mld2p4:
html/img1.png html/img32.png html/img52.png html/img58.png html/img71.png html/img83.png html/node11.html html/node12.html html/node3.html pdf pdf/Makefile pdf/abstract.tex pdf/background.tex pdf/bibliography.tex pdf/building.tex pdf/distribution.tex pdf/errors.tex pdf/figures pdf/gettingstarted.tex pdf/intro.tex pdf/license.tex pdf/overview.tex pdf/precs.tex pdf/title.tex pdf/tmp pdf/userguide.tex pdf/userhtml.tex pdf/userinterface.tex src src/Makefile src/abstract.tex src/background.tex src/bibliography.tex src/building.tex src/distribution.tex src/errors.tex src/figures src/gettingstarted.tex src/intro.tex src/license.tex src/overview.tex src/precs.tex src/title.tex src/tmp src/userguide.tex src/userhtml.tex src/userinterface.tex Docs: created SRC subdir.stopcriterion
Salvatore Filippone
16 years ago
parent
1b4d970e6f
commit
ef5a0f3d99
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## $Id: Makefile 1524 2007-01-17 17:06:06Z sfilippo $
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##---------------------------------------------------------------------------
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## LaTeX Makefile
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## Copyright (C) 1996-2001 Michael Forman Michael.Forman@Colorado.EDU
|
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##
|
||||
## This program is free software; you can redistribute it and/or
|
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## modify it under the terms of the GNU General Public License
|
||||
## as published by the Free Software Foundation; either version 2
|
||||
## of the License, or (at your option) any later version.
|
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##
|
||||
## This program is distributed in the hope that it will be useful,
|
||||
## but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
||||
## GNU General Public License for more details.
|
||||
##
|
||||
## You should have received a copy of the GNU General Public License
|
||||
## along with this program; if not, write to the Free Software
|
||||
## Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
|
||||
##
|
||||
## This copyright applies to this Makefile, and all perl scripts.
|
||||
## The GPL does NOT apply to the actual content of the paper or thesis.
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##---------------------------------------------------------------------------
|
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## 01.Dec,1996 forman Initial Makefile
|
||||
## 01.Jun,1997 forman Added support for print, preview, and bibtex
|
||||
## 01.Jan,1998 stones tmp and lib directories to reduce clutter
|
||||
## 05.Feb,1998 forman Added the following functions:
|
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## .PHONY to prevent errors
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||||
## generic TEXFILE definition with following patsubst's
|
||||
## vpath definitions
|
||||
## gzip, gunzip, tar, ci
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## search for \bibliography in tex file
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||||
## conditional ifeq omits bibtex if unnecessary
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## documentation added
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## 30.Nov,1998 forman Added TOPFILE and SECFILE definitions to support
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## texfiles with "input" commands.
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## 23.Dec,1998 marshats Added RCSFILES and ci/co capabilites for multiple files
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## Now only runs makeindex if $(IDX) file exists.
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## 15.Mar,1999 forman Added 'make wc' for papers with word quotas.
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## 24.Aug,1999 forman Converted Makefile to use pdftex as the primary
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## complier. Now generates true pdf and ps files.
|
||||
## 09.Feb,2000 forman Simplified the documentation. Removed direct
|
||||
## compilation of tex into DVI and PS. Conversion
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## is done with PDF2PS now.
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||||
## 08.Aug,2000 forman Added define for figures directories.
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## 18.Aug,2000 forman Removed the redundant code in the $(pdflatex) and
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## $(pdflatex-bibtex) definitions by splitting them up
|
||||
## into several smaller definitions which are called
|
||||
## from a single set of "if-then" statements.
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||||
## 18.Aug,2000 forman Added support for glosstex and makeindex.
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||||
## 19.Aug,2000 forman To save space, all files in tmp are now links.
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||||
## 19.Aug,2000 forman Removed all postscript commands in favor of pdf.
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||||
##---------------------------------------------------------------------------
|
||||
##
|
||||
## This Makefile expects the following directory structure:
|
||||
##
|
||||
## Makefile This file
|
||||
## *.tex Put latex files in root directory.
|
||||
## RCS/ Create an RCS directory for "ci" and "co".
|
||||
## doc/ Document directory. Includes gpl.txt.
|
||||
## lib/ Put all cls, sty, idx, gdf, and bib files in lib.
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## figures/ Put all figures in the figures directory.
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||||
## tmp/ Never put anything in tmp -- it gets cleaned out.
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##
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||||
##---------------------------------------------------------------------------
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||||
##
|
||||
## Normal Usage:
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## make Run pdflatex
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||||
## make clean Remove all files in tmp and the pdf link in root.
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## make preview Preview the compiled file
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||||
## make ci Check in the RCSFILES
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||||
## make co Check out the RCSFILES
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##
|
||||
## Advanced Usage:
|
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## make gzip Recursively gzip all the files in the root directory
|
||||
## make gunzip Reverse the above process
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## make tar Tar and gzip the working directory
|
||||
## make wc Count the number of words in your report
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##
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##---------------------------------------------------------------------------
|
||||
## WARNING:
|
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## If "make ci" fails, "make co" will nuke your files!
|
||||
## Don't put anything in tmp, "make clean" will delete everything!
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##---------------------------------------------------------------------------
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||||
#
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||||
|
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TOPFILE = userguide.tex
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HTMLFILE = userhtml.tex
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SECFILE = title.tex abstract.tex overview.tex distribution.tex \
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building.tex background.tex gettingstarted.tex userinterface.tex \
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errors.tex bibliography.tex license.tex
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||||
FIGDIR = figures
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||||
|
||||
XPDFFLAGS =
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ACROFLAGS = #-- macos
|
||||
#ACROFLAGS = -geometry 1234x1168+0+0 #-- new-sydney-wide
|
||||
#ACROFLAGS = -geometry 1000x1000+0+0 #-- sydney-wide
|
||||
#ACROFLAGS = -geometry 750x1000+0+0 #-- sydney
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||||
#ACROFLAGS = -geometry 1200x1200+0+0 #-- home-wide
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||||
#ACROFLAGS = -geometry 900x1200+0+0 #-- home
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||||
|
||||
##---------------------------------------------------------------------------
|
||||
## Change nothing below here (unless you're really really good).
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||||
#----------------------------------------------------------------------------
|
||||
|
||||
##
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## Accounting
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#
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||||
START = $(shell date)
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||||
WDIR = $(notdir $(shell pwd))
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||||
DATE = $(shell date +%Y-%m-%d)
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||||
|
||||
##
|
||||
## Programs
|
||||
#
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||||
LATEX = latex
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||||
LTX2HTML = latex2html
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||||
PDFLATEX = pdflatex
|
||||
ACRO = evince
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||||
XPDF = xpdf
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||||
WC = wc
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||||
PDF2PS = pdf2ps
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PDF2TEXT = pdftotext
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||||
MAKEIDX = makeindex
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||||
GLOSSTEX = glosstex
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||||
BIBTEX = bibtex
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FILTER = ../bin/texfilter
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||||
FILTER = cat
|
||||
CLEANIDX = ../bin/cleanidx
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|
||||
##
|
||||
## Files
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||||
#
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TEXFILES = $(TOPFILE) $(SECFILE) $(HTMLFILE)
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||||
RCSFILES = $(TEXFILES) Makefile
|
||||
|
||||
BASEFILE = $(patsubst %.tex,%,$(TOPFILE))
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IDX = $(join $(BASEFILE),.idx)
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PDF = $(join $(BASEFILE),.pdf)
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||||
PS = $(join $(BASEFILE),.ps)
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||||
GXS = $(join $(BASEFILE),.gxs)
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||||
GLX = $(join $(BASEFILE),.glx)
|
||||
BASEHTML = $(patsubst %.tex,%,$(HTMLFILE))
|
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HTML = $(join $(HTMLFILE),.html)
|
||||
HTMLDIR = ../html
|
||||
HTMLFLAGS = -noaddress
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|
||||
FIGURES = $(sort $(wildcard $(FIGDIR)/*))
|
||||
GLOFILES:= $(sort $(wildcard lib/*.gdf))
|
||||
GLOFILES:= $(patsubst lib/%,%,$(GLOFILES))
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||||
BIBFILES:= $(shell grep ^.bibliography{ $(TOPFILE)|sed "s/,/ /g"|sed "s/%.*//g")
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BIBFILES:= $(patsubst \bibliography{%,%,$(BIBFILES))
|
||||
BIBFILES:= $(patsubst %},%,$(BIBFILES))
|
||||
BIBFILES:= $(sort $(addsuffix .bib,$(BIBFILES)))
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|
||||
LIBFILES = $(sort $(wildcard lib/*))
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|
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TEXLNFIL = $(addprefix $(PWD)/,$(TEXFILES))
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|
||||
#============================================================================
|
||||
|
||||
all: pdf html
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||||
|
||||
pdf: $(PDF)
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||||
html: $(HTML)
|
||||
|
||||
$(PDF): $(TEXFILES) $(LIBFILES) $(FIGURES) Makefile
|
||||
$(header)
|
||||
$(initialize)
|
||||
$(pdflatex-filter)
|
||||
ifneq ($(BIBFILES),)
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$(bibtex)
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||||
$(pdflatex-filter)
|
||||
endif
|
||||
ifneq ($(GLOFILES),)
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||||
$(glosstex)
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||||
$(pdflatex-filter)
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||||
endif
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||||
# $(makeindex)
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$(pdflatex-filter)
|
||||
$(finish)
|
||||
|
||||
$(HTML): $(TEXFILES) $(LIBFILES) $(FIGURES) Makefile
|
||||
$(header)
|
||||
$(initialize)
|
||||
$(latex-filter)
|
||||
ifneq ($(BIBFILES),)
|
||||
$(bibtex)
|
||||
$(latex-filter)
|
||||
endif
|
||||
ifneq ($(GLOFILES),)
|
||||
$(glosstex)
|
||||
$(latex-filter)
|
||||
endif
|
||||
# $(makeindex)
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||||
$(latex-filter)
|
||||
$(ltx2html-filter)
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||||
|
||||
$(PS): $(PDF)
|
||||
$(PDF2PS) $< $(PS)
|
||||
|
||||
ps: $(PS)
|
||||
|
||||
#============================================================================
|
||||
|
||||
.PHONY: clean clean-tmp preview print gzip gunzip tar ci
|
||||
|
||||
clean:
|
||||
rm -f $(PDF) $(PS)
|
||||
cd tmp ; rm -f *
|
||||
|
||||
clean-tmp:
|
||||
cd tmp ; rm -f *
|
||||
|
||||
#----------------------------------------------------------------------------
|
||||
|
||||
preview: $(PDF)
|
||||
$(ACRO) $(ACROFLAGS) $< &
|
||||
|
||||
xpreview: $(PDF)
|
||||
$(XPDF) $(XPDFFLAGS) $< &
|
||||
|
||||
wc: $(PDF)
|
||||
$(PDF2TEXT) $< | $(WC)
|
||||
|
||||
#----------------------------------------------------------------------------
|
||||
|
||||
ci: $(RCSFILES)
|
||||
$(shell for i in $(RCSFILES) ; do ci -u $$i ; done)
|
||||
|
||||
co: $(RCSFILES)
|
||||
$(shell for i in $(RCSFILES) ; do co -l $$i ; done)
|
||||
|
||||
gzip: clean-tmp
|
||||
gzip -r ./
|
||||
gunzip Makefile.gz
|
||||
|
||||
gunzip:
|
||||
gunzip -r ./
|
||||
|
||||
tar: clean-tmp
|
||||
cd .. ; \
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||||
tar cvf $(WDIR)-$(DATE).tar $(WDIR) ; \
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||||
gzip $(WDIR)-$(DATE).tar
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||||
|
||||
info:
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||||
$(header)
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||||
|
||||
#============================================================================
|
||||
|
||||
define header
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||||
@echo
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||||
@echo "#---------------------------------------------------------------------"
|
||||
@echo "MAKEFILE = LaTeX PDF Makefile"
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||||
@echo "AUTHOR = Alfredo Buttari"
|
||||
@echo 'ID = $$Id: Makefile 1524 2007-01-17 17:06:06Z sfilippo $ '
|
||||
@echo "#---------------------------------------------------------------------"
|
||||
@echo
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||||
@echo "ACRO = $(ACRO) $(ACROFLAGS) $(PDF)"
|
||||
@echo "XPDF = $(XPDF) $(XPDFFLAGS) $(PDF)"
|
||||
@echo "GV = $(GV) $(GVFLAGS) $(PS)"
|
||||
@echo "LPR = $(LPR) $(LPRFLAGS) $(PS)"
|
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@echo
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||||
@echo "WDIR = $(WDIR)"
|
||||
@echo "DATE = $(DATE)"
|
||||
@echo
|
||||
@echo "TOPFILE = $(TOPFILE)"
|
||||
@echo "SECFILE = $(SECFILE)"
|
||||
@echo "TEXFILES = $(TEXFILES)"
|
||||
@echo "PDF = $(PDF)"
|
||||
@echo "PS = $(PS)"
|
||||
@echo "BIBFILES = $(BIBFILES)"
|
||||
@echo "GLOFILES = $(GLOFILES)"
|
||||
@echo "IDXFILES = $(IDXFILES)"
|
||||
@echo
|
||||
endef
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||||
# @echo "FIGURES = $(FIGURES)"
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|
||||
#----------------------------------------------------------------------------
|
||||
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||||
define initialize
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@if test ! -d tmp; then mkdir tmp; fi
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@ln -sf $(TEXLNFIL) tmp
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@ln -sf $(PWD)/lib/* tmp
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@ln -sf $(PWD)/$(FIGDIR) tmp
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||||
endef
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||||
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||||
#----------------------------------------------------------------------------
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define pdflatex
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@echo
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||||
@echo "----- pdflatex -------------------------------------------------------"
|
||||
@echo -n "Starting: "; date
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||||
@echo
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||||
cd tmp; $(PDFLATEX) $(TOPFILE)
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endef
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define latex
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||||
@echo
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||||
@echo "----- latex -------------------------------------------------------"
|
||||
@echo -n "Starting: "; date
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||||
@echo
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||||
cd tmp; $(LATEX) $(HTMLFILE)
|
||||
endef
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||||
|
||||
#----------------------------------------------------------------------------
|
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|
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define pdflatex-filter
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@echo
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||||
@echo "----- latex -------------------------------------------------------"
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||||
@echo -n "Starting: "; date
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||||
@echo
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||||
cd tmp; ($(PDFLATEX) $(TOPFILE) 2>&1) | $(FILTER)
|
||||
endef
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|
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define latex-filter
|
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@echo
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||||
@echo "----- latex -------------------------------------------------------"
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||||
@echo -n "Starting: "; date
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||||
@echo
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cd tmp; ($(LATEX) $(HTMLFILE) 2>&1) | $(FILTER)
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||||
endef
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|
||||
define ltx2html-filter
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@echo
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||||
@echo "----- latex -------------------------------------------------------"
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||||
@echo -n "Starting: "; date
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||||
@echo
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||||
cd tmp; ($(LTX2HTML) $(HTMLFLAGS) -dir ../$(HTMLDIR) $(HTMLFILE) 2>&1) | $(FILTER)
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||||
endef
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||||
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||||
#----------------------------------------------------------------------------
|
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|
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define bibtex
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@echo
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@echo "----- bibtex ---------------------------------------------------------"
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||||
@echo -n "Starting: "; date
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@echo
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cd tmp; $(BIBTEX) $(BASEFILE)
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||||
endef
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||||
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||||
#----------------------------------------------------------------------------
|
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|
||||
define glosstex
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@echo
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||||
@echo "----- glosstex -------------------------------------------------------"
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||||
@echo -n "Starting: "; date
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||||
@echo
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cd tmp; $(GLOSSTEX) $(BASEFILE) $(GLOFILES)
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cd tmp; $(MAKEIDX) $(GXS) -o $(GLX) -s glosstex.ist
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||||
endef
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||||
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||||
#----------------------------------------------------------------------------
|
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|
||||
define makeindex
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||||
@echo
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||||
@echo "----- makeindex ------------------------------------------------------"
|
||||
@echo -n "Starting: "; date
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||||
@echo
|
||||
cd tmp; mv $(IDX) $(IDX)-; $(CLEANIDX) < $(IDX)- > $(IDX)
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cd tmp; $(MAKEIDX) $(IDX)
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||||
endef
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||||
#----------------------------------------------------------------------------
|
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define finish
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||||
@ln -sf tmp/$@ .
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||||
@echo
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||||
@echo "----- finish ---------------------------------------------------------"
|
||||
@echo -n "Start: "$(START); echo
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@echo -n "Finish: "; date
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@echo -n "Output: "; ls -l -o tmp/$@
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@echo -n "Target: "
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||||
endef
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||||
|
||||
define ltx2html-finish
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@ln -sf tmp/$@ .
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||||
@echo
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||||
@echo "----- finish ---------------------------------------------------------"
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||||
@echo -n "Start: "$(START); echo
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@echo -n "Finish: "; date
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||||
@echo -n "Output: "; ls -l -o tmp/$@
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@echo -n "Target: "
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||||
endef
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@ -0,0 +1,27 @@
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\section*{Abstract}
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\addcontentsline{toc}{section}{Abstract}
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\textsc{MLD2P4 (Multi-Level Domain Decomposition Parallel Preconditioners Package based on
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PSBLAS}) is a package of parallel algebraic multi-level preconditioners.
