From cc4cf7930fa8cb31df5f9b17a80f9ac406ba36d0 Mon Sep 17 00:00:00 2001 From: lilia Date: Thu, 11 Dec 2014 13:52:31 +0100 Subject: [PATCH] 11-12-2014 v01 --- krylov_multi_reviewed.tex | 28 +- review.txt | 3 +- strong_scaling_150x150x150.eps | 750 --------------------------------- weak_scaling_280K.eps | 738 -------------------------------- weak_scaling_280k.eps | 740 -------------------------------- weak_scaling_50K.eps | 740 -------------------------------- weak_scaling_50k.eps | 740 -------------------------------- 7 files changed, 25 insertions(+), 3714 deletions(-) delete mode 100644 strong_scaling_150x150x150.eps delete mode 100644 weak_scaling_280K.eps delete mode 100644 weak_scaling_280k.eps delete mode 100644 weak_scaling_50K.eps delete mode 100644 weak_scaling_50k.eps diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex index 4cad7d7..936cdb6 100644 --- a/krylov_multi_reviewed.tex +++ b/krylov_multi_reviewed.tex @@ -320,10 +320,12 @@ The main key points of our Krylov multisplitting method to solve a large sparse \section{Experiments} \label{sec:04} -In order to illustrate the interest of our algorithm, we have compared our -algorithm with the GMRES method which is a commonly used method in many -situations. We have chosen to focus on only one problem which is very simple to -implement: a 3 dimension Poisson problem. +%%% MODIFIE *********************** +%%%******************************** +In order to illustrate the interest of our Krylov multisplitting algorithm, we have compared its performances with those of a classical block Jacobi multisplitting method and those of the GMRES method which is a commonly used method in many situations. +%%%******************************** +%%%******************************** + We have chosen to focus on only one problem which is very simple to implement: a 3 dimension Poisson problem. \begin{equation} \left\{ @@ -342,11 +344,27 @@ obtained for a 3D Poisson problem, the number of iterations is high. Using a preconditioner it is possible to reduce the number of iterations but preconditioners are not scalable when using many cores. + +%%% MODIFIE *********************** +%%%******************************** +We have performed some experiments on an infiniband cluster of 3 nodes of Intel Xeon CPU E5620 2.40 GHz and 12 GB of memory. For the GMRES code (alone and in both multisplitting versions) the restart parameter is fixed to 16. The precision of the GMRES version is fixed to 1e-6. For the multisplitting versions, there are two precisions, one for the external solver which is fixed to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It should be noted that a high precision is used but we also fixed a maximum number of iterations for each internal step. In practice, we limit the number of iterations in the internal step to 10. So an internal iteration is finished when the precision is reached or when the maximum internal number of iterations is reached. The precision and the maximum number of iterations of CGNR method used by our Krylov multisplitting algorithm are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis S is fixed to 10 vectors. + +\begin{figure}[htbp] +\centering + \includegraphics[width=0.7\textwidth]{strong_scaling_150x150x150} +\caption{Number of iterations per second with the same parameters as in Table~\ref{tab1} (weak scaling) with only 2 clusters} +\label{fig:01} +\end{figure} + +%%%******************************** +%%%******************************** + + %Doing many experiments with many cores is not easy and requires to access to a supercomputer with several hours for developing a code and then improving it. In the following we present some experiments we could achieve out on the Hector architecture, a UK's high-end computing resource, funded by the UK Research Councils~\cite{hector}. This is a Cray XE6 supercomputer, equipped with two -16-core AMD Opteron 2.3 Ghz and 32 GB of memory. Machines are interconnected +16-core AMD Opteron 2.3 GHz and 32 GB of memory. Machines are interconnected with a 3D torus. Table~\ref{tab1} shows the result of the experiments. The first column shows diff --git a/review.txt b/review.txt index 0c4d434..55007e2 100644 --- a/review.txt +++ b/review.txt @@ -91,7 +91,8 @@ The error is corrected. Reviewer #5: In this paper, the authors have implemented a Krylov multisplitting method to solve sparse linear systems on large-scale computing platforms. The technical approach and analysis of this paper is reasonable and the paper is clear, logical, and understandable. ,---- However, the paper does not take into considerate account relevant current and past research on the topic. -`---- +`--- +%Doing many experiments with many cores is not easy and requires to access to a supercomputer with several hours for developing a code and then improving it.