” object, which is almost
|
||||||
|
always acting only on the local part of the distributed matrix.
|
||||||
|
The parallel actions required to connect the various solver objects
|
||||||
|
are most often already provided by the block-Jacobi or the additive
|
||||||
|
Schwarz smoothers. To define a new solver, the developer will then
|
||||||
|
have to define its components and methods, perhaps taking one of the
|
||||||
|
predefined solvers as a starting point, if possible.
|
||||||
|
</BIG></BIG></BIG>
|
||||||
|
<P>
|
||||||
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Once the new smoother/solver class has been developed, to use it in
|
||||||
|
the context of the multilevel preconditioners it is necessary to:
|
||||||
|
</BIG></BIG></BIG>
|
||||||
|
<UL>
|
||||||
|
<LI>declare in the application program a variable of the new type;
|
||||||
|
</LI>
|
||||||
|
<LI>pass that variable as the argument to the <code>set</code> routine as in the
|
||||||
|
following:
|
||||||
|
<DIV ALIGN="CENTER">
|
||||||
|
<code>call p%set(smoother,info [,ilev,ilmax,pos])</code>
|
||||||
|
<BR><code>call p%set(solver,info [,ilev,ilmax,pos])</code>
|
||||||
|
|
||||||
|
</DIV>
|
||||||
|
</LI>
|
||||||
|
<LI>link the code implementing the various methods into the application executable.
|
||||||
|
</LI>
|
||||||
|
</UL><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
The new solver object is then dynamically included in the
|
||||||
|
preconditioner structure, and acts as a <SPAN CLASS="textit">mold</SPAN> to which the
|
||||||
|
preconditioner will conform, even though the MLD2P4 library has not
|
||||||
|
been modified to account for this new development.
|
||||||
|
</BIG></BIG></BIG>
|
||||||
|
<P>
|
||||||
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">It is possible to define new values for the keyword <code>WHAT</code> in the
|
||||||
|
<code>set</code> routine; if the library code does not recognize a keyword,
|
||||||
|
it passes it down the composition hierarchy (levels containing
|
||||||
|
smoothers containing in turn solvers), so that it can be eventually caught by
|
||||||
|
the new solver.
|
||||||
|
</BIG></BIG></BIG>
|
||||||
|
<P>
|
||||||
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">An example is provided in the source code distribution under the
|
||||||
|
folder <code>tests/newslv</code>. In this example we are implementing a new
|
||||||
|
incomplete factorization variant (which is simply the ILU(0)
|
||||||
|
factorization under a new name). Because of the specifics of this case, it is
|
||||||
|
possible to reuse the basic structure of the ILU solver, with its
|
||||||
|
L/D/U components and the methods needed to apply the solver; only a
|
||||||
|
few methods, such as the description and most importantly the build,
|
||||||
|
need to be ovverridden (rewritten).
|
||||||
|
</BIG></BIG></BIG>
|
||||||
|
<P>
|
||||||
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The interfaces for the calls shown above are defined using
|
||||||
|
</BIG></BIG></BIG>
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<DIV ALIGN="CENTER"><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG><TABLE CELLPADDING=3>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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||||||
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<code>smoother</code> </BIG></BIG></BIG></TD>
|
||||||
|
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> <code>class(mld_x_base_smoother_type)</code> </BIG></BIG></BIG></TD>
|
||||||
|
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||||||
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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||||||
|
</BIG></BIG></BIG></TD>
|
||||||
|
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> The user-defined new smoother to be employed in the
|
||||||
|
preconditioner.</BIG></BIG></BIG></TD>
|
||||||
|
</TR>
|
||||||
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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||||||
|
<code>solver</code> </BIG></BIG></BIG></TD>
|
||||||
|
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> <code>class(mld_x_base_solver_type)</code> </BIG></BIG></BIG></TD>
|
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|
</TR>
|
||||||
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
</BIG></BIG></BIG></TD>
|
||||||
|
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> The user-defined new solver to be employed in the
|
||||||
|
preconditioner.
|
||||||
|
</BIG></BIG></BIG></TD>
|
||||||
|
</TR>
|
||||||
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</TABLE><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG></DIV><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
The other arguments are defined in the way described in
|
||||||
|
Sec. <A HREF="node20.html#sec:precset">6.2</A>. As an example, in the <code>tests/newslv</code>
|
||||||
|
code we define a new object of type <code>mld_d_tlu_solver_type</code>, and
|
||||||
|
we pass it as follows:
|
||||||
|
</BIG></BIG></BIG><PRE>
|
||||||
|
! sparse matrix and preconditioner
|
||||||
|
type(psb_dspmat_type) :: a
|
||||||
|
type(mld_dprec_type) :: prec
|
||||||
|
type(mld_d_tlu_solver_type) :: tlusv
|
||||||
|
|
||||||
|
......