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It implements various versions of one-level additive and of multi-level additive
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and hybrid Schwarz algorithms. In the multi-level case, a purely algebraic approach
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is applied to generate coarse-level corrections, so that no geometric background is needed
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concerning the matrix to be preconditioned. The matrix is required to be square, real
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or complex, with a symmetric sparsity pattern.
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MLD2P4 has been designed to provide scalable and easy-to-use preconditioners in the
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context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms)
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computational framework and can be used in conjuction with the Krylov solvers
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available in this framework. MLD2P4 enables the user to easily specify different aspects
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of a generic algebraic multilevel Schwarz preconditioner, thus allowing to search
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for the ``best'' preconditioner for the problem at hand.
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The package has been designed employing object-oriented techniques,
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using Fortran 95, with interfaces to additional third party libraries
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such as UMFPACK, SuperLU and SuperLU\_Dist, that
|
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can be exploited in building multi-level preconditioners. The parallel
|
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implementation is based on a Single Program Multiple Data (SPMD)
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paradigm for distributed-memory architectures; the inter-process data
|
||||
communication is based on MPI and is managed mainly through PSBLAS.
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This guide provides a brief description of the functionalities and
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the user interface of MLD2P4.
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@ -0,0 +1,348 @@
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\section{Multi-level Domain Decomposition Background\label{sec:background}}
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\markboth{\textsc{MLD2P4 User's and Reference Guide}}
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{\textsc{\ref{sec:background} Multi-level Domain Decomposition Background}}
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\emph{Domain Decomposition} (DD) preconditioners, coupled with Krylov iterative
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solvers, are widely used in the parallel solution of large and sparse linear systems.
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These preconditioners are based on the divide and conquer technique: the matrix
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to be preconditioned is divided into submatrices, a ``local'' linear system
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involving each submatrix is (approximately) solved, and the local solutions are used
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to build a preconditioner for the whole original matrix. This process
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often corresponds to dividing a physical domain associated to the original matrix
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into subdomains, e.g. in a PDE discretization, to (approximately) solving the
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subproblems corresponding to the subdomains and to building an approximate
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solution of the original problem from the local solutions
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\cite{Cai_Widlund_92,dd1_94,dd2_96}.
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\emph{Additive Schwarz} preconditioners are DD preconditioners using overlapping
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submatrices, i.e.\ with some common rows, to couple the local information
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related to the submatrices (see, e.g., \cite{dd2_96}).
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The main motivation for choosing Additive Schwarz preconditioners is their
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intrinsic parallelism. A drawback of these
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preconditioners is that the number of iterations of the preconditioned solvers
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generally grows with the number of submatrices. This may be a serious limitation
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on parallel computers, since the number of submatrices usually matches the number
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of available processors. Optimal convergence rates, i.e.\ iteration numbers
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independent of the number of submatrices, can be obtained by correcting the
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preconditioner through a suitable approximation of the original linear system
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in a coarse space, which globally couples the information related to the single
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submatrices.
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\emph{Two-level Schwarz} preconditioners are obtained
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by combining basic (one-level) Sch\-warz preconditioners with a coarse-level
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correction. In this context, the one-level preconditioner is often
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||||
called `smoother'. Different two-level preconditioners are obtained by varying the
|
||||
choice of the smoother and of the coarse-level correction, and the
|
||||
way they are combined \cite{dd2_96}. The same reasoning can be applied starting
|
||||
from the coarse-level system, i.e.\ a coarse-space correction can be built
|
||||
from this system, thus obtaining \emph{multi-level} preconditioners.
|
||||
|
||||
It is worth noting that optimal preconditioners do not necessarily correspond
|
||||
to minimum execution times. Indeed, to obtain effective multi-level preconditioners
|
||||
a tradeoff between optimality of convergence and the cost of building and applying
|
||||
the coarse-space corrections must be achieved. The choice of the number of levels,
|
||||
i.e.\ of the coarse-space corrections, also affects the effectiveness of the
|
||||
preconditioners. One more goal is to get convergence rates as less sensitive
|
||||
as possible to variations in the matrix coefficients.
|
||||
|
||||
Two main approaches can be used to build coarse-space corrections. The geometric approach
|
||||
applies coarsening strategies based on the knowledge of some physical grid associated
|
||||
to the matrix and requires the user to define grid transfer operators from the fine
|
||||
to the coarse levels and vice versa. This may result difficult for complex geometries;
|
||||
furthermore, suitable one-level preconditioners may be required to get efficient
|
||||
interplay between fine and coarse levels, e.g.\ when matrices with highly varying coefficients
|
||||
are considered. The algebraic approach builds coarse-space corrections using only matrix
|
||||
information. It performs a fully automatic coarsening and enforces the interplay between
|
||||
the fine and coarse levels by suitably choosing the coarse space and the coarse-to-fine
|
||||
interpolation \cite{StubenGMD69_99}.
|
||||
|
||||
MLD2P4 uses a pure algebraic approach for building the sequence of coarse matrices
|
||||
starting from the original matrix. The algebraic approach is based on the \emph{smoothed
|
||||
aggregation} algorithm \cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}. A decoupled version
|
||||
of this algorithm is implemented, where the smoothed aggregation is applied locally
|
||||
to each submatrix \cite{TUMINARO_TONG}. In the next two subsections we provide
|
||||
a brief description of the multi-level Schwarz preconditioners and of the smoothed
|
||||
aggregation technique as implemented in MLD2P4. For further details the user
|
||||
is referred to \cite{para_04,aaecc_07,apnum_07,dd2_96}.
|
||||
|
||||
|
||||
\subsection{Multi-level Schwarz Preconditioners\label{sec:multilevel}}
|
||||
|
||||
The Multilevel preconditioners implemented in MLD2P4 are obtained by combining
|
||||
AS preconditioners with coarse-space corrections; therefore
|
||||
we first provide a sketch of the AS preconditioners.
|
||||
|
||||
Given the linear system \Ref{system1},
|
||||
where $A=(a_{ij}) \in \Re^{n \times n}$ is a
|
||||
nonsingular sparse matrix with a symmetric nonzero pattern,
|
||||
let $G=(W,E)$ be the adjacency graph of $A$, where $W=\{1, 2, \ldots, n\}$
|
||||
and $E=\{(i,j) : a_{ij} \neq 0\}$ are the vertex set and the edge set of $G$,
|
||||
respectively. Two vertices are called adjacent if there is an edge connecting
|
||||
them. For any integer $\delta > 0$, a $\delta$-overlap
|
||||
partition of $W$ can be defined recursively as follows.
|
||||
Given a 0-overlap (or non-overlapping) partition of $W$,
|
||||
i.e.\ a set of $m$ disjoint nonempty sets $W_i^0 \subset W$ such that
|
||||
$\cup_{i=1}^m W_i^0 = W$, a $\delta$-overlap
|
||||
partition of $W$ is obtained by considering the sets
|
||||
$W_i^\delta \supset W_i^{\delta-1}$ obtained by including the vertices that
|
||||
are adjacent to any vertex in $W_i^{\delta-1}$.
|
||||
|
||||
Let $n_i^\delta$ be the size of $W_i^\delta$ and $R_i^{\delta} \in
|
||||
\Re^{n_i^\delta \times n}$ the restriction operator that maps
|
||||
a vector $v \in \Re^n$ onto the vector $v_i^{\delta} \in \Re^{n_i^\delta}$
|
||||
containing the components of $v$ corresponding to the vertices in
|
||||
$W_i^\delta$. The transpose of $R_i^{\delta}$ is a
|
||||
prolongation operator from $\Re^{n_i^\delta}$ to $\Re^n$.
|
||||
The matrix $A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
||||
\Re^{n_i^\delta \times n_i^\delta}$ can be considered
|
||||
as a restriction of $A$ corresponding to the set $W_i^{\delta}$.
|
||||
|
||||
The \emph{classical one-level AS} preconditioner is defined by
|
||||
\[
|
||||
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
(A_i^\delta)^{-1} R_i^{\delta},
|
||||
\]
|
||||
where $A_i^\delta$ is assumed to be nonsingular. Its application
|
||||
to a vector $v \in \Re^n$ within a Krylov solver requires the following
|
||||
three steps:
|
||||
\begin{enumerate}
|
||||
\item restriction of $v$ as $v_i = R_i^{\delta} v$, $i=1,\ldots,m$;
|
||||
\item solution of the linear systems $A_i^\delta w_i = v_i$,
|
||||
$i=1,\ldots,m$;
|
||||
\item prolongation and sum of the $w_i$'s, i.e. $w = \sum_{i=1}^m (R_i^{\delta})^T w_i$.
|
||||
\end{enumerate}
|
||||
Note that the linear systems at step 2 are usually solved approximately,
|
||||
e.g.\ using incomplete LU factorizations such as ILU($p$), MILU($p$) and
|
||||
ILU($p,t$) \cite[Chapter 10]{Saad_book}.
|
||||
|
||||
A variant of the classical AS preconditioner that outperforms it
|
||||
in terms of convergence rate and of computation and communication
|
||||
time on parallel distributed-memory computers is the so-called \emph{Restricted AS
|
||||
(RAS)} preconditioner~\cite{CAI_SARKIS,EFSTATHIOU}. It
|
||||
is obtained by zeroing the components of $w_i$ corresponding to the
|
||||
overlapping vertices when applying the prolongation. Therefore,
|
||||
RAS differs from classical AS by the prolongation operators,
|
||||
which are substituted by $(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$,
|
||||
where $\tilde{R}_i^0$ is obtained by zeroing the rows of $R_i^\delta$
|
||||
corresponding to the vertices in $W_i^\delta \backslash W_i^0$:
|
||||
\[
|
||||
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
(A_i^\delta)^{-1} R_i^{\delta}.
|
||||
\]
|
||||
Analogously, the AS variant called \emph{AS with Harmonic extension (ASH)}
|
||||
is defined by
|
||||
\[ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
(A_i^\delta)^{-1} \tilde{R}_i^0.
|
||||
\]
|
||||
We note that for $\delta=0$ the three variants of the AS preconditioner are
|
||||
all equal to the block-Jacobi preconditioner.
|
||||
|
||||
As already observed, the convergence rate of the one-level Schwarz
|
||||
preconditioned iterative solvers deteriorates as the number $m$ of partitions
|
||||
of $W$ increases \cite{dd1_94,dd2_96}. To reduce the dependency
|
||||
of the number of iterations on the degree of parallelism we may
|
||||
introduce a global coupling among the overlapping partitions by defining
|
||||
a coarse-space approximation $A_C$ of the matrix $A$.
|
||||
In a pure algebraic setting, $A_C$ is usually built with
|
||||
a Galerkin approach. Given a set $W_C$ of \emph{coarse vertices},
|
||||
with size $n_C$, and a suitable restriction operator
|
||||
$R_C \in \Re^{n_C \times n}$, $A_C$ is defined as
|
||||
\[
|
||||
A_C=R_C A R_C^T
|
||||
\]
|
||||
and the coarse-level correction matrix to be combined with a generic
|
||||
one-level AS preconditioner $M_{1L}$ is obtained as
|
||||
\[
|
||||
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
\]
|
||||
where $A_C$ is assumed to be nonsingular. The application of $M_{C}^{-1}$
|
||||
to a vector $v$ corresponds to a restriction, a solution and
|
||||
a prolongation step; the solution step, involving the matrix $A_C$,
|
||||
may be carried out also approximately.
|
||||
|
||||
The combination of $M_{C}$ and $M_{1L}$ may be
|
||||
performed in either an additive or a multiplicative framework.
|
||||
In the former case, the \emph{two-level additive} Schwarz preconditioner
|
||||
is obtained:
|
||||
\[
|
||||
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
\]
|
||||
Applying $M_{2L-A}^{-1}$ to a vector $v$ within a Krylov solver
|
||||
corresponds to applying $M_{C}^{-1}$
|
||||
and $M_{1L}^{-1}$ to $v$ independently and then summing up
|
||||
the results.
|
||||
|
||||
In the multiplicative case, the combination can be
|
||||
performed by first applying the smoother $M_{1L}^{-1}$ and then
|
||||
the coarse-level correction operator $M_{C}^{-1}$:
|
||||
\[
|
||||
\begin{array}{l}
|
||||
w = M_{1L}^{-1} v, \\
|
||||
z = w + M_{C}^{-1} (v-Aw);
|
||||
\end{array}
|
||||
\]
|
||||
this corresponds to the following \emph{two-level hybrid pre-smoothed}
|
||||
Schwarz preconditioner:
|
||||
\[
|
||||
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
\]
|
||||
On the other hand, by applying the smoother after the coarse-level correction,
|
||||
i.e.\ by computing
|
||||
\[
|
||||
\begin{array}{l}
|
||||
w = M_{C}^{-1} v , \\
|
||||
z = w + M_{1L}^{-1} (v-Aw) ,
|
||||
\end{array}
|
||||
\]
|
||||
the \emph{two-level hybrid post-smoothed}
|
||||
Schwarz preconditioner is obtained:
|
||||
\[
|
||||
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
\]
|
||||
One more variant of two-level hybrid preconditioner is obtained by applying
|
||||
the smoother before and after the coarse-level correction. In this case, the
|
||||
preconditioner is symmetric if $A$, $M_{1L}$ and $M_{C}$ are symmetric.
|
||||
|
||||
As previously noted, on parallel computers the number of submatrices usually matches
|
||||
the number of available processors. When the size of the system to be preconditioned
|
||||
is very large, the use of many processors, i.e.\ of many small submatrices, often
|
||||
leads to a large coarse-level system, whose solution may be computationally expensive.
|
||||
On the other hand, the use of few processors often leads to local sumatrices that
|
||||
are too expensive to be processed on single processors, because of memory and/or
|
||||
computing requirements. Therefore, it seems natural to use a recursive approach,
|
||||
in which the coarse-level correction is re-applied starting from the current
|
||||
coarse-level system. The corresponding preconditioners, called \emph{multi-level}
|
||||
preconditioners, can significantly reduce the computational cost of preconditioning
|
||||
with respect to the two-level case (see \cite[Chapter 3]{dd2_96}).
|
||||
Additive and hybrid multilevel preconditioners
|
||||
are obtained as direct extensions of the two-level counterparts.
|
||||
For a detailed descrition of them, the reader is
|
||||
referred to \cite[Chapter 3]{dd2_96}.
|
||||
The algorithm for the application of a multi-level hybrid
|
||||
post-smoothed preconditioner $M$ to a vector $v$, i.e.\ for the
|
||||
computation of $w=M^{-1}v$, is reported, for
|
||||
example, in Figure~\ref{fig:mlhpost_alg}. Here the number of levels
|
||||
is denoted by $nlev$ and the levels are numbered in increasing order starting
|
||||
from the finest one, i.e.\ the finest level is level 1; the coarse matrix
|
||||
and the corresponding basic preconditioner at each level $l$ are denoted by $A_l$ and
|
||||
$M_l$, respectively, with $A_1=A$.
|
||||
%
|
||||
\begin{figure}[t]
|
||||
\begin{center}
|
||||
\framebox{
|
||||
\begin{minipage}{.85\textwidth} {\small
|
||||
\begin{tabbing}
|
||||
\quad \=\quad \=\quad \=\quad \\[-1mm]
|
||||
%
|
||||
%! assign the finest matrix\\
|
||||
%$A_1 \leftarrow A$;\\[1mm]
|
||||
%! define the number of levels $nlev$ \\[1mm]
|
||||
%! define $nlev-1$ prolongators\\
|
||||
%$R_l^T, l=2, \ldots, nlev$;\\[1mm]
|
||||
%! define $nlev-1$ coarser matrices\\
|
||||
%$A_l \leftarrow R_lA_{l-1}R_l^T, \; l=2, \ldots, nlev$;\\[1mm]
|
||||
%! define the $nlev-1$ basic Schwarz preconditioners\\
|
||||
%$M_l$, basic preconditioner for $A_l \; l=1, \ldots, nlev-1$;\\[1mm]
|
||||
%$! assign a vector $v$\\
|
||||
%
|
||||
$v_1 = v$; \\[2mm]
|
||||
\textbf{for $l=2, nlev$ do}\\[1mm]
|
||||
\> ! transfer $v_{l-1}$ to the next coarser level\\
|
||||
\> $v_l = R_lv_{l-1}$ \\[1mm]
|
||||
\textbf{endfor} \\[2mm]
|
||||
! apply the coarsest-level correction\\[1mm]
|
||||
$y_{nlev} = A_{nlev}^{-1} v_{nlev}$\\[2mm]
|
||||
\textbf{for $l=nlev -1 , 1, -1$ do}\\[1mm]
|
||||
\> ! transfer $y_{l+1}$ to the next finer level\\
|
||||
\> $y_l = R_{l+1}^T y_{l+1}$;\\[1mm]
|
||||
\> ! compute the residual at the current level\\
|
||||
\> $r_l = v_l-A_l^{-1} y_l$;\\[1mm]
|
||||
\> ! apply the basic Schwarz preconditioner to the residual\\
|
||||
\> $r_l = M_l^{-1} r_l$\\[1mm]
|
||||
\> ! update $y_l$\\
|
||||
\> $y_l = y_l+r_l$\\
|
||||
\textbf{endfor} \\[1mm]
|
||||
$w = y_1$;
|
||||
\end{tabbing}
|
||||
}
|
||||
\end{minipage}
|
||||
}
|
||||
\caption{Application of the multi-level hybrid post-smoothed preconditioner.\label{fig:mlhpost_alg}}
|
||||
\end{center}
|
||||
\end{figure}
|
||||
%
|
||||
|
||||
|
||||
\subsection{Smoothed Aggregation\label{sec:aggregation}}
|
||||
|
||||
In order to define the restriction operator $R_C$, which is used to compute
|
||||
the coarse-level matrix $A_C$, MLD2P4 uses the \emph{smoothed aggregation}
|
||||
algorithm described in \cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}.