- **************************************************** diff --git a/strong_scaling_150x150x150.eps b/strong_scaling_150x150x150.eps deleted file mode 100644 index 37192a4..0000000 --- a/strong_scaling_150x150x150.eps +++ /dev/null @@ -1,750 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%Title: strong_scaling_150x150x150.eps -%%Creator: gnuplot 4.6 patchlevel 6 -%%CreationDate: Wed Dec 10 17:34:25 2014 -%%DocumentFonts: (atend) -%%BoundingBox: 50 50 410 302 -%%EndComments -%%BeginProlog -/gnudict 256 dict def -gnudict begin -% -% The following true/false flags may be edited by hand if desired. -% The 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MFshow grestore} {MFshow} ifelse } bind def -/MRshow { currentpoint stroke M - exch dup MFwidth neg 3 -1 roll R - Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def -/MCshow { currentpoint stroke M - exch dup MFwidth -2 div 3 -1 roll R - Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def -/XYsave { [( ) 1 2 true false 3 ()] } bind def -/XYrestore { [( ) 1 2 true false 4 ()] } bind def -Level1 SuppressPDFMark or -{} { -/SDict 10 dict def -systemdict /pdfmark known not { - userdict /pdfmark systemdict /cleartomark get put -} if -SDict begin [ - /Title (weak_scaling_280k.eps) - /Subject (gnuplot plot) - /Creator (gnuplot 4.6 patchlevel 6) - /Author (lilia) -% /Producer (gnuplot) -% /Keywords () - /CreationDate (Thu Dec 11 10:46:23 2014) - /DOCINFO pdfmark -end -} ifelse -end -%%EndProlog -%%Page: 1 1 -gnudict begin -gsave -doclip -50 50 translate -0.050 0.050 scale -0 setgray -newpath -(Helvetica) findfont 140 scalefont setfont -BackgroundColor 0 lt 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dl1 2 dl2] LC4 DL} def -/LT5 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC5 DL} def -/LT6 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC6 DL} def -/LT7 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC7 DL} def -/LT8 {PL [2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 4 dl2] LC8 DL} def -/Pnt {stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore} def -/Dia {stroke [] 0 setdash 2 copy vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke - Pnt} def -/Pls {stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box {stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt} def -/Crs {stroke [] 0 setdash exch hpt sub exch vpt add M - hpt2 vpt2 neg V currentpoint stroke M - hpt2 neg 0 R hpt2 vpt2 V stroke} def -/TriU {stroke [] 0 setdash 2 copy vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke - Pnt} def -/Star 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[] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill - Bsquare} bind def -/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill - Bsquare} bind def -/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def -/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy vpt Square fill Bsquare} bind def -/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def -/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def -/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def -/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def -/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def -/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def -/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def -/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def -/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def -/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def -/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def -/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def -/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def -/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def -/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def -/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def -/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def -/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def -/DiaE {stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke} def -/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke} def -/TriUE {stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke} def -/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke} def -/PentE {stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore} def -/CircE {stroke [] 0 setdash - hpt 0 360 arc stroke} def -/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def -/DiaW {stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V Opaque stroke} def -/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V Opaque stroke} def -/TriUW {stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V Opaque stroke} def -/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V Opaque stroke} def -/PentW {stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - Opaque