|
||||||
|
!
|
||||||
|
! prepare the preconditioner: an ML with defaults, but with TLU solver at
|
||||||
|
! intermediate levels. All other parameters are at default values.
|
||||||
|
!
|
||||||
|
call prec%init('ML', info)
|
||||||
|
call prec%hierarchy_build(a,desc_a,info)
|
||||||
|
nlv = prec%get_nlevs()
|
||||||
|
call prec%set(tlusv, info,ilev=1,ilmax=max(1,nlv-1))
|
||||||
|
call prec%smoothers_build(a,desc_a,info)
|
||||||
|
</PRE><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
</BIG></BIG></BIG>
|
||||||
|
<DIV CLASS="navigation"><HR>
|
||||||
|
<!--Navigation Panel-->
|
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|
<A NAME="tex2html501"
|
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|
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HREF="node2.html">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
||||||
|
<BR>
|
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HREF="node32.html">Method: free_wrk</A>
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<H1><A NAME="SECTION000100000000000000000"></A><A NAME="sec:errors"></A>
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<BR>
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Error Handling
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</H1><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
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||||||
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<P>
|
||||||
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The error handling in MLD2P4 is based on the PSBLAS error
|
||||||
|
handling. Error conditions are signaled via an integer argument
|
||||||
|
<code>info</code>; whenever an error condition is detected, an error trace
|
||||||
|
stack is built by the library up to the top-level, user-callable
|
||||||
|
routine. This routine will then decide, according to the user
|
||||||
|
preferences, whether the error should be handled by terminating the
|
||||||
|
program or by returning the error condition to the user code, which
|
||||||
|
will then take action, and whether
|
||||||
|
an error message should be printed. These options may be set by using
|
||||||
|
the PSBLAS error handling routines; for further details see the PSBLAS
|
||||||
|
User's Guide [<A
|
||||||
|
HREF="node36.html#PSBLASGUIDE">13</A>].
|
||||||
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<H1><A NAME="SECTION000110000000000000000"></A><A NAME="sec:license"></A>
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|
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License
|
||||||
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</H1><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
|
<P>
|
||||||
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The MLD2P4 is freely distributable under the following copyright
|
||||||
|
terms: </BIG></BIG></BIG><PRE>
|
||||||
|
|
||||||
|
MLD2P4 version 2.1
|
||||||
|
MultiLevel Domain Decomposition Parallel Preconditioners Package
|
||||||
|
based on PSBLAS (Parallel Sparse BLAS version 3.5)
|
||||||
|
|
||||||
|
(C) Copyright 2008, 2010, 2012, 2015, 2017
|
||||||
|
|
||||||
|
Salvatore Filippone Cranfield University, Cranfield, UK
|
||||||
|
Pasqua D'Ambra IAC-CNR, Naples, IT
|
||||||
|
Daniela di Serafino University of Campania L. Vanvitelli, Caserta, IT
|
||||||
|
|
||||||
|
Redistribution and use in source and binary forms, with or without
|
||||||
|
modification, are permitted provided that the following conditions
|
||||||
|
are met:
|
||||||
|
1. Redistributions of source code must retain the above copyright
|
||||||
|
notice, this list of conditions and the following disclaimer.
|
||||||
|
2. Redistributions in binary form must reproduce the above copyright
|
||||||
|
notice, this list of conditions, and the following disclaimer in the
|
||||||
|
documentation and/or other materials provided with the distribution.
|
||||||
|
3. The name of the MLD2P4 group or the names of its contributors may
|
||||||
|
not be used to endorse or promote products derived from this
|
||||||
|
software without specific written permission.
|
||||||
|
|
||||||
|
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
|
||||||
|
“AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
|
||||||
|
TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
|
||||||
|
PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE MLD2P4 GROUP OR ITS CONTRIBUTORS
|
||||||
|
BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
|
||||||
|
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
|
||||||
|
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
|
||||||
|
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
|
||||||
|
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
|
||||||
|
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
|
||||||
|
POSSIBILITY OF SUCH DAMAGE.