|
||||
The basic idea of this algorithm is to build a coarse set of vertices
|
||||
$W_C$ by suitably grouping the vertices of $W$ into disjoint subsets
|
||||
(aggregates), and to define the coarse-to-fine space transfer operator $R_C^T$ by
|
||||
applying a suitable smoother to a simple piecewise constant
|
||||
prolongation operator, to improve the quality of the coarse-space correction.
|
||||
|
||||
Three main steps can be identified in the smoothed aggregation procedure:
|
||||
\begin{enumerate}
|
||||
\item coarsening of the vertex set $W$, to obtain $W_C$;
|
||||
\item construction of the prolongator $R_C^T$;
|
||||
\item application of $R_C$ and $R_C^T$ to build $A_C$.
|
||||
\end{enumerate}
|
||||
%\textbf{NOTA: Controllare cosa fa trilinos dopo il primo passo.}
|
||||
|
||||
To perform the coarsening step, we have implemented the aggregation algorithm sketched
|
||||
in \cite{apnum_07}. According to \cite{VANEK_MANDEL_BREZINA}, a modification of
|
||||
this algorithm has been actually considered,
|
||||
in which each aggregate $N_r$ is made of vertices of $W$ that are \emph{strongly coupled}
|
||||
to a certain root vertex $r \in W$, i.e.\
|
||||
\[ N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
|
||||
\cup \left\{ r \right\} ,
|
||||
\]
|
||||
for a given $\theta \in [0,1]$.
|
||||
Since this algorithm has a sequential nature, a \emph{decoupled} version of
|
||||
it has been chosen, where each processor $i$ independently applies the algorithm to
|
||||
the set of vertices $W_i^0$ assigned to it in the initial data distribution. This
|
||||
version is embarrassingly parallel, since it does not require any data communication.
|
||||
On the other hand, it may produce non-uniform aggregates near boundary vertices,
|
||||
i.e.\ near vertices adjacent to vertices in other processors, and is strongly
|
||||
dependent on the number of processors and on the initial partitioning of the matrix $A$.
|
||||
Nevertheless, this algorithm has been chosen for the implementation in MLD2P4,
|
||||
since it has been shown to produce good results in practice
|
||||
\cite{aaecc_07,apnum_07,TUMINARO_TONG}.
|
||||
|
||||
The prolongator $P_C=R_C^T$ is built starting from a \emph{tentative prolongator}
|
||||
$P \in \Re^{n \times n_C}$, defined as
|
||||
\begin{equation}
|
||||
P=(p_{ij}), \quad p_{ij}=
|
||||
\left\{ \begin{array}{ll}
|
||||
1 & \quad \mbox{if} \; i \in V^j_C \\
|
||||
0 & \quad \mbox{otherwise}
|
||||
\end{array} \right. .
|
||||
\label{eq:tent_prol}
|
||||
\end{equation}
|
||||
$P_C$ is obtained by
|
||||
applying to $P$ a smoother $S \in \Re^{n \times n}$:
|
||||
\begin{equation}
|
||||
P_C = S P,
|
||||
\label{eq:smoothed_prol}
|
||||
\end{equation}
|
||||
in order to remove oscillatory components from the range of the prolongator
|
||||
and hence to improve the convergence properties of the multi-level
|
||||
Schwarz method \cite{BREZINA_VANEK,StubenGMD69_99}.
|
||||
A simple choice for $S$ is the damped Jacobi smoother:
|
||||
\begin{equation}
|
||||
S = I - \omega D^{-1} A ,
|
||||
\label{eq:jac_smoother}
|
||||
\end{equation}
|
||||
where the value of $\omega$ can be chosen
|
||||
using some estimate of the spectral radius of $D^{-1}A$ \cite{BREZINA_VANEK}.
|
||||
%
|
||||
%\textbf{NOTA: filtering di $A$ nello smoothing, da implementare?}
|
||||
%
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "userguide"
|
||||
%%% End:
|
||||
@ -0,0 +1,207 @@
|
||||
%\section{Bibliography\label{sec:bib}}
|
||||
\begin{thebibliography}{99}
|
||||
\addcontentsline{toc}{section}{\refname}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{References}}
|
||||
|
||||
%\let\refname\relax
|
||||
|
||||
%
|
||||
%\bibitem{PARA04FOREST}
|
||||
%G.~Bella, S.~Filippone, A.~De Maio, A., Testa, M.:
|
||||
%A Simulation Model for Forest Fires.
|
||||
%In: Dongarra, J., Madsen, K., Wasniewski, J. (eds.):
|
||||
%Proceedings of PARA~04 Workshop on State of the Art
|
||||
%in Scientific Computing. Lecture Notes in Computer Science, 3732. Berlin:
|
||||
%Springer, 2005
|
||||
%
|
||||
\bibitem{BREZINA_VANEK}
|
||||
M.~Brezina, P.~Van{\v e}k,
|
||||
{\em A Black-Box Iterative Solver Based on a Two-Level Schwarz Method},
|
||||
Computing, 63, 1999, 233--263.
|
||||
%
|
||||
\bibitem{para_04}
|
||||
A.~Buttari, P.~D'Ambra, D.~di Serafino, S.~Filippone,
|
||||
{\em Extending PSBLAS to Build Parallel Schwarz Preconditioners},
|
||||
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.
|
||||
%
|
||||
\bibitem{aaecc_07} A.~Buttari, P.~D'Ambra, D.~di~Serafino, S.~Filippone,
|
||||
{\em 2LEV-D2P4: a package of high-performance preconditioners
|
||||
for scientific and engineering applications},
|
||||
Applicable Algebra in Engineering, Communications and Computing,
|
||||
18, 3, 2007, 223--239.
|
||||
%Published online: 13 February 2007, {\tt http://dx.doi.org/10.1007/s00200-007-0035-z}
|
||||
%
|
||||
\bibitem{apnum_07} P.~D'Ambra, S.~Filippone, D.~di~Serafino,
|
||||
{\em On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners},
|
||||
Applied Numerical Mathematics, Elsevier Science,
|
||||
57, 11-12, 2007, 1181-1196.
|
||||
%published online 3 February 2007, {\tt
|
||||
% http://dx.doi.org/10.1016/j.apnum.2007.01.006}
|
||||
|
||||
%% \bibitem{DOUGLAS}
|
||||
%% R.E.~Bank and C.C.~Douglas,
|
||||
%% {\em SMMP: Sparse Matrix Multiplication Package},
|
||||
%% Advances in Computational Mathematics, 1993, 1, 127-137.
|
||||
%% (See also {\tt http://www.mgnet.org/~douglas/ccd-codes.html})
|
||||
%
|
||||
%
|
||||
%% \bibitem{CAI_SAAD}
|
||||
%% X.~C.~Cai and Y.~Saad,
|
||||
%% {\em Overlapping Domain Decomposition Algorithms for General Sparse Matrices},
|
||||
%% Numerical Linear Algebra with Applications, 3(3), pp.~221--237, 1996.
|
||||
%
|
||||
\bibitem{CAI_SARKIS}
|
||||
X.~C.~Cai, M.~Sarkis,
|
||||
{\em A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems},
|
||||
SIAM Journal on Scientific Computing, 21, 2, 1999, 792--797.
|
||||
%
|
||||
\bibitem{Cai_Widlund_92}
|
||||
X.~C.~Cai, O.~B.~Widlund,
|
||||
{\em Domain Decomposition Algorithms for Indefinite Elliptic Problems},
|
||||
SIAM Journal on Scientific and Statistical Computing, 13, 1, 1992, 243--258.
|
||||
%
|
||||
\bibitem{dd1_94}
|
||||
T.~Chan and T.~Mathew,
|
||||
{\em Domain Decomposition Algorithms},
|
||||
in A.~Iserles, editor, Acta Numerica 1994, 61--143.
|
||||
Cambridge University Press.
|
||||
%
|
||||
\bibitem{UMFPACK}
|
||||
T.A.~Davis,
|
||||
{\em Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal
|
||||
Method with a Column Pre-ordering Strategy},
|
||||
ACM Transactions on Mathematical Software, 30, 2004, 196--199.
|
||||
(See also {\tt http://www.cise.ufl.edu/~davis/})
|
||||
%
|
||||
\bibitem{SUPERLU}
|
||||
J.W.~Demmel, S.C.~Eisenstat, J.R.~Gilbert, X.S.~Li and J.W.H.~Liu,
|
||||
A supernodal approach to sparse partial pivoting,
|
||||
SIAM Journal on Matrix Analysis and Applications, 20, 3, 1999, 720--755.
|
||||
%
|
||||
\bibitem{blas3}
|
||||
J.~J.~Dongarra, J.~Du Croz, I.~S.~Duff, S.~Hammarling,
|
||||
\emph{A set of Level 3 Basic Linear Algebra Subprograms},
|
||||
ACM Transactions on Mathematical Software, 16, 1990, 1--17.
|
||||
%
|
||||
\bibitem{blas2}
|
||||
J.~J.~Dongarra, J.~Du Croz, S.~Hammarling, R.~J.~Hanson,
|
||||
\emph{An extended set of FORTRAN Basic Linear Algebra Subprograms},
|
||||
ACM Transactions on Mathematical Software, 14, 1988, 1--17.
|
||||
%
|
||||
\bibitem{BLACS}
|
||||
J.~J.~Dongarra and R.~C.~Whaley,
|
||||
{\em A User's Guide to the BLACS v.~1.1},
|
||||
Lapack Working Note 94, Tech.\ Rep.\ UT-CS-95-281, University of
|
||||
Tennessee, March 1995 (updated May 1997).
|
||||
%
|
||||
%\bibitem{sblas_97}
|
||||
%I.~Duff, M.~Marrone, G.~Radicati and C.~Vittoli,
|
||||
%{\em Level 3 Basic Linear Algebra Subprograms for Sparse Matrices:
|
||||
%a User Level Interface},
|
||||
%ACM Transactions on Mathematical Software, 23(3), pp.~379--401, 1997.
|
||||
%
|
||||
%\bibitem{sblas_02}
|
||||
%I.~Duff, M.~Heroux and R.~Pozo,
|
||||
%{\em An Overview of the Sparse Basic Linear
|
||||
%Algebra Subprograms: the New Standard from the BLAS Technical Forum},
|
||||
%ACM Transactions on Mathematical Software, 28(2), pp.~239--267, 2002.
|
||||
%
|
||||
\bibitem{EFSTATHIOU}
|
||||
E.~Efstathiou, J.~G.~Gander,
|
||||
{\em Why Restricted Additive Schwarz Converges Faster than Additive Schwarz},
|
||||
BIT Numerical Mathematics, 43, 2003, 945--959.
|
||||
%
|
||||
\bibitem{PSBLASGUIDE}
|
||||
S.~Filippone, A.~Buttari,
|
||||
{\em PSBLAS-2.3 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library}, 2008,
|
||||
available from \texttt{http://www.ce.uniroma2.it/psblas/}.
|
||||
%
|
||||
\bibitem{psblas_00}
|
||||
S.~Filippone, M.~Colajanni,
|
||||
{\em PSBLAS: A Library for Parallel Linear Algebra
|
||||
Computation on Sparse Matrices},
|
||||
ACM Transactions on Mathematical Software, 26, 4, 2000, 527--550.
|
||||
%
|
||||
\bibitem{MPI2}
|
||||
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},
|
||||
MIT Press, 1998.
|
||||
%
|
||||
\bibitem{blas1}
|
||||
C.~L.~Lawson, R.~J.~Hanson, D.~Kincaid, F.~T.~Krogh,
|
||||
\emph{Basic Linear Algebra Subprograms for FORTRAN usage},
|
||||
ACM Transactions on Mathematical Software, 5, 1979, 308--323.
|
||||
%
|
||||
\bibitem{SUPERLUDIST}
|
||||
X.~S.~Li, J.~W.~Demmel, {\em SuperLU\_DIST: A Scalable Distributed-memory
|
||||
Sparse Direct Solver for Unsymmetric Linear Systems},
|
||||
ACM Transactions on Mathematical Software, 29, 2, 2003, 110--140.
|
||||
%
|
||||
%\bibitem{KIVA3PSBLAS}
|
||||
%S.~Filippone, P.~D'Ambra, M.~Colajanni,
|
||||
%{\em Using a Parallel Library of Sparse Linear Algebra in a Fluid Dynamics
|
||||
%Applications Code on Linux Clusters},
|
||||
%in G.~Joubert, A.~Murli, F.~Peters, M.~Vanneschi, editors,
|
||||
%Parallel Computing - Advances \& Current Issues,
|
||||
%pp.~441--448, Imperial College Press, 2002.
|
||||
%
|
||||
%\bibitem{METIS}
|
||||
%Karypis, G. and Kumar, V.,
|
||||
%{\em {METIS}: Unstructured Graph Partitioning and Sparse Matrix
|
||||
% Ordering System}.
|
||||
%Minneapolis, MN 55455: University of Minnesota, Department of
|
||||
% Computer Science, 1995.
|
||||
%Internet Address: {\verb|http://www.cs.umn.edu/~karypis|}.
|
||||
%\bibitem{BLAS1}
|
||||
%Lawson, C., Hanson, R., Kincaid, D. and Krogh, F.,
|
||||
% Basic {L}inear {A}lgebra {S}ubprograms for {F}ortran usage,
|
||||
%{ACM Trans. Math. Softw.} vol.~{5}, 38--329, 1979.
|
||||
%
|
||||
%\bibitem{machiels}
|
||||
%{Machiels, L. and Deville, M.}
|
||||
%{\em Fortran 90: An entry to object-oriented programming for the solution
|
||||
% of partial differential equations.}
|
||||
%{ACM Trans. Math. Softw.} vol.~{23}, 32--49.
|
||||
%\bibitem{metcalf}
|
||||
%{Metcalf, M., Reid, J. and Cohen, M.}
|
||||
%{\em Fortran 95/2003 explained.}
|
||||
%{Oxford University Press}, 2004.
|
||||
%
|
||||
\bibitem{Saad_book}
|
||||
Y.~Saad,
|
||||
\emph{Iterative methods for sparse linear systems}, 2nd edition,
|
||||
SIAM, 2003
|
||||
|
||||
\bibitem{dd2_96}
|
||||
B.~Smith, P.~Bjorstad, W.~Gropp,
|
||||
{\em Domain Decomposition: Parallel Multilevel Methods for Elliptic
|
||||
Partial Differential Equations},
|
||||
Cambridge University Press, 1996.
|
||||
%
|
||||
\bibitem{MPI1}
|
||||
M.~Snir, S.~Otto, S.~Huss-Lederman, D.~Walker, J.~Dongarra,
|
||||
{\em MPI: The Complete Reference. Volume 1 - The MPI Core}, second edition,
|
||||
MIT Press, 1998.
|
||||
%%
|
||||
\bibitem{StubenGMD69_99}
|
||||
K.~St\"{u}ben,
|
||||
{\em Algebraic Multigrid (AMG): an Introduction with Applications},
|
||||
in A.~Sch\"{u}ller, U.~Trottenberg, C.~Oosterlee, editors, Multigrid,
|
||||
Academic Press, 2000.
|
||||
%
|
||||
\bibitem{TUMINARO_TONG}
|
||||
R.~S.~Tuminaro, C.~Tong,
|
||||
{\em Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines},
|
||||
in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.
|
||||
%
|
||||
\bibitem{VANEK_MANDEL_BREZINA}
|
||||
P.~Van{\v e}k, J.~Mandel and M.~Brezina,
|
||||
{\em Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems},
|
||||
Computing, 56, 1996, 179-196.
|
||||
%
|
||||
|
||||
\end{thebibliography}
|
||||
@ -0,0 +1,242 @@
|
||||
\section{Configuring and Building MLD2P4\label{sec:building}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:building} Configuring and Building MLD2P4}}
|
||||
To build MLD2P4 it is necessary to set up a Makefile with appropriate
|
||||
values for your system; this is done by means of the \verb|configure|
|
||||
script. The distribution also includes the autoconf and automake
|
||||
sources employed to generate the script, but usually this is not needed
|
||||
to build the software.
|
||||
|
||||
MLD2P4 is implemented almost entirely in Fortran~95, with some
|
||||
interfaces to external libraries in C; the Fortran compiler
|
||||
must support the Fortran~95 standard plus the extension TR15581, which
|
||||
enhances the usability of \verb|ALLOCATABLE| variables. Most modern
|
||||
Fortran compilers support this language level. In particular, this is
|
||||
supported by the GNU Fortran compiler as of version 4.2.0; however we
|
||||
recommend to use the latest available release (4.3.1 at the time of
|
||||
this writing).
|
||||
The software defines data types and interfaces for
|
||||
real and complex data, in both single and double precision.
|
||||
|
||||
\subsection{Prerequisites}
|
||||
|
||||
The following base libraries are needed:
|
||||
\begin{description}
|
||||
\item[BLAS] \cite{blas3,blas2,blas1} Many vendors provide optimized versions
|
||||
of the Basic Linear Algebra Subprograms; if no vendor version is
|
||||
available for a given platform, the ATLAS software
|
||||
(\verb!http://math-atlas.sourceforge.net/!)