stroke grestore} def -/CircW {stroke [] 0 setdash - hpt 0 360 arc Opaque stroke} def -/BoxFill {gsave Rec 1 setgray fill grestore} def -/Density { - /Fillden exch def - currentrgbcolor - /ColB exch def /ColG exch def /ColR exch def - /ColR ColR Fillden mul Fillden sub 1 add def - /ColG ColG Fillden mul Fillden sub 1 add def - /ColB ColB Fillden mul Fillden sub 1 add def - ColR ColG ColB setrgbcolor} def -/BoxColFill {gsave Rec PolyFill} def -/PolyFill {gsave Density fill grestore grestore} def -/h {rlineto rlineto rlineto gsave closepath fill grestore} bind def -% -% PostScript Level 1 Pattern Fill routine for rectangles -% Usage: x y w h s a XX PatternFill -% x,y = lower left corner of box to be filled -% w,h = width and height of box -% a = angle in degrees between lines and x-axis -% XX = 0/1 for no/yes cross-hatch -% -/PatternFill {gsave /PFa [ 9 2 roll ] def - PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add 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pattern fill definitions -% -/Level2PatternFill { -/Tile8x8 {/PaintType 2 /PatternType 1 /TilingType 1 /BBox [0 0 8 8] /XStep 8 /YStep 8} - bind def -/KeepColor {currentrgbcolor [/Pattern /DeviceRGB] setcolorspace} bind def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke} ->> matrix makepattern -/Pat1 exch def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke - 0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke} ->> matrix makepattern -/Pat2 exch def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L - 8 8 L 8 0 L 0 0 L fill} ->> matrix makepattern -/Pat3 exch def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L - 0 12 M 12 0 L stroke} ->> matrix makepattern -/Pat4 exch def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L - 0 -4 M 12 8 L stroke} ->> matrix makepattern -/Pat5 exch def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L - 0 12 M 8 -4 L 4 12 M 10 0 L stroke} ->> matrix makepattern -/Pat6 exch def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L - 0 -4 M 8 12 L 4 -4 M 10 8 L stroke} ->> matrix makepattern -/Pat7 exch def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L - 12 0 M -4 8 L 12 4 M 0 10 L stroke} ->> matrix makepattern -/Pat8 exch def -<< Tile8x8 - /PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L - -4 0 M 12 8 L -4 4 M 8 10 L stroke} ->> matrix makepattern -/Pat9 exch def -/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def -/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def -/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def -/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def -/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def -/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def -/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def -} def -% -% -%End of PostScript Level 2 code -% -/PatternBgnd { - TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse -} def -% -% Substitute for Level 2 pattern fill codes with -% grayscale if Level 2 support is not selected. -% -/Level1PatternFill { -/Pattern1 {0.250 Density} bind def -/Pattern2 {0.500 Density} bind def -/Pattern3 {0.750 Density} bind def -/Pattern4 {0.125 Density} bind def -/Pattern5 {0.375 Density} bind def -/Pattern6 {0.625 Density} bind def -/Pattern7 {0.875 Density} bind def -} def -% -% Now test for support of Level 2 code -% -Level1 {Level1PatternFill} {Level2PatternFill} ifelse -% -/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont -dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall -currentdict end definefont pop -/MFshow { - { dup 5 get 3 ge - { 5 get 3 eq {gsave} {grestore} ifelse } - {dup dup 0 get findfont exch 1 get scalefont setfont - [ currentpoint ] exch dup 2 get 0 exch R dup 5 get 2 ne {dup dup 6 - get exch 4 get {Gshow} {stringwidth pop 0 R} ifelse }if dup 5 get 0 eq - {dup 3 get {2 get neg 0 exch R pop} {pop aload pop M} ifelse} {dup 5 - get 1 eq {dup 2 get exch dup 3 get exch 6 get stringwidth pop -2 div - dup 0 R} {dup 6 get stringwidth pop -2 div 0 R 6 get - show 2 index {aload pop M neg 3 -1 roll neg R pop pop} {pop pop pop - pop aload pop M} ifelse }ifelse }ifelse } - ifelse } - forall} def -/Gswidth {dup type /stringtype eq {stringwidth} {pop (n) stringwidth} ifelse} def -/MFwidth {0 exch { dup 5 get 3 ge { 5 get 3 eq { 0 } { pop } ifelse } - {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont - 6 get Gswidth pop add} {pop} ifelse} ifelse} forall} def -/MLshow { currentpoint stroke M - 0 exch R - Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def -/MRshow { currentpoint stroke M - exch dup MFwidth neg 3 -1 roll R - Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def -/MCshow { currentpoint stroke M - exch dup MFwidth -2 div 3 -1 roll R - Blacktext {gsave 0 setgray MFshow grestore} 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