|
||||||
|
</PRE><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
|
|
||||||
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|
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|
||||||
|
<H2><A NAME="SECTION000120000000000000000">
|
||||||
|
Bibliography</A>
|
||||||
|
</H2><DL COMPACT><DD>
|
||||||
|
|
||||||
|
<P>
|
||||||
|
<P></P><DT><A NAME="MUMPS">1</A>
|
||||||
|
<DD>
|
||||||
|
P. R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. L'Excellent, C. Weisbecker,
|
||||||
|
<EM>Improving multifrontal methods by means of block low-rank representations</EM>,
|
||||||
|
SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474.
|
||||||
|
See also <TT>http://mumps.enseeiht.fr</TT>.
|
||||||
|
<P></P><DT><A NAME="BREZINA_VANEK">2</A>
|
||||||
|
<DD>
|
||||||
|
M. Brezina, P. Vanek,
|
||||||
|
<EM>A Black-Box Iterative Solver Based on a Two-Level Schwarz Method</EM>,
|
||||||
|
Computing, 63, 1999, 233-263.
|
||||||
|
<P></P><DT><A NAME="Briggs2000">3</A>
|
||||||
|
<DD>
|
||||||
|
W. L. Briggs, V. E. Henson, S. F. McCormick,
|
||||||
|
<EM>A Multigrid Tutorial, Second Edition</EM>,
|
||||||
|
SIAM, 2000.
|
||||||
|
<P></P><DT><A NAME="para_04">4</A>
|
||||||
|
<DD>
|
||||||
|
A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone,
|
||||||
|
<EM>Extending PSBLAS to Build Parallel Schwarz Preconditioners</EM>,
|
||||||
|
in J. Dongarra, K. Madsen, J. Wasniewski, editors,
|
||||||
|
Proceedings of PARA 04 Workshop on State of the Art
|
||||||
|
in Scientific Computing, Lecture Notes in Computer Science,
|
||||||
|
Springer, 2005, 593-602.
|
||||||
|
<P></P><DT><A NAME="aaecc_07">5</A>
|
||||||
|
<DD>
|
||||||
|
A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone,
|
||||||
|
<EM>2LEV-D2P4: a package of high-performance preconditioners
|
||||||
|
for scientific and engineering applications</EM>,
|
||||||
|
Applicable Algebra in Engineering, Communications and Computing,
|
||||||
|
18 (3) 2007, 223-239.
|
||||||
|
<P></P><DT><A NAME="CAI_SARKIS">6</A>
|
||||||
|
<DD>
|
||||||
|
X. C. Cai, M. Sarkis,
|
||||||
|
<EM>A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems</EM>,
|
||||||
|
SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797.
|
||||||
|
<P></P><DT><A NAME="apnum_07">7</A>
|
||||||
|
<DD>
|
||||||
|
P. D'Ambra, S. Filippone, D. di Serafino,
|
||||||
|
<EM>On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners</EM>,
|
||||||
|
Applied Numerical Mathematics, Elsevier Science,
|
||||||
|
57 (11-12), 2007, 1181-1196.
|
||||||
|
<P></P><DT><A NAME="MLD2P4_TOMS">8</A>
|
||||||
|
<DD>
|
||||||
|
P. D'Ambra, D. di Serafino, S. Filippone,
|
||||||
|
<SPAN CLASS="textit">MLD2P4: a Package of Parallel Multilevel
|
||||||
|
Algebraic Domain Decomposition Preconditioners
|
||||||
|
in Fortran 95</SPAN>, ACM Trans. Math. Softw., 37(3), 2010, art. 30.
|
||||||
|
<P></P><DT><A NAME="UMFPACK">9</A>
|
||||||
|
<DD>
|
||||||
|
T. A. Davis,
|
||||||
|
<EM>Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal
|
||||||
|
Method with a Column Pre-ordering Strategy</EM>,
|
||||||
|
ACM Transactions on Mathematical Software, 30, 2004, 196-199.
|
||||||
|
(See also <TT>http://www.cise.ufl.edu/~davis/</TT>)
|
||||||
|
<P></P><DT><A NAME="SUPERLU">10</A>
|
||||||
|
<DD>
|
||||||
|
J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu,
|
||||||
|
<EM>A supernodal approach to sparse partial pivoting</EM>,
|
||||||
|
SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755.
|
||||||
|
<P></P><DT><A NAME="blas3">11</A>
|
||||||
|
<DD>
|
||||||
|
J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling,
|
||||||
|
<SPAN CLASS="textit">A set of Level 3 Basic Linear Algebra Subprograms</SPAN>,
|
||||||
|
ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17.
|
||||||
|
<P></P><DT><A NAME="blas2">12</A>
|
||||||
|
<DD>
|
||||||
|
J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson,
|
||||||
|
<SPAN CLASS="textit">An extended set of FORTRAN Basic Linear Algebra Subprograms</SPAN>,
|
||||||
|
ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17.