|
||||
may be employed. The reference BLAS from Netlib
|
||||
(\verb|http://www.netlib.org/blas|) are meant to define the standard
|
||||
behaviour of the BLAS interface, so they are not optimized for any
|
||||
particular plaftorm, and should only be used as a last
|
||||
resort. Note that BLAS computations form a relatively small part of
|
||||
the MLD2P4/PSBLAS computations; they are however critical when using
|
||||
preconditioners based on the UMFPACK or SuperLU third party
|
||||
libraries.
|
||||
\item[MPI] \cite{MPI2,MPI1} A version of MPI is available on most
|
||||
high-performance computing systems; only version 1.1 is required.
|
||||
\item[BLACS] \cite{BLACS} The Basic Linear Algebra Communication Subprograms
|
||||
are available in source form from \verb|http://www.netlib.org/blacs|;
|
||||
some vendors include them in their parallel computing
|
||||
support libraries.
|
||||
\item[PSBLAS] \cite{PSBLASGUIDE,psblas_00} Parallel Sparse BLAS is
|
||||
available from \\ \verb|http://www.ce.uniroma2.it/psblas|; version 2.3
|
||||
(or later) is required. Indeed, all the prerequisites
|
||||
listed so far are also prerequisites of PSBLAS.
|
||||
To build the MLD2P4 library it is necessary to get access to
|
||||
the source PSBLAS directory employed to build the version under use; after
|
||||
the MLD2P4 build process completes, only the compiled form of the
|
||||
PSBLAS library is necessary to build user applications.
|
||||
\end{description}
|
||||
|
||||
Please note that the four previous libraries must have Fortran
|
||||
interfaces compatible with MLD2P4;
|
||||
usually this means that they should all be built with the same
|
||||
compiler as MLD2P4.
|
||||
|
||||
\subsection{Optional third party libraries}
|
||||
|
||||
We provide interfaces to the following third-party software libraries;
|
||||
note that these are optional, but if you enable them some defaults
|
||||
for multilevel preconditioners may change to reflect their presence.
|
||||
|
||||
\begin{description}
|
||||
\item[UMFPACK] \cite{UMFPACK}
|
||||
A sparse direct factorization package available from \\
|
||||
\verb|http://www.cise.ufl.edu/research/sparse/umfpack/|;
|
||||
provides serial factorization and triangular system solution for double
|
||||
precision real and complex data. We have tested
|
||||
versions 4.4 and 5.1.
|
||||
\item[SuperLU] \cite{SUPERLU}
|
||||
A sparse direct factorization package available from \\
|
||||
\verb|http://crd.lbl.gov/~xiaoye/SuperLU/|; provides serial
|
||||
factorization and triangular system solution for single and double precision,
|
||||
real and complex data. We have tested versions 3.0 and 3.1.
|
||||
\item[SuperLU\_Dist] \cite{SUPERLUDIST}
|
||||
A sparse direct factorization package available
|
||||
from the same site as SuperLU; provides parallel factorization and
|
||||
triangular system solution for double precision real and complex data.
|
||||
We have tested version 2.1.
|
||||
\end{description}
|
||||
|
||||
\subsection{Configuration options}
|
||||
|
||||
To build MLD2P4 the first step is to use the \verb|configure| script
|
||||
in the main directory to generate the necessary makefile(s).
|
||||
|
||||
As a minimal example consider the following:
|
||||
\begin{verbatim}
|
||||
./configure --with-psblas=/home/user/PSBLAS/psblas-2.3
|
||||
\end{verbatim}
|
||||
which assumes that the various MPI compilers and support libraries are
|
||||
available in the standard directories on the system, and specifies
|
||||
only the PSBLAS build directory (note that the latter directory must
|
||||
be specified with an {\em absolute} path).
|
||||
The full set of options may be looked at by issuing the command
|
||||
\verb|./configure --help|, which produces:
|
||||
\begin{verbatim}
|
||||
`configure' configures MLD2P4 1.0 to adapt to many kinds of systems.
|
||||
|
||||
Usage: ./configure [OPTION]... [VAR=VALUE]...
|
||||
|
||||
To assign environment variables (e.g., CC, CFLAGS...), specify them as
|
||||
VAR=VALUE. See below for descriptions of some of the useful variables.
|
||||
|
||||
Defaults for the options are specified in brackets.
|
||||
|
||||
Configuration:
|
||||
-h, --help display this help and exit
|
||||
--help=short display options specific to this package
|
||||
--help=recursive display the short help of all the included packages
|
||||
-V, --version display version information and exit
|
||||
-q, --quiet, --silent do not print `checking...' messages
|
||||
--cache-file=FILE cache test results in FILE [disabled]
|
||||
-C, --config-cache alias for `--cache-file=config.cache'
|
||||
-n, --no-create do not create output files
|
||||
--srcdir=DIR find the sources in DIR [configure dir or `..']
|
||||
|
||||
Installation directories:
|
||||
--prefix=PREFIX install architecture-independent files in PREFIX
|
||||
[/usr/local]
|
||||
--exec-prefix=EPREFIX install architecture-dependent files in EPREFIX
|
||||
[PREFIX]
|
||||
|
||||
By default, `make install' will install all the files in
|
||||
`/usr/local/bin', `/usr/local/lib' etc. You can specify
|
||||
an installation prefix other than `/usr/local' using `--prefix',
|
||||
for instance `--prefix=$HOME'.
|
||||
|
||||
For better control, use the options below.
|
||||
|
||||
Fine tuning of the installation directories:
|
||||
--bindir=DIR user executables [EPREFIX/bin]
|
||||
--sbindir=DIR system admin executables [EPREFIX/sbin]
|
||||
--libexecdir=DIR program executables [EPREFIX/libexec]
|
||||
--sysconfdir=DIR read-only single-machine data [PREFIX/etc]
|
||||
--sharedstatedir=DIR modifiable architecture-independent data [PREFIX/com]
|
||||
--localstatedir=DIR modifiable single-machine data [PREFIX/var]
|
||||
--libdir=DIR object code libraries [EPREFIX/lib]
|
||||
--includedir=DIR C header files [PREFIX/include]
|
||||
--oldincludedir=DIR C header files for non-gcc [/usr/include]
|
||||
--datarootdir=DIR read-only arch.-independent data root [PREFIX/share]
|
||||
--datadir=DIR read-only architecture-independent data [DATAROOTDIR]
|
||||
--infodir=DIR info documentation [DATAROOTDIR/info]
|
||||
--localedir=DIR locale-dependent data [DATAROOTDIR/locale]
|
||||
--mandir=DIR man documentation [DATAROOTDIR/man]
|
||||
--docdir=DIR documentation root [DATAROOTDIR/doc/mld2p4]
|
||||
--htmldir=DIR html documentation [DOCDIR]
|
||||
--dvidir=DIR dvi documentation [DOCDIR]
|
||||
--pdfdir=DIR pdf documentation [DOCDIR]
|
||||
--psdir=DIR ps documentation [DOCDIR]
|
||||
|
||||
Optional Packages:
|
||||
--with-PACKAGE[=ARG] use PACKAGE [ARG=yes]
|
||||
--without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no)
|
||||
--with-psblas The source directory for PSBLAS, for example,
|
||||
--with-psblas=/opt/packages/psblas-2.3
|
||||
--with-libs List additional link flags here. For example,
|
||||
--with-libs=-lspecial_system_lib or
|
||||
--with-libs=-L/path/to/libs
|
||||
--with-clibs additional CLIBS flags to be added: will prepend
|
||||
to CLIBS
|
||||
--with-flibs additional FLIBS flags to be added: will prepend
|
||||
to FLIBS
|
||||
--with-library-path additional LIBRARYPATH flags to be added: will
|
||||
prepend to LIBRARYPATH
|
||||
--with-include-path additional INCLUDEPATH flags to be added: will
|
||||
prepend to INCLUDEPATH
|
||||
--with-module-path additional MODULE_PATH flags to be added: will
|
||||
prepend to MODULE_PATH
|
||||
--with-umfpack=LIBNAME Specify the library name for UMFPACK library.
|
||||
Default: "-lumfpack -lamd"
|
||||
--with-umfpackdir=DIR Specify the directory for UMFPACK library and
|
||||
includes.
|
||||
--with-superlu=LIBNAME Specify the library name for SUPERLU library.
|
||||
Default: "-lslu"
|
||||
--with-superludir=DIR Specify the directory for SUPERLU library and
|
||||
includes.
|
||||
--with-superludist=LIBNAME
|
||||
Specify the libname for SUPERLUDIST library.
|
||||
Requires you also specify SuperLU. Default: "-lslud"
|
||||
--with-superludistdir=DIR
|
||||
Specify the directory for SUPERLUDIST library and
|
||||
includes.
|
||||
|
||||
Some influential environment variables:
|
||||
FC Fortran compiler command
|
||||
FCFLAGS Fortran compiler flags
|
||||
LDFLAGS linker flags, e.g. -L<lib dir> if you have libraries in a
|
||||
nonstandard directory <lib dir>
|
||||
LIBS libraries to pass to the linker, e.g. -l<library>
|
||||
CC C compiler command
|
||||
CFLAGS C compiler flags
|
||||
CPPFLAGS C/C++/Objective C preprocessor flags, e.g. -I<include dir> if
|
||||
you have headers in a nonstandard directory <include dir>
|
||||
CPP C preprocessor
|
||||
MPICC MPI C compiler command
|
||||
|
||||
Use these variables to override the choices made by `configure' or to help
|
||||
it to find libraries and programs with nonstandard names/locations.
|
||||
|
||||
Report bugs to <bugreport@mld2p4.it>.
|
||||
\end{verbatim}
|
||||
Thus, a sample build with libraries in installation
|
||||
directories specifics to the GNU 4.3 compiler suite might be as
|
||||
follows, specifying only the UMFPACK external package:
|
||||
\begin{verbatim}
|
||||
./configure --with-psblas=/home/user/psblas-2.3/ \
|
||||
--with-libs="-L/usr/local/BLAS/gnu43 -L/usr/local/BLACS/gnu43" \
|
||||
--with-blacs=-lmpiblacs --with-umfpackdir=/usr/local/UMFPACK/gnu43
|
||||
\end{verbatim}
|
||||
Once the configure script has completed execution, it will have
|
||||
generated the file \verb|Make.inc| which will then be used by all
|
||||
Makefiles in the directory tree.
|
||||
|
||||
To build the library the user will now enter
|
||||
\begin{verbatim}
|
||||
make
|
||||
\end{verbatim}
|
||||
followed (optionally) by
|
||||
\begin{verbatim}
|
||||
make install
|
||||
\end{verbatim}
|
||||
|
||||
\subsection{Example and test programs\label{sec:ex_and_test}}
|
||||
The package contains the \verb|examples| and \verb|tests| directories;
|
||||
both of them are further divided into \verb|fileread| and
|
||||
\verb|pargen| subdirectories. Their purpose is as follows:
|
||||
\begin{description}
|
||||
\item[\tt examples] contains a set of simple example programs with a
|
||||
predefined choice of preconditioners, selectable via integer
|
||||
values. These are intended to get an acquaintance with the
|
||||
multilevel preconditioners.
|
||||
\item[\tt tests] contains a set of more sophisticated examples that
|
||||
will allow the user, via the input files in the \verb|runs|
|
||||
subdirectories, to experiment with the full range of preconditioners
|
||||
implemented in the library.
|
||||
\end{description}
|
||||
The \verb|fileread| directories contain sample programs that read
|
||||
sparse matrices from files, according to the Matrix Market or the
|
||||
Harwell-Boeing storage format; the \verb|pdegen| instead generate
|
||||
matrices in full parallel mode from the discretization of a sample PDE.
|
||||
@ -0,0 +1,19 @@
|
||||
\section{Code Distribution\label{sec:distribution}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:distribution} Code Distribution}}
|
||||
|
||||
\noindent
|
||||
MLD2P4 is available from the web site
|
||||
\begin{quotation}
|
||||
\texttt{http://www.mld2p4.it}
|
||||
\end{quotation}
|
||||
where contact points for further information can be also found.
|
||||
To report bugs or ask general usage questions, please, send an email to
|
||||
\texttt{bugreport@mld2p4.it}.
|
||||
|
||||
|
||||
The software is available under a modified BSD license, as specified
|
||||
in Appendix~\ref{sec:license}; please note that some of the optional
|
||||
third party libraries may be licensed under a different and more
|
||||
stringent license, most notably the GPL, and this should be taken into
|
||||
account when treating derived works.
|
||||
@ -0,0 +1,20 @@
|
||||
\section{Error Handling\label{sec:errors}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:errors} Error handling}}
|
||||
|
||||
The error handling in MLD2P4 is based on the PSBLAS (version 2) error
|
||||
handling. Error conditions are signaled via an integer argument
|
||||
\verb|info|; 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 \cite{PSBLASGUIDE}.
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "userguide"
|
||||
%%% End:
|
||||
@ -0,0 +1,308 @@
|
||||
\section{Getting Started\label{sec:started}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:started} Getting Started}}
|
||||
|
||||
We describe the basics for building and applying MLD2P4 one-level and multi-level
|
||||
Schwarz preconditioners with the Krylov solvers included in PSBLAS \cite{PSBLASGUIDE}.
|
||||
The following steps are required:
|
||||
\begin{enumerate}
|
||||
\item \emph{Declare the preconditioner data structure}. It is a derived data type,
|
||||
\verb|mld_|\-\emph{x}\verb|prec_| \verb|type|, where \emph{x} may be \verb|s|, \verb|d|, \verb|c|
|
||||
or \verb|z|, according to the basic data type of the sparse matrix
|
||||
(\verb|s| = real single precision; \verb|d| = real double precision;
|
||||
\verb|c| = complex single precision; \verb|z| = complex double precision).
|
||||
This data structure is accessed by the user only through the MLD2P4 routines,
|
||||
following an object-oriented approach.
|
||||
\item \emph{Allocate and initialize the preconditioner data structure, according to
|
||||
a preconditioner type chosen by the user}. This is performed by the routine
|
||||
\verb|mld_precinit|, which also sets defaults for each preconditioner
|
||||
type selected by the user. The defaults associated to each preconditioner
|
||||
type are given in Table~\ref{tab:precinit}, where the strings used by
|
||||
\verb|mld_precinit| to identify the preconditioner types are also given.
|
||||
Note that these strings are valid also if uppercase letters are substituted by
|
||||
corresponding lowercase ones.
|
||||
\item \emph{Modify the selected preconditioner type, by properly setting
|
||||
preconditioner parameters.} This is performed by the routine \verb|mld_precset|.
|
||||
This routine must be called only if the user wants to modify the default values
|
||||
of the parameters associated to the selected preconditioner type, to obtain a variant
|
||||
of the preconditioner. Examples of use of \verb|mld_precset| are given in
|
||||
Section~\ref{sec:examples}; a complete list of all the
|
||||
preconditioner parameters and their allowed and default values is provided in
|
||||
Section~\ref{sec:userinterface}, Tables~\ref{tab:p_type}-\ref{tab:p_coarse}.
|
||||
\item \emph{Build the preconditioner for a given matrix.} This is performed by
|
||||
the routine \verb|mld_precbld|.
|
||||
\item \emph{Apply the preconditioner at each iteration of a Krylov solver.}
|
||||
This is performed by the routine \verb|mld_precaply|. When using the PSBLAS Krylov solvers,
|
||||
this step is completely transparent to the user, since \verb|mld_precaply| is called
|
||||
by the PSBLAS routine implementing the Krylov solver (\verb|psb_krylov|).
|
||||
\item \emph{Free the preconditioner data structure}. This is performed by
|
||||
the routine \verb|mld_| \verb|precfree|. This step is complementary to step 1 and should
|
||||
be performed when the preconditioner is no more used.
|
||||
\end{enumerate}
|
||||
A detailed description of the above routines is given in Section~\ref{sec:userinterface}.
|
||||
Examples showing the basic use of MLD2P4 are reported in Section~\ref{sec:examples}.
|
||||
|
||||
Note that the Fortran 95 module \verb|mld_prec_mod|, containing the definition of the
|
||||
preconditioner data type and the interfaces to the routines of MLD2P4,
|
||||
must be used in any program calling such routines.
|
||||
The modules \verb|psb_base_mod|, for the sparse matrix and communication descriptor
|
||||
data types, and \verb|psb_krylov_mod|, for interfacing with the
|
||||
Krylov solvers, must be also used (see Section~\ref{sec:examples}).
|
||||
|
||||
\ \\
|
||||
\textbf{Remark 1.} The coarsest-level solver used by the default two-level
|
||||
preconditioner has been chosen by taking into account that, on parallel
|
||||
machines, it often leads to the smallest execution time when applied to
|
||||
linear systems coming from finite-difference discretizations of basic
|
||||
elliptic PDE problems, considered as standard tests for multi-level Schwarz
|
||||
preconditioners \cite{aaecc_07,apnum_07}. However, this solver does
|
||||
not necessarily correspond to the smallest number of iterations of the
|
||||
preconditioned Krylov method, which is usually obtained by applying
|
||||
a direct solver to the coarsest-level system, e.g.\ based on the LU
|
||||
factorization (see Section~\ref{sec:userinterface}
|
||||
for the coarsest-level solvers available in MLD2P4).
|
||||
|
||||
\ \\
|
||||
\textbf{Remark 2.} The include path for MLD2P4 must override
|
||||
those for PSBLAS, e.g.\ the latter must come first in the sequence
|
||||
passed to the compiler, as the MLD2P4 version of the Krylov solver
|
||||
interfaces must override that of PSBLAS. This will change in the future
|
||||
when the support for the \verb|class| statement becomes widespread in Fortran
|
||||
compilers.
|
||||
|
||||
|
||||
\begin{table}[th]
|
||||
\begin{center}
|
||||
%{\small
|
||||
\begin{tabular}{|l|l|p{7.8cm}|}
|
||||
\hline
|
||||
\textsc{type} & \textsc{string} & \textsc{default preconditioner} \\ \hline
|
||||
No preconditioner &\verb|'NOPREC'|& Considered only to use the PSBLAS
|
||||
Krylov solvers with no preconditioner. \\ \hline
|
||||
Diagonal & \verb|'DIAG'| & --- \\ \hline
|
||||
Block Jacobi & \verb|'BJAC'| & Block Jacobi with ILU(0) on the local blocks.\\ \hline
|
||||
Additive Schwarz & \verb|'AS'| & Restricted Additive Schwarz (RAS),
|
||||
with overlap 1 and ILU(0) on the local blocks. \\ \hline
|
||||
Multilevel &\verb|'ML'| & Multi-level hybrid preconditioner (additive on the
|
||||
same level and multiplicative through the levels),
|
||||
with post-smoothing only.
|
||||
Number of levels: 2.
|
||||
Post-smoother: RAS with overlap 1 and ILU(0)
|
||||
on the local blocks.
|
||||
Aggregation: decoupled smoothed aggregation with
|
||||
threshold $\theta = 0$.
|
||||
Coarsest matrix: distributed among the processors.
|
||||
Coarsest-level solver:
|
||||
4 sweeps of the block-Jacobi solver,
|
||||
with LU (or ILU) factorization of the blocks
|
||||
(UMFPACK for the double precision versions and
|
||||
SuperLU for the single precision ones, if the packages
|
||||
have been installed; ILU(0), otherwise). \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
%}
|
||||
\end{center}
|
||||
|
||||
\caption{Preconditioner types, corresponding strings and default choices.
|
||||
\label{tab:precinit}}
|
||||
\end{table}
|
||||
|
||||
\subsection{Examples\label{sec:examples}}
|
||||
|
||||
The code reported in Figure~\ref{fig:ex_default} shows how to set and apply the default
|
||||
multi-level preconditioner available in the real double precision version
|
||||
of MLD2P4 (see Table~\ref{tab:precinit}). This preconditioner is chosen
|
||||
by simply specifying \verb|'ML'| as second argument of \verb|mld_precinit|
|
||||
(a call to \verb|mld_precset| is not needed) and is applied with the BiCGSTAB
|
||||
solver provided by PSBLAS. As previously observed, the modules \verb|psb_base_mod|,
|
||||
\verb|mld_prec_mod| and \verb|psb_krylov_mod| must be used by the example program.
|
||||
|
||||
The part of the code concerning the
|
||||
reading and assembling of the sparse matrix and the right-hand side vector, performed
|
||||
through the PSBLAS routines for sparse matrix and vector management, is not reported
|
||||
here for brevity; the statements concerning the deallocation of the PSBLAS
|
||||
data structure are neglected too.
|
||||
The complete code can be found in the example program file \verb|mld_dexample_ml.f90|,
|
||||
in the directory \verb|examples/fileread| of the MLD2P4 tree (see
|
||||
Section~\ref{sec:ex_and_test}).
|
||||
For details on the use of the PSBLAS routines, see the PSBLAS User's
|
||||
Guide \cite{PSBLASGUIDE}.
|
||||
|
||||
The setup and application of the default multi-level
|
||||
preconditioners for the real single precision and the complex, single and double
|
||||
precision, versions are obtained with straightforward modifications of the previous
|
||||
example (see Section~\ref{sec:userinterface} for details). If these versions are installed,
|
||||
the corresponding Fortran 95 codes are available in \verb|examples/fileread/|.
|
||||
|
||||
\begin{figure}[tbp]
|
||||
\begin{center}
|
||||
\begin{minipage}{.90\textwidth}
|
||||
{\small
|
||||
\begin{verbatim}
|
||||
use psb_base_mod
|
||||
use mld_prec_mod
|
||||
use psb_krylov_mod
|
||||
... ...
|
||||
!
|
||||
! sparse matrix
|
||||
type(psb_dspmat_type) :: A
|
||||
! sparse matrix descriptor
|
||||
type(psb_desc_type) :: desc_A
|
||||
! preconditioner
|
||||
type(mld_dprec_type) :: P
|
||||
! right-hand side and solution vectors
|
||||
real(kind(1.d0)) :: b(:), x(:)
|
||||
... ...
|
||||
!
|
||||
! initialize the parallel environment
|
||||
call psb_init(ictxt)
|
||||
call psb_info(ictxt,iam,np)
|
||||
... ...
|
||||
!
|
||||
! read and assemble the matrix A and the right-hand side b
|
||||
! using PSBLAS routines for sparse matrix / vector management
|
||||
... ...
|
||||
!
|
||||
! initialize the default multi-level preconditioner, i.e. hybrid
|
||||
! Schwarz, using RAS (with overlap 1 and ILU(0) on the blocks)
|
||||
! as post-smoother and 4 block-Jacobi sweeps (with UMFPACK LU
|
||||
! on the blocks) as distributed coarse-level solver
|
||||
call mld_precinit(P,'ML',info)
|
||||
!
|
||||
! build the preconditioner
|
||||
call mld_precbld(A,desc_A,P,info)
|
||||
!
|
||||
! set the solver parameters and the initial guess
|
||||
... ...
|
||||
!
|
||||
! solve Ax=b with preconditioned BiCGSTAB
|
||||
call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info)
|
||||
... ...
|
||||
!
|
||||
! deallocate the preconditioner
|
||||
call mld_precfree(P,info)
|
||||
!
|
||||
! deallocate other data structures
|
||||
... ...
|
||||
!
|
||||
! exit the parallel environment
|
||||
call psb_exit(ictxt)
|
||||
stop
|
||||
\end{verbatim}
|
||||
}
|
||||
\end{minipage}
|
||||
\caption{Setup and application of the default multi-level Schwarz preconditioner.
|
||||
\label{fig:ex_default}}
|
||||
\end{center}
|
||||
\end{figure}
|
||||
|
||||
Different versions of multi-level preconditioners can be obtained by changing
|
||||
the default values of the preconditioner parameters. The code reported in
|
||||
Figure~\ref{fig:ex_3lh} shows how to set a three-level hybrid Schwarz
|
||||
preconditioner, which uses block Jacobi with ILU(0) on the
|
||||
local blocks as post-smoother, has a coarsest matrix replicated on the processors,
|
||||
and solves the coarsest-level system with the LU factorization from UMFPACK~\cite{UMFPACK}.
|
||||
The number of levels is specified by using \verb|mld_precinit|; the other
|
||||
preconditioner parameters are set by calling \verb|mld_precset|. Note that
|
||||
the type of multilevel framework (i.e.\ multiplicative among the levels
|
||||
with post-smoothing only) is not specified since it is the default
|
||||
set by \verb|mld_precinit|.
|
||||
|
||||
Figure~\ref{fig:ex_3la} shows how to
|
||||
set a three-level additive Schwarz preconditioner,
|
||||
which uses RAS, with overlap 1 and ILU(0) on the blocks,
|
||||
as pre- and post-smoother, and applies five block-Jacobi sweeps, with
|
||||
the UMFPACK LU factorization on the blocks, as distributed coarsest-level
|
||||
solver. Again, \verb|mld_precset| is used only to set
|
||||
non-default values of the parameters (see Tables~\ref{tab:p_type}-\ref{tab:p_coarse}).
|
||||
In both cases, the construction and the application of the preconditioner
|
||||
are carried out as for the default multi-level preconditioner.
|
||||
The code fragments shown in in Figures~\ref{fig:ex_3lh}-\ref{fig:ex_3la} are
|
||||
included in the example program file \verb|mld_dexample_ml.f90| too.
|
||||
|
||||
Finally, Figure~\ref{fig:ex_1l} shows the setup of a one-level
|
||||
additive Schwarz preconditioner, i.e.\ RAS with overlap 2. The corresponding
|
||||
example program is available in \verb|mld_dexample_| \verb|1lev.f90|.
|
||||
|
||||
For all the previous preconditioners, example programs where the sparse matrix and
|
||||
the right-hand side are generated by discretizing a PDE with Dirichlet
|
||||
boundary conditions are also available in the directory \verb|examples/pdegen|.
|
||||
|
||||
\ \\
|
||||
\textbf{Remark 3.} Any PSBLAS-based program using the basic preconditioners
|
||||
implemented in PSBLAS 2.0, i.e.\ the diagonal and block-Jacobi ones,
|
||||
can use the diagonal and block-Jacobi preconditioners
|
||||
implemented in MLD2P4 without any change in the code.
|
||||
The PSBLAS-based program must be only recompiled
|
||||
and linked to the MLD2P4 library.
|
||||
\\
|
||||
|
||||
|
||||
\begin{figure}[tbh]
|
||||
\begin{center}
|
||||
\begin{minipage}{.90\textwidth}
|
||||
{\small
|
||||
\begin{verbatim}
|
||||
... ...
|
||||
! set a three-level hybrid Schwarz preconditioner, which uses
|
||||
! block Jacobi (with ILU(0) on the blocks) as post-smoother,
|
||||
! a coarsest matrix replicated on the processors, and the
|
||||
! LU factorization from UMFPACK as coarse-level solver
|
||||
call mld_precinit(P,'ML',info,nlev=3)
|
||||
call_mld_precset(P,mld_smoother_type_,'BJAC',info)
|
||||
call mld_precset(P,mld_coarse_mat_,'REPL',info)
|
||||
call mld_precset(P,mld_coarse_solve_,'UMF',info)
|
||||
... ...
|
||||
\end{verbatim}
|
||||
}
|
||||
\end{minipage}
|
||||
|
||||
\caption{Setup of a hybrid three-level Schwarz preconditioner.\label{fig:ex_3lh}}
|
||||
\end{center}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[tbh]
|
||||
\begin{center}
|
||||
\begin{minipage}{.90\textwidth}
|
||||
{\small
|
||||
\begin{verbatim}
|
||||
... ...
|
||||
! set a three-level additive Schwarz preconditioner, which uses
|
||||
! RAS (with overlap 1 and ILU(0) on the blocks) as pre- and
|
||||
! post-smoother, and 5 block-Jacobi sweeps (with UMFPACK LU
|
||||
! on the blocks) as distributed coarsest-level solver
|
||||
call mld_precinit(P,'ML',info,nlev=3)
|
||||
call mld_precset(P,mld_ml_type_,'ADD',info)
|
||||
call_mld_precset(P,mld_smoother_pos_,'TWOSIDE',info)
|
||||
call mld_precset(P,mld_coarse_sweeps_,5,info)
|
||||
... ...
|
||||
\end{verbatim}
|
||||
}
|
||||
\end{minipage}
|
||||
|
||||
\caption{Setup of an additive three-level Schwarz preconditioner.\label{fig:ex_3la}}
|
||||
\end{center}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[tbh]
|
||||
\begin{center}
|
||||
\begin{minipage}{.90\textwidth}
|
||||
{\small
|
||||
\begin{verbatim}
|
||||
... ...
|
||||
! set RAS with overlap 2 and ILU(0) on the local blocks
|
||||
call mld_precinit(P,'AS',info)
|
||||
call mld_precset(P,mld_sub_ovr_,2,info)
|
||||
... ...
|
||||
\end{verbatim}
|
||||
}
|
||||
\end{minipage}
|
||||
\caption{Setup of a one-level Schwarz preconditioner.\label{fig:ex_1l}}
|
||||
\end{center}
|
||||
\end{figure}
|
||||
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "userguide"
|
||||
%%% End:
|
||||
@ -0,0 +1,34 @@
|
||||
\section{Introduction}\label{sec:intro}
|
||||
\markboth{\underline{MLD2P4 User's and Reference Guide}}
|
||||
{\underline{\ref{sec:overview} Introduction}}
|
||||
|
||||
The MLD2P4 library provides ....
|
||||
|
||||
|
||||
\subsection{Programming model}
|
||||
|
||||
The MLD2P4 librarary is based on the Single Program Multiple Data
|
||||
(SPMD) programming model: each process participating in the
|
||||
computation performs the same actions on a chunk of data. Parallelism
|
||||
is thus data-driven.
|
||||
|
||||
Because of this structure, many subroutines coordinate their action
|
||||
across the various processes, thus providing an implicit
|
||||
synchronization point, and therefore \emph{must} be
|
||||
called simultaneously by all processes participating in the
|
||||
computation.
|
||||
However there are many cases where no synchronization, and indeed no
|
||||
communication among processes, is implied.
|
||||
|
||||
Throughout this user's guide each subroutine will be clearly indicated
|
||||
as:
|
||||
\begin{description}
|
||||
\item[Synchronous:] must be called simultaneously by all the
|
||||
processes in the relevant communication context;
|
||||
\item[Asynchronous:] may be called in a totally independent manner.
|
||||
\end{description}
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "userguide"
|
||||
%%% End:
|
||||
@ -0,0 +1,44 @@
|
||||
\section{License\label{sec:license}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:license} License}}
|
||||
|
||||
The MLD2P4 is freely distributable under the following copyright
|
||||
terms: {\small
|
||||
\begin{verbatim}
|
||||
MLD2P4 version 1.0
|
||||
MultiLevel Domain Decomposition Parallel Preconditioners Package
|
||||
based on PSBLAS (Parallel Sparse BLAS version 2.3)
|
||||
|
||||
(C) Copyright 2008
|
||||
|
||||
Salvatore Filippone University of Rome Tor Vergata
|
||||
Alfredo Buttari University of Rome Tor Vergata
|
||||
Pasqua D'Ambra ICAR-CNR, Naples
|
||||
Daniela di Serafino Second University of Naples
|
||||
|
||||
|
||||
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.
|
||||
\end{verbatim}
|
||||
}
|
||||
@ -0,0 +1,90 @@
|
||||
\section{General Overview\label{sec:overview}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:overview} General Overview}}
|
||||
|
||||
The \textsc{Multi-Level Domain Decomposition Parallel Preconditioners Package based on
|
||||
PSBLAS (MLD2P4}) provides \emph{multi-level Schwarz preconditioners}~\cite{dd2_96},
|
||||
to be used in the iterative solutions of sparse linear systems:
|
||||
\begin{equation}
|
||||
Ax=b,
|
||||
\label{system1}
|
||||
\end{equation}
|
||||
where $A$ is a square, real or complex, sparse matrix with a symmetric sparsity pattern.
|
||||
%
|
||||
%\textbf{NOTA: Caso non simmetrico, aggregazione con $(A+A^T)$ fatta!
|
||||
%Dovremmo implementare uno smoothed prolongator
|
||||
%adeguato e fare qualcosa di consistente anche con 1-lev Schwarz.}
|
||||
%
|
||||
These preconditioners have the following general features:
|
||||
\begin{itemize}
|
||||
\item both \emph{additive and hybrid multilevel} variants are implemented,
|
||||
i.e.\ variants that are additive among the levels and inside each level, and variants
|
||||
that are multiplicative among the levels and additive inside each level;
|
||||
the basic Additive Schwarz (AS) preconditioners are obtained by considering only one level;
|
||||
\item a \emph{purely algebraic} approach is used to
|
||||
generate a sequence of coarse-level corrections to a basic AS preconditioner, without
|
||||
explicitly using any information on the geometry of the original problem (e.g.\ the
|
||||
discretization of a PDE). The \emph{smoothed aggregation} technique is applied
|
||||
as algebraic coarsening strategy~\cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}.
|
||||
\end{itemize}
|
||||
|
||||
The package is written in \emph{Fortran~95}, following an
|
||||
\emph{object-oriented approach} through the exploitation of features
|
||||
such as abstract data type creation, functional
|
||||
overloading and dynamic memory management.
|
||||
% , while providing a smooth
|
||||
% path towards the integration in legacy application codes.
|
||||
The parallel implementation is based
|
||||
on a Single Program Multiple Data (SPMD) paradigm for distributed-memory architectures.
|
||||
Single and double precision implementations of MLD2P4 are available for both the
|
||||
real and the complex case, that can be used through a single interface.
|
||||
|
||||
|
||||
MLD2P4 has been designed to implement scalable and easy-to-use multilevel preconditioners
|
||||
in the context of the \emph{PSBLAS (Parallel Sparse BLAS)
|
||||
computational framework}~\cite{psblas_00}.
|
||||
PSBLAS is a library originally developed to address the parallel implementation of
|
||||
iterative solvers for sparse linear system, by providing basic linear algebra
|
||||
operators and data management facilities for distributed sparse matrices; it
|
||||
also includes parallel Krylov solvers, built on the top of the basic PSBLAS kernels.
|
||||
The preconditioners available in MLD2P4 can be used with these Krylov solvers.
|
||||
The choice of PSBLAS has been mainly motivated by the need of having
|
||||
a portable and efficient software infrastructure implementing ``de facto'' standard
|
||||
parallel sparse linear algebra kernels, to pursue goals such as performance,
|
||||
portability, modularity ed extensibility in the development of the preconditioner
|
||||
package. On the other hand, the implementation of MLD2P4 has led to some
|
||||
revisions and extentions of the PSBLAS kernels, leading to the
|
||||
recent PSBLAS 2.0 version~\cite{PSBLASGUIDE}. The inter-process comunication required
|
||||
by MLD2P4 is encapsulated into the PSBLAS routines, except few cases where
|
||||
MPI~\cite{MPI1} is explicitly called. Therefore, MLD2P4 can be run on any parallel
|
||||
machine where PSBLAS and MPI implementations are available.
|
||||
|
||||
MLD2P4 has a layered and modular software architecture where three main layers can be identified.
|
||||
The lower layer consists of the PSBLAS kernels, the middle one implements
|
||||
the construction and application phases of the preconditioners, and the upper one
|
||||
provides a uniform and easy-to-use interface to all the preconditioners.
|
||||
This architecture allows for different levels of use of the package:
|
||||
few black-box routines at the upper layer allow non-expert users to easily
|
||||
build any preconditioner available in MLD2P4 and to apply it within a PSBLAS Krylov solver.
|
||||
On the other hand, the routines of the middle and lower layer can be used and extended
|
||||
by expert users to build new versions of multi-level Schwarz preconditioners.
|
||||
We provide here a description of the upper-layer routines, but not of the
|
||||
medium-layer ones.
|
||||
|
||||
This guide is organized as follows. General information on the distribution of the source code
|
||||
is reported in Section~\ref{sec:distribution}, while details on the configuration
|
||||
and installation of the package are given in Section~\ref{sec:building}. A description of
|
||||
multi-level Schwarz preconditioners based on smoothed aggregation is provided
|
||||
in Section~\ref{sec:background}, to help the users in choosing among the different preconditioners
|
||||
implemented in MLD2P4. The basics for building and applying the preconditioners
|
||||
with the Krylov solvers implemented in PSBLAS are reported in Section~\ref{sec:started}, where the
|
||||
Fortran 95 codes of a few sample programs are also shown. A reference guide for
|
||||
the upper-layer routines of MLD2P4, that are the user interface, is provided
|
||||
in Section~\ref{sec:userinterface}. The error handling mechanism used by the package is briefly described
|
||||
in Section~\ref{sec:errors}. The copyright terms concerning the distribution and modification
|
||||
of MLD2P4 are reported in Appendix~\ref{sec:license}.