|
||||||
|
<P></P><DT><A NAME="PSBLASGUIDE">13</A>
|
||||||
|
<DD>
|
||||||
|
S. Filippone, A. Buttari,
|
||||||
|
<EM>PSBLAS 3.5.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library</EM>, 2012,
|
||||||
|
available from <TT>https://github.com/sfilippone/psblas3/tree/master/docs</TT>.
|
||||||
|
<P></P><DT><A NAME="PSBLAS3">14</A>
|
||||||
|
<DD>
|
||||||
|
S. Filippone, A. Buttari,
|
||||||
|
<EM>Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003</EM>.
|
||||||
|
ACM Transactions on on Mathematical Software, 38 (4), 2012, art. 23.
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|
<P></P><DT><A NAME="psblas_00">15</A>
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|
<DD>
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|
S. Filippone, M. Colajanni,
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|
<EM>PSBLAS: A Library for Parallel Linear Algebra
|
||||||
|
Computation on Sparse Matrices</EM>,
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||||||
|
ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550.
|
||||||
|
<P></P><DT><A NAME="MPI2">16</A>
|
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|
<DD>
|
||||||
|
W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir,
|
||||||
|
<EM>MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions</EM>,
|
||||||
|
MIT Press, 1998.
|
||||||
|
<P></P><DT><A NAME="blas1">17</A>
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<DD>
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|
C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh,
|
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|
<SPAN CLASS="textit">Basic Linear Algebra Subprograms for FORTRAN usage</SPAN>,
|
||||||
|
ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323.
|
||||||
|
<P></P><DT><A NAME="SUPERLUDIST">18</A>
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|
<DD>
|
||||||
|
X. S. Li, J. W. Demmel,
|
||||||
|
<EM>SuperLU_DIST: A Scalable Distributed-memory
|
||||||
|
Sparse Direct Solver for Unsymmetric Linear Systems</EM>,
|
||||||
|
ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140.
|
||||||
|
<P></P><DT><A NAME="Notay2008">19</A>
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|
<DD>
|
||||||
|
Y. Notay, P. S. Vassilevski,
|
||||||
|
<EM>Recursive Krylov-based multigrid cycles</EM>,
|
||||||
|
Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487.
|
||||||
|
<P></P><DT><A NAME="Saad_book">20</A>
|
||||||
|
<DD>
|
||||||
|
Y. Saad,
|
||||||
|
<EM>Iterative methods for sparse linear systems</EM>, 2nd edition, SIAM, 2003.
|
||||||
|
<P></P><DT><A NAME="dd2_96">21</A>
|
||||||
|
<DD>
|
||||||
|
B. Smith, P. Bjorstad, W. Gropp,
|
||||||
|
<EM>Domain Decomposition: Parallel Multilevel Methods for Elliptic
|
||||||
|
Partial Differential Equations</EM>,
|
||||||
|
Cambridge University Press, 1996.
|
||||||
|
<P></P><DT><A NAME="MPI1">22</A>
|
||||||
|
<DD>
|
||||||
|
M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra,
|
||||||
|
<EM>MPI: The Complete Reference. Volume 1 - The MPI Core</EM>, second edition,
|
||||||
|
MIT Press, 1998.
|
||||||
|
<P></P><DT><A NAME="Stuben_01">23</A>
|
||||||
|
<DD>
|
||||||
|
K. Stüben,
|
||||||
|
<EM>An Introduction to Algebraic Multigrid</EM>,
|
||||||
|
in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid,
|
||||||
|
Academic Press, 2001.
|
||||||
|
<P></P><DT><A NAME="TUMINARO_TONG">24</A>
|
||||||
|
<DD>
|
||||||
|
R. S. Tuminaro, C. Tong,
|
||||||
|
<EM>Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines</EM>, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.
|
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|
<P></P><DT><A NAME="VANEK_MANDEL_BREZINA">25</A>
|
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|
<DD>
|
||||||
|
P. Vanek, J. Mandel, M. Brezina,
|
||||||
|
<EM>Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems</EM>,
|
||||||
|
Computing, 56 (3) 1996, 179-196.
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|
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<P>
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Copyright © 1993, 1994, 1995, 1996,
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Computer Based Learning Unit, University of Leeds.
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<BR>
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Copyright © 1997, 1998, 1999,
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<A HREF="http://www.maths.mq.edu.au/~ross/">Ross Moore</A>,
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Mathematics Department, Macquarie University, Sydney.
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