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "userguide"
|
||||
%%% End:
|
||||
@ -0,0 +1,280 @@
|
||||
\section{Preconditioner routines}
|
||||
\label{sec:precs}
|
||||
\markboth{\underline{MLD2P4 User's and Reference Guide}}
|
||||
{\underline{\ref{sec:precs} Preconditioners}}
|
||||
|
||||
% \section{Preconditioners}
|
||||
\label{sec:psprecs}
|
||||
The MLD2P4 library contains the implementation of many preconditioning
|
||||
techniques. The preconditioners may be applied as normal ``base''
|
||||
preconditioners; alternatively multiple ``base'' preconditioners may
|
||||
be combined in a multilevel framework.
|
||||
|
||||
The base (one-level) preconditioners include:
|
||||
\begin{itemize}
|
||||
\item Diagonal Scaling
|
||||
\item Block Jacobi
|
||||
\item Additive Schwarz, Restricted Additive Schwarz and
|
||||
Additive Schwarz with Harmonic extensions;
|
||||
\end{itemize}
|
||||
The Jacobi and Additive Schwarz preconditioners can make use of the
|
||||
following solvers:
|
||||
\begin{itemize}
|
||||
\item Level-$p$ Incomplete LU factorization ($ILU(p)$);
|
||||
\item Threshold Incomplete LU factorization ($ILU(\tau,p)$);
|
||||
\item Complete LU factorization by means of the following optional
|
||||
external packages:
|
||||
\begin{itemize}
|
||||
\item UMFPACK;
|
||||
\item SuperLU;
|
||||
\item SuperLU\_Dist.
|
||||
\end{itemize}
|
||||
\end{itemize}
|
||||
|
||||
The supporting data type and subroutine interfaces are defined in the
|
||||
module \verb|mld_prec_mod|; the module also overrides the variables
|
||||
and tyep definitions of \verb|psb_prec_mod| so as to function as a
|
||||
drop-in replacement for the PSBLAS methods. Thus if the user does not
|
||||
wish to employ the additional MLD2P4 capabitlities, it is possible to
|
||||
migrate an existing PSBLAS program without any source code
|
||||
modifications, only a recompilation is needed.
|
||||
|
||||
%% We also provide a companion package of multi-level Additive
|
||||
%% Schwarz preconditioners called MD2P4; this is actually a family of
|
||||
%% preconditioners since there is the possibility to choose between
|
||||
%% many variants, and is currently in an experimental stateIts
|
||||
%% documentation is planned to appear after stabilization of the
|
||||
%% package, which will characterize release 2.1 of our library.
|
||||
|
||||
|
||||
|
||||
|
||||
\subroutine{mld\_precinit}{Initialize a preconditioner}
|
||||
|
||||
\syntax{call mld\_precinit}{prec, ptype, info}
|
||||
\syntax*{call mld\_precinit}{prec, ptype, info, nlev}
|
||||
|
||||
\begin{description}
|
||||
\item[Type:] Asynchronous.
|
||||
\item[\bf On Entry]
|
||||
\item[ptype] the type of preconditioner.
|
||||
Scope: {\bf global} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in}.\\
|
||||
Specified as: a character string, see usage notes.
|
||||
\item[nlev] Number of levels in a multilevel precondtioner.
|
||||
Scope: {\bf global} \\
|
||||
Type: {\bf optional}\\
|
||||
Specified as: an integer value, see usage notes.
|
||||
%% \item[rs]
|
||||
%% Scope: {\bf global} \\
|
||||
%% Type: {\bf optional}\\
|
||||
%% Specified as: a long precision real number.
|
||||
\item[\bf On Exit]
|
||||
|
||||
\item[prec]
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: a preconditioner data structure \precdata.
|
||||
\item[info]
|
||||
Scope: {\bf global} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf out}.\\
|
||||
Error code: if no error, 0 is returned.
|
||||
\end{description}
|
||||
\subsection*{Usage Notes}
|
||||
%% The PSBLAS 2.0 contains a number of preconditioners, ranging from a
|
||||
%% simple diagonal scaling to 2-level domain decomposition. These
|
||||
%% preconditioners may use the SuperLU or the UMFPACK software, if
|
||||
%% installed; see~\cite{SUPERLU,UMFPACK}.
|
||||
Legal inputs to this subroutine are interpreted depending on the
|
||||
$ptype$ string as follows\footnote{The string is case-insensitive}:
|
||||
\begin{description}
|
||||
\item[NONE] No preconditioning, i.e. the preconditioner is just a copy
|
||||
operator.
|
||||
\item[DIAG] Diagonal scaling; each entry of the input vector is
|
||||
multiplied by the reciprocal of the sum of the absolute values of
|
||||
the coefficients in the corresponding row of matrix $A$;
|
||||
\item[BJAC] Precondition by a factorization of the
|
||||
block-diagonal of matrix $A$, where block boundaries are determined
|
||||
by the data allocation boundaries for each process; requires no
|
||||
communication.
|
||||
\item[AS] Additive Schwarz; default is to apply the Restricted
|
||||
Additive Schwarz variant, with an $ILU(0)$ factorization
|
||||
\item[ML] Multilevel preconditioner.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\subroutine{mld\_precset}{Set preconditioner features}
|
||||
|
||||
\syntax{call mld\_precset}{prec, what, val, info, ilev}
|
||||
|
||||
|
||||
\begin{description}
|
||||
\item[Type:] Asynchronous.
|
||||
\item[\bf On Entry]
|
||||
\item[prec] the preconditioner.\\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: an already initialized precondtioner data structure \precdata\\
|
||||
\item[what] The feature to be set. \\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in}.\\
|
||||
Specified as: an integer constants. Symbolic names are available in
|
||||
the library module, see usage notes for legal values.
|
||||
\item[val] The value to set the chosen feature to. \\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in}.\\
|
||||
Specified as: an integer, double precision or character variable.
|
||||
Symbolic names for some choices are available in the library module,
|
||||
see usage notes for legal values.
|
||||
\item[ilev] The level of a multilevel preconditioner to which the
|
||||
feature choice should apply.\\
|
||||
Scope: {\bf global} \\
|
||||
Type: {\bf optional}\\
|
||||
Specified as: an integer value, see usage notes.
|
||||
\end{description}
|
||||
|
||||
\begin{description}
|
||||
\item[\bf On Return]
|
||||
\item[prec] the preconditioner.\\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: a precondtioner data structure \precdata\\
|
||||
\item[info] Error code.\\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required} \\
|
||||
Intent: {\bf out}.\\
|
||||
An integer value; 0 means no error has been detected.
|
||||
\end{description}
|
||||
|
||||
\subsection*{Usage Notes}
|
||||
Legal inputs to this subroutine are interpreted depending on the value
|
||||
of \verb|what| input as follows
|
||||
\begin{description}
|
||||
\item[mld\_coarse\_mat\_]
|
||||
\end{description}
|
||||
|
||||
|
||||
\subroutine{mld\_precbld}{Builds a preconditioner}
|
||||
|
||||
\syntax{call mld\_precbld}{a, desc\_a, prec, info}
|
||||
|
||||
\begin{description}
|
||||
\item[Type:] Synchronous.
|
||||
\item[\bf On Entry]
|
||||
\item[a] the system sparse matrix.
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in}, target.\\
|
||||
Specified as: a sparse matrix data structure \spdata.
|
||||
\item[prec] the preconditioner.\\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: an already initialized precondtioner data structure \precdata\\
|
||||
\item[desc\_a] the problem communication descriptor.
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in}, target.\\
|
||||
Specified as: a communication descriptor data structure \descdata.
|
||||
%% \item[upd]
|
||||
%% Scope: {\bf global} \\
|
||||
%% Type: {\bf optional}\\
|
||||
%% Intent: {\bf in}.\\
|
||||
%% Specified as: a character.
|
||||
\end{description}
|
||||
|
||||
\begin{description}
|
||||
\item[\bf On Return]
|
||||
\item[prec] the preconditioner.\\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: a precondtioner data structure \precdata\\
|
||||
\item[info] Error code.\\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required} \\
|
||||
Intent: {\bf out}.\\
|
||||
An integer value; 0 means no error has been detected.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\subroutine{mld\_precaply}{Preconditioner application routine}
|
||||
|
||||
\syntax{call mld\_precaply}{prec,x,y,desc\_a,info,trans,work}
|
||||
\syntax*{call mld\_precaply}{prec,x,desc\_a,info,trans}
|
||||
|
||||
\begin{description}
|
||||
\item[Type:] Synchronous.
|
||||
\item[\bf On Entry]
|
||||
\item[prec] the preconditioner.
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in}.\\
|
||||
Specified as: a preconditioner data structure \precdata.
|
||||
\item[x] the source vector.
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: a double precision array.
|
||||
\item[desc\_a] the problem communication descriptor.
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in}.\\
|
||||
Specified as: a communication data structure \descdata.
|
||||
\item[trans]
|
||||
Scope: {\bf } \\
|
||||
Type: {\bf optional}\\
|
||||
Intent: {\bf in}.\\
|
||||
Specified as: a character.
|
||||
\item[work] an optional work space
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf optional}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: a double precision array.
|
||||
\end{description}
|
||||
|
||||
\begin{description}
|
||||
\item[\bf On Return]
|
||||
\item[y] the destination vector.
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: a double precision array.
|
||||
\item[info] Error code.\\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required} \\
|
||||
Intent: {\bf out}.\\
|
||||
An integer value; 0 means no error has been detected.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\subroutine{mld\_prec\_descr}{Prints a description of current preconditioner}
|
||||
|
||||
\syntax{call mld\_prec\_descr}{prec}
|
||||
|
||||
\begin{description}
|
||||
\item[Type:] Asynchronous.
|
||||
\item[\bf On Entry]
|
||||
\item[prec] the preconditioner.
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in}.\\
|
||||
Specified as: a preconditioner data structure \precdata.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "userguide"
|
||||
%%% End:
|
||||
@ -0,0 +1,72 @@
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Contents: The title page
|
||||
% $Id: title.tex 1999 2007-10-29 15:25:27Z sfilippo $
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\ifcase\pdfoutput % We're not running pdftex
|
||||
{\Large\bfseries MLD2P4\\[.8ex] User's and Reference Guide}\\
|
||||
\emph{\large A guide for the Multi-Level Domain Decomposition \\[.6ex]
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS}
|
||||
{\bfseries Pasqua D'Ambra}\\
|
||||
ICAR-CNR, Naples, Italy\\[3ex]
|
||||
{\bfseries Daniela di Serafino}\\
|
||||
Second University of Naples, Italy\\[3ex]
|
||||
{\bfseries Salvatore Filippone} \\
|
||||
University of Rome ``Tor Vergata'', Italy
|
||||
%\\[10ex]
|
||||
%\today
|
||||
Software version: 1.0\\
|
||||
%\today
|
||||
July 24, 2008
|
||||
\or
|
||||
\pdfbookmark{MLD2P4 User's and Reference Guide}{title}
|
||||
\newlength{\centeroffset}
|
||||
%\setlength{\centeroffset}{-0.5\oddsidemargin}
|
||||
%\addtolength{\centeroffset}{0.5\evensidemargin}
|
||||
%\addtolength{\textwidth}{-\centeroffset}
|
||||
\thispagestyle{empty}
|
||||
\vspace*{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
{\Huge\bfseries MLD2P4\\[.8ex] User's and Reference Guide
|
||||
}
|
||||
\noindent\rule[-1ex]{\textwidth}{5pt}\\[2.5ex]
|
||||
\hfill\emph{\Large A guide for the Multi-Level Domain Decomposition \\[.6ex]
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS}
|
||||
\end{minipage}}
|
||||
|
||||
\vspace{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
{\large\bfseries Pasqua D'Ambra}\\
|
||||
\large ICAR-CNR, Naples, Italy\\[3ex]
|
||||
{\large\bfseries Daniela di Serafino}\\
|
||||
\large Second University of Naples, Italy\\[3ex]
|
||||
{\large\bfseries Salvatore Filippone} \\
|
||||
\large University of Rome ``Tor Vergata'', Italy
|
||||
%\\[10ex]
|
||||
%\today
|
||||
\end{minipage}}
|
||||
|
||||
\vspace{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
\large Software version: 1.0\\
|
||||
%\today
|
||||
\large July 24, 2008
|
||||
\end{minipage}}
|
||||
%\addtolength{\textwidth}{\centeroffset}
|
||||
\vspace{\stretch{2}}
|
||||
\fi
|
||||
|
||||
\endinput
|
||||
|
||||
%
|
||||
|
||||
% Local Variables:
|
||||
% TeX-master: "userguide"
|
||||
% mode: latex
|
||||
% mode: flyspell
|
||||
% End:
|
||||
@ -0,0 +1,173 @@
|
||||
\documentclass[a4paper,twoside,11pt]{article}
|
||||
\usepackage{pstricks}
|
||||
\usepackage{fancybox}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{ifpdf}
|
||||
% \usepackage{minitoc}
|
||||
% \setcounter{minitocdepth}{2}
|
||||
\usepackage[bookmarks=true,
|
||||
bookmarksnumbered=true,
|
||||
bookmarksopen=false,
|
||||
plainpages=false,
|
||||
pdfpagelabels,
|
||||
colorlinks,
|
||||
citecolor=red,
|
||||
linkcolor=blue]{hyperref}
|
||||
\usepackage{ifthen}
|
||||
\usepackage{graphicx}
|
||||
\newtheorem{theorem}{Theorem}
|
||||
\newtheorem{corollary}{Corollary}
|
||||
\usepackage{rotating}
|
||||
%\newboolean{mtc}
|
||||
%\setboolean{mtc}{true}
|
||||
|
||||
\pdfoutput=1
|
||||
\relax
|
||||
\pdfcompresslevel=0 %-- 0 = none, 9 = best
|
||||
\pdfinfo{ %-- Info dictionary of PDF output /Author (PD, DdS, SF)
|
||||
/Title (MultiLevel Domain Decomposition Parallel Preconditioners Package
|
||||
based on PSBLAS, V. 1.0)
|
||||
/Subject (MultiLevel Domain Decomposition Parallel Preconditioners Package)
|
||||
/Keywords (Parallel Numerical Software, Algebraic Multilevel Preconditioners, Sparse Iterative Solvers, PSBLAS, MPI)
|
||||
/Creator (pdfLaTeX)
|
||||
/Producer ($Id: userguide.tex 2008-04-08 Pasqua D'Ambra, Daniela di Serafino,
|
||||
Salvatore Filippone$)
|
||||
}
|
||||
\pdfcatalog{ %-- Catalog dictionary of PDF output.
|
||||
% /URI (http://ce.uniroma2.it/psblas)
|
||||
}
|
||||
|
||||
\setlength\textwidth{1.15\textwidth}
|
||||
\setlength\oddsidemargin{0.3in}
|
||||
\setlength\evensidemargin{0.2in}
|
||||
% \newlength{\centeroffset}
|
||||
% \setlength{\centeroffset}{0.5\oddsidemargin}
|
||||
% \addtolength{\centeroffset}{0.5\evensidemargin}
|
||||
% \addtolength{\textwidth}{-\centeroffset}
|
||||
\pagestyle{myheadings}
|
||||
|
||||
\newcounter{subroutine}[subsection]
|
||||
\newcounter{example}[subroutine]
|
||||
\makeatletter
|
||||
\def\subroutine{\@ifstar{\@subroutine}{\clearpage\@subroutine}}%
|
||||
\def\@subroutine#1#2{%
|
||||
\stepcounter{subroutine}%
|
||||
\section*{\flushleft #1---#2 \endflushleft}%
|
||||
\addcontentsline{toc}{subsection}{#1}%
|
||||
\markright{#1}}%
|
||||
\newcommand{\subsubroutine}[2]{%
|
||||
\stepcounter{subroutine}%
|
||||
\subsection*{\flushleft #1---#2 \endflushleft}%
|
||||
\addcontentsline{toc}{subsubsection}{#1}%
|
||||
\markright{#1}}%
|
||||
\newcommand{\examplename}{Example}
|
||||
\newcommand{\syntaxname}{Syntax}
|
||||
\def\syntax{\@ifstar{\@ssyntax}{\@syntax}}%
|
||||
\def\@syntax{\nobreak\section*{\syntaxname}%
|
||||
\@ssyntax}%
|
||||
\def\@ssyntax#1#2{%
|
||||
\nobreak
|
||||
\setbox\@tempboxa\hbox{#1\ {\em $($#2$)$}}%
|
||||
\ifdim \wd\@tempboxa >\hsize
|
||||
\setbox\@tempboxa\hbox{\em $($#2$)$}
|
||||
\ifdim\wd\@tempboxa >\hsize
|
||||
\begin{flushright}#1\ \em$($#2$)$\end{flushright}%
|
||||
\else
|
||||
\hbox to\hsize{#1\hfil}%
|
||||
\hbox to\hsize{\hfil\box\@tempboxa}%
|
||||
\fi
|
||||
\else
|
||||
\hbox to\hsize{\hfil\box\@tempboxa\hfil}%
|
||||
\fi\par\vskip\baselineskip}
|
||||
\makeatother
|
||||
\newcommand{\example}{\stepcounter{example}%
|
||||
\section*{\examplename~\theexample}}
|
||||
\def\bsideways{\sidewaystable}
|
||||
\def\esideways{\endsidewaystable}
|
||||
|
||||
\newcommand{\precdata}{\hyperlink{precdata}{{\tt mld\_prec\_type}}}
|
||||
\newcommand{\descdata}{\hyperlink{descdata}{{\tt psb\_desc\_type}}}
|
||||
\newcommand{\spdata}{\hyperlink{spdata}{{\tt psb\_spmat\_type}}}
|
||||
\newcommand{\Ref}[1]{\mbox{(\ref{#1})}}
|
||||
|
||||
\begin{document}
|
||||
\pdfbookmark{MLD2P4 User's and Reference Guide}{title}
|
||||
\newlength{\centeroffset}
|
||||
%\setlength{\centeroffset}{-0.5\oddsidemargin}
|
||||
%\addtolength{\centeroffset}{0.5\evensidemargin}
|
||||
%\addtolength{\textwidth}{-\centeroffset}
|
||||
\thispagestyle{empty}
|
||||
\vspace*{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
{\Huge\bfseries MLD2P4\\[.8ex] User's and Reference Guide
|
||||
}
|
||||
\noindent\rule[-1ex]{\textwidth}{5pt}\\[2.5ex]
|
||||
\hfill\emph{\Large A guide for the Multi-Level Domain Decomposition \\[.6ex]
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS}
|
||||
\end{minipage}}
|
||||
|
||||
\vspace{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
{\large\bfseries Pasqua D'Ambra}\\
|
||||
\large ICAR-CNR, Naples, Italy\\[3ex]
|
||||
{\large\bfseries Daniela di Serafino}\\
|
||||
\large Second University of Naples, Italy\\[3ex]
|
||||
{\large\bfseries Salvatore Filippone} \\
|
||||
\large University of Rome ``Tor Vergata'', Italy
|
||||
%\\[10ex]
|
||||
%\today
|
||||
\end{minipage}}
|
||||
|
||||
\vspace{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
\large Software version: 1.0\\
|
||||
%\today
|
||||
\large July 24, 2008
|
||||
\end{minipage}}
|
||||
%\addtolength{\textwidth}{\centeroffset}
|
||||
\vspace{\stretch{2}}
|
||||
\clearpage
|
||||
\ \\
|
||||
\thispagestyle{empty}
|
||||
\clearpage
|
||||
|
||||
\pagenumbering{roman} % Roman numbering
|
||||
\setcounter{page}{1} % Abstract start on page i
|
||||
|
||||
\include{abstract}
|
||||
\cleardoublepage
|
||||
|
||||
\begingroup
|
||||
\renewcommand*{\thepage}{toc}
|
||||
%\pagenumbering{roman} % Roman numbering
|
||||
%\setcounter{page}{1} % Abstract start on page ii
|
||||
\tableofcontents
|
||||
\endgroup
|
||||
\cleardoublepage
|
||||
|
||||
\pagenumbering{arabic} % Arabic numbering
|
||||
\setcounter{page}{1} % Chapters start on page 1
|
||||
|
||||
\include{overview}
|
||||
\include{distribution}
|
||||
\include{building}
|
||||
\include{background}
|
||||
\include{gettingstarted}
|
||||
\include{userinterface}
|
||||
\include{errors}
|
||||
\clearpage
|
||||
\appendix
|
||||
\include{license}
|
||||
\cleardoublepage
|
||||
\include{bibliography}
|
||||
|
||||
|
||||
\end{document}
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: 'userguide'
|
||||
%%% End:
|
||||
@ -0,0 +1,149 @@
|
||||
\documentclass[a4paper,twoside,11pt]{article}
|
||||
\usepackage{pstricks}
|
||||
\usepackage{fancybox}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{ifpdf}
|
||||
% \usepackage{minitoc}
|
||||
% \setcounter{minitocdepth}{2}
|
||||
\usepackage[bookmarks=true,
|
||||
bookmarksnumbered=true,
|
||||
bookmarksopen=false,
|
||||
plainpages=false,
|
||||
pdfpagelabels,
|
||||
colorlinks,
|
||||
linkcolor=blue]{hyperref}
|
||||
\usepackage{ifthen}
|
||||
\usepackage{graphicx}
|
||||
\newtheorem{theorem}{Theorem}
|
||||
\newtheorem{corollary}{Corollary}
|
||||
\usepackage{rotating}
|
||||
%\newboolean{mtc}
|
||||
%\setboolean{mtc}{true}
|
||||
|
||||
\pdfoutput=0
|
||||
% \relax
|
||||
% \pdfcompresslevel=0 %-- 0 = none, 9 = best
|
||||
% \pdfinfo{ %-- Info dictionary of PDF output /Author (PD, DdS, SF)
|
||||
% /Title (MultiLevel Domain Decomposition Parallel Preconditioners Package
|
||||
% based on PSBLAS, V. 1.0)
|
||||
% /Subject (MultiLevel Domain Decomposition Parallel Preconditioners Package)
|
||||
% /Keywords (Parallel Numerical Software, Algebraic Multilevel Preconditioners, Sparse Iterative Solvers, PSBLAS, MPI)
|
||||
% /Creator (pdfLaTeX)
|
||||
% /Producer ($Id: userguide.tex 2008-04-08 Pasqua D'Ambra, Daniela di Serafino,
|
||||
% Salvatore Filippone$)
|
||||
% }
|
||||
% \pdfcatalog{ %-- Catalog dictionary of PDF output.
|
||||
% % /URI (http://ce.uniroma2.it/psblas)
|
||||
% }
|
||||
|
||||
\setlength\textwidth{1.15\textwidth}
|
||||
% \setlength\evensidemargin{.7in}
|
||||
% \newlength{\centeroffset}
|
||||
% \setlength{\centeroffset}{0.5\oddsidemargin}
|
||||
% \addtolength{\centeroffset}{0.5\evensidemargin}
|
||||
% \addtolength{\textwidth}{-\centeroffset}
|
||||
\pagestyle{myheadings}
|
||||
|
||||
\newcounter{subroutine}[subsection]
|
||||
\newcounter{example}[subroutine]
|
||||
\makeatletter
|
||||
\def\subroutine{\@ifstar{\@subroutine}{\clearpage\@subroutine}}%
|
||||
\def\@subroutine#1#2{%
|
||||
\stepcounter{subroutine}%
|
||||
\section*{\flushleft #1---#2 \endflushleft}%
|
||||
\addcontentsline{toc}{subsection}{#1}%
|
||||
\markright{#1}}%
|
||||
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|
||||
\stepcounter{subroutine}%
|
||||
\subsection*{\flushleft #1---#2 \endflushleft}%
|
||||
\addcontentsline{toc}{subsubsection}{#1}%
|
||||
\markright{#1}}%
|
||||
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|
||||
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|
||||
\def\syntax{\@ifstar{\@ssyntax}{\@syntax}}%
|
||||
\def\@syntax{\nobreak\section*{\syntaxname}%
|
||||
\@ssyntax}%
|
||||
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|
||||
\nobreak
|
||||
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|
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|
||||
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|
||||
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|
||||
\begin{flushright}#1\ \em$($#2$)$\end{flushright}%
|
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|
||||
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|
||||
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|
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|
||||
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|
||||
\hbox to\hsize{\hfil\box\@tempboxa\hfil}%
|
||||
\fi\par\vskip\baselineskip}
|
||||
\makeatother
|
||||
\newcommand{\example}{\stepcounter{example}%
|
||||
\section*{\examplename~\theexample}}
|
||||
\def\bsideways{\begin{table}}
|
||||
\def\esideways{\end{table}}
|
||||
|
||||
\newcommand{\precdata}{\hyperlink{precdata}{{\tt mld\_prec\_type}}}
|
||||
\newcommand{\descdata}{\hyperlink{descdata}{{\tt psb\_desc\_type}}}
|
||||
\newcommand{\spdata}{\hyperlink{spdata}{{\tt psb\_spmat\_type}}}
|
||||
\newcommand{\Ref}[1]{\mbox{(\ref{#1})}}
|
||||
|
||||
\begin{document}
|
||||
{\Large\bfseries MLD2P4\\[.8ex] User's and Reference Guide}\\[\baselineskip]
|
||||
\emph{\large A guide for the Multi-Level Domain Decomposition
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS}\\[3ex]
|
||||
{\bfseries Pasqua D'Ambra}\\
|
||||
ICAR-CNR, Naples, Italy\\
|
||||
{\bfseries Daniela di Serafino}\\
|
||||
Second University of Naples, Italy\\
|
||||
{\bfseries Salvatore Filippone} \\
|
||||
University of Rome ``Tor Vergata'', Italy\\[2ex]
|
||||
%\\[10ex]
|
||||
%\today
|
||||
Software version: 1.0\\
|
||||
%\today
|
||||
July 24, 2008
|
||||
\clearpage
|
||||
\ \\
|
||||
\thispagestyle{empty}
|
||||
\clearpage
|
||||
|
||||
\pagenumbering{roman} % Roman numbering
|
||||
\setcounter{page}{1} % Abstract start on page i
|
||||
|
||||
\include{abstract}
|
||||
\cleardoublepage
|
||||
|
||||
\begingroup
|
||||
\renewcommand*{\thepage}{toc}
|
||||
%\pagenumbering{roman} % Roman numbering
|
||||
%\setcounter{page}{1} % Abstract start on page ii
|
||||
\tableofcontents
|
||||
\endgroup
|
||||
\cleardoublepage
|
||||
|
||||
\pagenumbering{arabic} % Arabic numbering
|
||||
\setcounter{page}{1} % Chapters start on page 1
|
||||
|
||||
\include{overview}
|
||||
\include{distribution}
|
||||
\include{building}
|
||||
\include{background}
|
||||
\include{gettingstarted}
|
||||
\include{userinterface}
|
||||
%\include{advanced}
|
||||
\include{errors}
|
||||
%\include{listofroutines}
|
||||
\cleardoublepage
|
||||
\appendix
|
||||
\include{license}
|
||||
\cleardoublepage
|
||||
\include{bibliography}
|
||||
|
||||
|
||||
\end{document}
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: 'userguide'
|
||||
%%% End:
|
||||
@ -0,0 +1,443 @@
|
||||
\section{User Interface\label{sec:userinterface}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:userinterface} User Interface}}
|
||||
|
||||
|
||||
The basic user interface of MLD2P4 consists of six routines. The four routines \verb|mld_| \verb|precinit|,
|
||||
\verb|mld_precset|, \verb|mld_precbld| and \verb|mld_precaply| encapsulate all the functionalities
|
||||
for the setup and the application of any one-level and multi-level
|
||||
preconditioner implemented in the package.
|
||||
The routine \verb|mld_precfree| deallocates the preconditioner data structure, while
|
||||
\verb|mld_precdescr| prints a description of the preconditioner setup by the user.
|
||||
|
||||
For each routine, the same user interface is overloaded with
|
||||
respect to the real/complex case and the single/double precision;
|
||||
arguments with appropriate data types must be passed to the routine,
|
||||
i.e.
|
||||
\begin{itemize}
|
||||
\item the sparse matrix data structure, containing the matrix to be
|
||||
preconditioned, must be of type \verb|mld_|\emph{x}\verb|spmat_type|
|
||||
with \emph{x} = \verb|s| for real single precision, \emph{x} = \verb|d|
|
||||
for real double precision, \emph{x} = \verb|c| for complex single precision,
|
||||
\emph{x} = \verb|z| for complex double precision;
|
||||
\item the preconditioner data structure must be of type
|
||||
\verb|mld_|\emph{x}\verb|prec_type|, with \emph{x} =
|
||||
\verb|s|, \verb|d|, \verb|c|, \verb|z|, according to the sparse
|
||||
matrix data structure;
|
||||
\item the arrays containing the vectors $v$ and $w$ involved in
|
||||
the preconditioner application $w=M^{-1}v$ must be of type
|
||||
\emph{type}\verb|(|\emph{kind\_parameter}\verb|)|, with \emph{type} =
|
||||
\verb|real|, \verb|complex| and \emph{kind\_parameter} = \verb|kind(1.e0)|,
|
||||
\verb|kind(1.d0)|, according to the sparse matrix and preconditioner
|
||||
data structure; note that the PSBLAS module \verb|psb_base_mod|
|
||||
provides the constants \verb|psb_spk_|
|
||||
= \verb|kind(1.e0)| and \verb|psb_dpk_| = \verb|kind(1.d0)|;
|
||||
\item real parameters defining the preconditioner must be declared
|
||||
according to the precision of the sparse matrix and preconditioner
|
||||
data structures (see Section~\ref{sec:precset}).
|
||||
\end{itemize}
|
||||
A description of each routine is given in the remainder of this section.
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precinit\label{sec:precinit}}
|
||||
|
||||
\begin{center}
|
||||
\verb|mld_precinit(p,ptype,info)| \\
|
||||
\verb|mld_precinit(p,ptype,info,nlev)| \\
|
||||
\end{center}
|
||||
|
||||
\noindent
|
||||
This routine allocates and initializes the preconditioner data structure,
|
||||
according to the preconditioner type chosen by the user.
|
||||
|
||||
{\vskip2\baselineskip\noindent\large\bfseries Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\
|
||||
& The preconditioner data structure. Note that \emph{x}
|
||||
must be chosen according to the real/complex, single/double
|
||||
precision version of MLD2P4 under use.\\
|
||||
\verb|ptype| & \verb|character(len=*), intent(in)|.\\
|
||||
& The type of preconditioner. Its values are specified
|
||||
in Table~\ref{tab:precinit}.\\
|
||||
& Note that the strings are case insensitive.\\
|
||||
\verb|info| & \verb|integer, intent(out)|.\\
|
||||
& Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\
|
||||
\verb|nlev| & \verb|integer, optional, intent(in)|.\\
|
||||
& The number of levels of the multilevel preconditioner.
|
||||
If \verb|nlev| is not present and \verb|ptype|=\verb|'ML'|, \verb|'ml'|,
|
||||
then \verb|nlev|=2 is assumed. Otherwise, \verb|nlev| is ignored.\\
|
||||
\end{tabular}
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precset\label{sec:precset}}
|
||||
|
||||
\begin{center}
|
||||
\verb|mld_precset(p,what,val,info)|\\
|
||||
\end{center}
|
||||
|
||||
\noindent
|
||||
This routine sets the parameters defining the preconditioner. More
|
||||
precisely, the parameter identified by \verb|what| is assigned the value
|
||||
contained in \verb|val|.
|
||||
|
||||
{\vskip2\baselineskip\noindent\large\bfseries Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\
|
||||
& The preconditioner data structure. Note that \emph{x} must
|
||||
be chosen according to the real/complex, single/double precision
|
||||
version of MLD2P4 under use.\\
|
||||
\verb|what| & \verb|integer, intent(in)|. \\
|
||||
& The number identifying the parameter to be set.
|
||||
A mnemonic constant has been associated to each of these
|
||||
numbers, as reported in Tables~\ref{tab:p_type}-\ref{tab:p_coarse}.\\
|
||||
\verb|val | & \verb|integer| \emph{or} \verb|character(len=*)| \emph{or}
|
||||
\verb|real(psb_spk_)| \emph{or} \verb|real(psb_dpk_)|,
|
||||
\verb|intent(in)|.\\
|
||||
& The value of the parameter to be set. The list of allowed
|
||||
values and the corresponding data types is given in
|
||||
Tables~\ref{tab:p_type}-\ref{tab:p_coarse}.
|
||||
When the value is of type \verb|character(len=*)|,
|
||||
it is also treated as case insensitive.\\
|
||||
\verb|info| & \verb|integer, intent(out)|.\\
|
||||
& Error code. If no error, 0 is returned. See Section~\ref{sec:errors}
|
||||
for details.\\
|
||||
%
|
||||
%\verb|ilev| & \verb|integer, optional, intent(in)|.\\
|
||||
% & For the multilevel preconditioner, the level at which the
|
||||
% preconditioner parameter has to be set.
|
||||
% The levels are numbered in increasing
|
||||
% order starting from the finest one, i.e.\ level 1 is the finest level.
|
||||
% If \verb|ilev| is not present, the parameter identified by \verb|what|
|
||||
% is set at all the appropriate levels (see Table~\ref{tab:params}).
|
||||
\end{tabular}
|
||||
|
||||
\ \\
|
||||
A variety of (one-level and multi-level) preconditioners can be obtained
|
||||
by a suitable setting of the preconditioner parameters. These parameters
|
||||
can be logically divided into four groups, i.e.\ parameters defining
|
||||
\begin{enumerate}
|
||||
\item the type of multi-level preconditioner;
|
||||
\item the one-level preconditioner used as smoother;
|
||||
\item the aggregation algorithm;
|
||||
\item the coarse-space correction at the coarsest level.
|
||||
\end{enumerate}
|
||||
A list of the parameters that can be set, along with their allowed and
|
||||
default values, is given in Tables~\ref{tab:p_type}-\ref{tab:p_coarse}.
|
||||
For a detailed description of the meaning of the parameters, please
|
||||
refer to Section~\ref{sec:background}.
|
||||
%
|
||||
%Note that the routine allows to set different features of the
|
||||
%preconditioner at each level through the use of \verb|ilev|.
|
||||
%This should be done by users with experience in the field of
|
||||
%multi-level preconditioners. Non-expert users are recommended
|
||||
%to call \verb| mld_precset| without specifying \verb|ilev|.
|
||||
|
||||
\bsideways
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|p{2cm}|l|p{7cm}|}
|
||||
\hline
|
||||
\verb|what| & \textsc{data type} & \verb|val| & \textsc{default} &
|
||||
\textsc{comments} \\ \hline
|
||||
%\multicolumn{5}{|c|}{\emph{type of the multi-level preconditioner}}\\ \hline
|
||||
\verb|mld_ml_type_| & \verb|character(len=*)|
|
||||
& \texttt{'ADD'} \ \ \ \texttt{'MULT'}
|
||||
& \texttt{'MULT'}
|
||||
& Basic multi-level framework: additive or multiplicative
|
||||
among the levels (always additive inside a level). \\ \hline
|
||||
\verb|mld_smoother_type_|& \verb|character(len=*)|
|
||||
& \texttt{'DIAG'} \ \ \ \texttt{'BJAC'} \ \ \ \texttt{'AS'}
|
||||
& \texttt{'AS'}
|
||||
& Basic one-level preconditioner (i.e.\ smoother): diagonal,
|
||||
block Jacobi, AS. \\ \hline
|
||||
\verb|mld_smoother_pos_| & \verb|character(len=*)|
|
||||
& \texttt{'PRE'} \ \ \ \texttt{'POST'} \ \ \ \texttt{'TWOSIDE'}
|
||||
& \texttt{'POST'}
|
||||
& ``Position'' of the smoother: pre-smoother, post-smoother,
|
||||
pre- and post-smoother. \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\caption{Parameters defining the type of multi-level preconditioner.
|
||||
\label{tab:p_type}}
|
||||
\esideways
|
||||
|
||||
\bsideways
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|p{3.2cm}|l|p{7cm}|}
|
||||
\hline
|
||||
\verb|what| & \textsc{data type} & \verb|val| & \textsc{default} &
|
||||
\textsc{comments} \\ \hline
|
||||
%\multicolumn{5}{|c|}{\emph{basic one-level preconditioner (smoother)}} \\ \hline
|
||||
\verb|mld_sub_ovr_| & \verb|integer|
|
||||
& any~int.~num.~$\ge 0$
|
||||
& 1
|
||||
& Number of overlap layers. \\ \hline
|
||||
\verb|mld_sub_restr_| & \verb|character(len=*)|
|
||||
& \texttt{'HALO'} \hspace{2.5cm} \texttt{'NONE'}
|
||||
& \texttt{'HALO'}
|
||||
& Type of restriction operator:
|
||||
\texttt{'HALO'} for taking into account the overlap, \texttt{'NONE'}
|
||||
for neglecting it. \\ \hline
|
||||
\verb|mld_sub_prol_| & \verb|character(len=*)|
|
||||
& \texttt{'SUM'} \hspace{2.5cm} \texttt{'NONE'}
|
||||
& \texttt{'NONE'}
|
||||
& Type of prolongation operator:
|
||||
\texttt{'SUM'} for adding the contributions from the overlap, \texttt{'NONE'}
|
||||
for neglecting them. \\ \hline
|
||||
\verb|mld_sub_solve_| & \verb|character(len=*)|
|
||||
& \texttt{'ILU'} \hspace{2.5cm} \texttt{'MILU'} \hspace{2.5cm} \texttt{'ILUT'}
|
||||
\hspace{2.5cm} \texttt{'UMF'} \hspace{2.5cm} \texttt{'SLU'}
|
||||
& \texttt{'UMF'}
|
||||
& Local solver: ILU($p$), MILU($p$), ILU($p,t$), LU from UMFPACK, LU from SuperLU
|
||||
(plus triangular solve). \\ \hline
|
||||
\verb|mld_sub_fillin_| & \verb|integer|
|
||||
& Any~int.~num.~$\ge 0$
|
||||
& 0
|
||||
& Fill-in level $p$ of the incomplete LU factorizations. \\ \hline
|
||||
\verb|mld_sub_iluthrs_| & \verb|real(|\emph{kind\_parameter}\verb|)|
|
||||
& Any~real~num.~$\ge 0$
|
||||
& 0
|
||||
& Drop tolerance $t$ in the ILU($p,t$) factorization. \\ \hline
|
||||
\verb|mld_sub_ren_| & \verb|character(len=*)|
|
||||
& \texttt{'RENUM\_NONE'} \texttt{'RENUM\_GLOBAL'} %, \texttt{'RENUM_GPS'}
|
||||
& \texttt{'RENUM\_NONE'}
|
||||
& Row and column reordering of the local submatrices: no reordering,
|
||||
reordering according to the global numbering of the rows and columns of
|
||||
the whole matrix. \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\caption{Parameters defining the one-level preconditioner used as smoother.
|
||||
\label{tab:p_smoother}}
|
||||
\esideways
|
||||
|
||||
\bsideways
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|p{2.3cm}|p{2.6cm}|p{7cm}|}
|
||||
\hline
|
||||
\verb|what| & \textsc{data type} & \verb|val| & \textsc{default} &
|
||||
\textsc{comments} \\ \hline
|
||||
%\multicolumn{5}{|c|}{\emph{aggregation algorithm}} \\ \hline
|
||||
\verb|mld_aggr_alg_| & \verb|character(len=*)|
|
||||
& \texttt{'DEC'}
|
||||
& \texttt{'DEC'}
|
||||
& Aggregation algorithm. Currently, only the decoupled aggregation is available. \\ \hline
|
||||
\verb|mld_aggr_kind_| & \verb|character(len=*)|
|
||||
& \texttt{'SMOOTH'} \hspace{2.5cm} \texttt{'RAW'}
|
||||
& \texttt{'SMOOTH'}
|
||||
& Type of aggregation: smoothed, raw (i.e.\ using the tentative prolongator). \\ \hline
|
||||
\verb|mld_aggr_thresh_| & \verb|real(|\emph{kind\_parameter}\verb|)|
|
||||
& Any~real~num. $\in [0, 1]$
|
||||
& 0
|
||||
& Threshold $\theta$ in the aggregation algorithm. \\ \hline
|
||||
\verb|mld_aggr_eig_| & \verb|character(len=*)|
|
||||
& \texttt{'A\_NORMI'}
|
||||
& \texttt{'A\_NORMI'}
|
||||
& Estimate of the eigenvalue $D^{-1}A$ with largest modulus,
|
||||
to build the damping parameter $\omega$ in the smoothed aggregation.
|
||||
Currently, only the infinity norm of
|
||||
the matrix is available. \\ \hline
|
||||
\verb|mld_aggr_damp_| & \verb|real(|\emph{kind\_parameter}\verb|)|
|
||||
& Any~real~num.
|
||||
& $4/(3||D^{-1}A||_\infty)$
|
||||
& Damping parameter $\omega$ in the smoothed aggregation algorithm.
|
||||
If the user specifies a negative value, then $\omega$
|
||||
is set to its default value;
|
||||
otherwise, $\omega$ is set to the value provided by the
|
||||
user. In the latter case no estimate of the eigenvalue of
|
||||
$D^{-1}A$ with largest modulus is computed.\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\caption{Parameters defining the aggregation algorithm.
|
||||
\label{tab:p_aggregation}}
|
||||
\esideways
|
||||
|
||||
\bsideways
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|p{3.2cm}|l|p{7cm}|}
|
||||
\hline
|
||||
\verb|what| & \textsc{data type} & \verb|val| & \textsc{default} &
|
||||
\textsc{comments} \\ \hline
|
||||
%\multicolumn{5}{|c|}{\emph{coarse-space correction at the coarsest level}}\\ \hline
|
||||
\verb|mld_coarse_mat_| & \verb|character(len=*)|
|
||||
& \texttt{'DISTR'} \hspace{2.5cm} \texttt{'REPL'}
|
||||
& \texttt{'DISTR'}
|
||||
& Coarsest matrix: distributed among the processors or
|
||||
replicated on each of them. \\ \hline
|
||||
\verb|mld_coarse_solve_| & \verb|character(len=*)|
|
||||
& \texttt{'BJAC'} \hspace{2.5cm} \texttt{'UMF'} \hspace{2.5cm}
|
||||
\texttt{'SLU'} \hspace{2.5cm} \texttt{'SLUDIST'}
|
||||
& \texttt{'BJAC'}
|
||||
& Solver used at the coarsest level: block Jacobi, sequential
|
||||
LU from UMFPACK, sequential LU from SuperLU,
|
||||
distributed LU from SuperLU\_Dist.
|
||||
\texttt{'BJAC'} and \texttt{'SLUDIST'} require the coarsest
|
||||
matrix to be distributed, while \texttt{'UMF'} and
|
||||
\texttt{'SLU'} require it to be replicated. \\ \hline
|
||||
\verb|mld_coarse_subsolve_| & \verb|character(len=*)|
|
||||
& \texttt{'ILU'} \hspace{2.5cm} \texttt{'MILU'}
|
||||
\hspace{2.5cm} \texttt{'ILUT'}
|
||||
\hspace{2.5cm} \texttt{'UMF'} \hspace{2.5cm} \texttt{'SLU'}
|
||||
& \texttt{'UMF'}
|
||||
& Solver for the diagonal blocks of the coarse matrix,
|
||||
in case the block Jacobi solver
|
||||
is chosen as coarsest-level solver: ILU($p$), MILU($p$),
|
||||
ILU($p,t$), LU from UMFPACK,
|
||||
LU from SuperLU, plus triangular solve. \\ \hline
|
||||
\verb|mld_coarse_sweeps_|& \verb|integer|
|
||||
& Any~int.~num.~$> 0$
|
||||
& 4
|
||||
& Number of Block-Jacobi sweeps when 'BJAC' is used as
|
||||
coarsest-level solver. \\ \hline
|
||||
\verb|mld_coarse_fillin_| & \verb|integer|
|
||||
& Any~int.~num.~$\ge 0$
|
||||
& 0
|
||||
& Fill-in level $p$ of the incomplete LU factorizations. \\ \hline
|
||||
\verb|mld_coarse_iluthrs_| & \verb|real(|\emph{kind\_parameter}\verb|)|
|
||||
& Any~real.~num.~$\ge 0$
|
||||
& 0
|
||||
& Drop tolerance $t$ in the ILU($p,t$) factorization. \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\caption{Parameters defining the coarse-space correction at the coarsest
|
||||
level.\label{tab:p_coarse}}
|
||||
\esideways
|
||||
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precbld\label{sec:precbld}}
|
||||
|
||||
\begin{center}
|
||||
\verb|mld_precbld(a,desc_a,p,info)|\\
|
||||
\end{center}
|
||||
|
||||
\noindent
|
||||
This routine builds the preconditioner according to the requirements made by
|
||||
the user through the routines \verb|mld_precinit| and \verb|mld_precset|.
|
||||
|
||||
{\vskip2\baselineskip\noindent\large\bfseries Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|a| & \verb|type(psb_|\emph{x}\verb|spmat_type), intent(in)|. \\
|
||||
& The sparse matrix structure containing the local part of the
|
||||
matrix to be preconditioned. Note that \emph{x} must be chosen according
|
||||
to the real/complex,
|
||||
single/double precision version of MLD2P4 under use.
|
||||
See the PSBLAS User's Guide for details \cite{PSBLASGUIDE}.\\
|
||||
\verb|desc_a| & \verb|type(psb_desc_type), intent(in)|. \\
|
||||
& The communication descriptor of \verb|a|. See the PSBLAS User's Guide for
|
||||
details \cite{PSBLASGUIDE}.\\
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\
|
||||
& The preconditioner data structure. Note that \emph{x} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
\verb|info| & \verb|integer, intent(out)|.\\
|
||||
& Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\
|
||||
\end{tabular}
|
||||
|
||||
\clearpage
|
||||
\subsection{Subroutine mld\_precaply\label{sec:precaply}}
|
||||
|
||||
\begin{center}
|
||||
\verb|mld_precaply(p,x,y,desc_a,info)|\\
|
||||
\verb|mld_precaply(p,x,y,desc_a,info,trans,work)|\\
|
||||
\end{center}
|
||||
|
||||
\noindent
|
||||
This routine computes $y = op(M^{-1})\, x$, where $M$ is a previously built
|
||||
preconditioner, stored into \verb|p|, and $op$
|
||||
denotes the preconditioner itself or its transpose, according to
|
||||
the value of \verb|trans|.
|
||||
Note that, when MLD2P4 is used with a Krylov solver from PSBLAS,
|
||||
\verb|mld_precaply| is called within the PSBLAS routine \verb|mld_krylov|
|
||||
and hence it is completely transparent to the user.
|
||||
|
||||
{\vskip2\baselineskip\noindent\large\bfseries Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\
|
||||
& The preconditioner data structure, containing the local part of $M$.
|
||||
Note that \emph{x} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
\verb|x| & \emph{type}\verb|(|\emph{kind\_parameter}\verb|), dimension(:), intent(in)|.\\
|
||||
& The local part of the vector $x$. Note that \emph{type} and
|
||||
\emph{kind\_parameter} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
\verb|y| & \emph{type}\verb|(|\emph{kind\_parameter}\verb|), dimension(:), intent(out)|.\\
|
||||
& The local part of the vector $y$. Note that \emph{type} and
|
||||
\emph{kind\_parameter} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
\verb|desc_a| & \verb|type(psb_desc_type), intent(in)|. \\
|
||||
& The communication descriptor associated to the matrix to be
|
||||
preconditioned.\\
|
||||
\verb|info| & \verb|integer, intent(out)|.\\
|
||||
& Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\
|
||||
\verb|trans| & \verb|character(len=1), optional, intent(in).|\\
|
||||
& If \verb|trans| = \verb|'N','n'| then $op(M^{-1}) = M^{-1}$;
|
||||
if \verb|trans| = \verb|'T','t'| then $op(M^{-1}) = M^{-T}$
|
||||
(transpose of $M^{-1})$; if \verb|trans| = \verb|'C','c'| then $op(M^{-1}) = M^{-C}$
|
||||
(conjugate transpose of $M^{-1})$.\\
|
||||
\verb|work| & \emph{type}\verb|(|\emph{kind\_parameter}\verb|), dimension(:), optional, target|.\\
|
||||
& Workspace. Its size should be at
|
||||
least \verb|4 * psb_cd_get_local_| \verb|cols(desc_a)| (see the PSBLAS User's Guide).
|
||||
Note that \emph{type} and \emph{kind\_parameter} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
\end{tabular}
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precfree\label{sec:precfree}}
|
||||
|
||||
\begin{center}
|
||||
\verb|mld_precfree(p,info)|\\
|
||||
\end{center}
|
||||
|
||||
\noindent
|
||||
This routine deallocates the preconditioner data structure.
|
||||
|
||||
{\vskip2\baselineskip\noindent\large\bfseries Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{10.5cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\
|
||||
& The preconditioner data structure. Note that \emph{x} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
\verb|info| & \verb|integer, intent(out)|.\\
|
||||
& Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\
|
||||
\end{tabular}
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precdescr\label{sec:precdescr}}
|
||||
|
||||
\begin{center}
|
||||
\verb|mld_precdescr(p,info)|\\
|
||||
\verb|mld_precdescr(p,info,iout)|\\
|
||||
\end{center}
|
||||
|
||||
\noindent
|
||||
This routine prints a description of the preconditioner to the standard output or
|
||||
to a file. It must be called after \verb|mld_precbld| has been called.
|
||||
|
||||
{\vskip2\baselineskip\noindent\large\bfseries Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(in)|.\\
|
||||
& The preconditioner data structure. Note that \emph{x} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
\verb|info| & \verb|integer, intent(out)|.\\
|
||||
& Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\
|
||||
\verb|iout| & \verb|integer, intent(in), optional|.\\
|
||||
& The id of the file where the preconditioner description
|
||||
will be printed; the default is the standard output.\\
|
||||
\end{tabular}
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "userguide"
|
||||
%%% End:
|
||||
Loading…
Reference in New Issue