source: palm/trunk/TUTORIAL/SOURCE/parallelization.tex @ 965

% $Id: parallelization.tex 945 2012-07-17 15:43:01Z raasch $
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\institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
\date{last update: \today}
\event{PALM Seminar}
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\title[Parallelization]{Parallelization}
\author{Siegfried Raasch}

\begin{document}

% Folie 1
\begin{frame}
   \titlepage
\end{frame}

\section{Parallelization}
\subsection{Parallelization}

% Folie 2
\begin{frame}
   \frametitle{Basics of Parallelization}
   \tikzstyle{yellow} = [rectangle,  fill=yellow!20, text width=0.6\textwidth, font=\scriptsize]
   \scriptsize
   \textbf{Parallelization:}
   \begin{itemize}
      \item<2-> All processor elements (PE, core) are carrying out the same program (SIMD).
      \item<3-> Each PE of a parallel computer operates on a different set of data.
   \end{itemize}

   \ \\
   \onslide<4->\textbf{Realization:}
   \begin{columns}[T]
      \begin{column}{0.45\textwidth}
         \onslide<5->each PE solves the equations for a different subdomain of the total domain
         \begin{center}
            \includegraphics[width=0.5\textwidth]{parallelization_figures/subdomain.png}
         \end{center}
         \onslide<7->each PE only knows the variable values from its subdomain, communication / data exchange between PEs is necessary\\
         \onslide<9->\textbf{message passing model (MPI)}
      \end{column}
      \begin{column}{0.45\textwidth}
         \onslide<6->program loops are parallelized, i.e. each processor solves for a subset of the total index range
         \begin{center}
            \begin{tikzpicture}[auto, node distance=0]
               \node [yellow] (1) {%
               \texttt{!\$OMP DO}\\
               \texttt{\quad \quad DO  i = 1, 100}\\
               \quad \quad \quad $\vdots$\\
               \texttt{\quad \quad ENDDO}};
            \end{tikzpicture}
         \end{center}
         \onslide<8-> parallelization can easily be done by the compiler, if all PEs have access to all variables (shared memory)\\
         \onslide<10-> \textbf{shared memory model (OpenMP)}
      \end{column}
   \end{columns}
\end{frame}

% Folie 3
\begin{frame}
   \frametitle{Basic Architectures of Massively Parallel Computers}
   \tikzstyle{info} = [rectangle, text width=0.25\textwidth, font=\scriptsize]
   
   
   \begin{center}
      \begin{tikzpicture}

         \node (center) at (0,1) {};
         \onslide<2-> \node (Network) at (-3.5,1) [ellipse,fill=green!20] {Network};
         \node (dis_mem) at (-3.5,-1) [text width=0.28\textwidth] {\footnotesize \textbf{distributed} memory\\(Cray-T3E)};
         \onslide<3-> \node (add_mem) at (3.5,1) [rectangle, draw] {adressable memory};
         \node (sha_mem) at (3.5,-1) [text width=0.35\textwidth] {\footnotesize \textbf{shared} memory\\(SGI-Altix, multicore PCs)};
         \onslide<7-> \node (MPI) at (-3.5,-3) [ellipse,fill=yellow!90] {MPI};
         \onslide<8-> \node (OpenMP) at (3.5,-3) [ellipse,fill=yellow!90] {OpenMP};         
         \onslide<6-> \node (clustered_systems) at (0,-3) [draw, text width=0.15\textwidth] {clustered systems};
         \node (cs_info) at (0,-4.2) [text width=0.4\textwidth] {\footnotesize (IBM-Regatta, Linux-Cluster,
NEC-SX, SGI-ICE, Cray-XT4)};

% Adressable memory node (big)

         \onslide<3-> \node (p1) at (2,-0.05) [draw,circle, scale=0.9] {\scriptsize p};
         \node (p2) at (2.6,-0.05) [draw,circle, scale=0.9] {\scriptsize p};
         \node (p3) at (3.2,-0.05) [draw,circle, scale=0.9] {\scriptsize p};
         \node (p4) at (3.8,-0.05) [draw,circle, scale=0.9] {\scriptsize p};
         \node (p5) at (4.4,-0.05) [draw,circle, scale=0.9] {\scriptsize p};
         \node (p6) at (5,-0.05) [draw,circle, scale=0.9] {\scriptsize p};
         
         \draw[-] (3.5,0.7) -- (3.5,0.4);
         \draw[-] (2,0.4) -- (5,0.4);
         \draw[-] (2,0.4) -- (p1);
         \draw[-] (2.6,0.4) -- (p2);
         \draw[-] (3.2,0.4) -- (p3);         
         \draw[-] (3.8,0.4) -- (p4);
         \draw[-] (4.4,0.4) -- (p5);         
         \draw[-] (5,0.4) -- (p6);
         
% Adressable memory node (small)    
         \onslide<4->
         \draw[->, thick] (1.5,0.2) -- (0.4,0.2) ;    
         \node at (0,0.2) [scale=0.2] {%
         \begin{tikzpicture}
            
            \node (add_mem_small) at (3.5,0.9) [ultra thick, rectangle, draw, minimum width=3cm] {};
            \node (p1_small) at (2,-0.05) [ultra thick, draw,circle, scale=0.9] {};
            \node (p2_small) at (2.6,-0.05) [ultra thick, draw,circle, scale=0.9] {};
            \node (p3_small) at (3.2,-0.05) [ultra thick, draw,circle, scale=0.9] {};
            \node (p4_small) at (3.8,-0.05) [ultra thick, draw,circle, scale=0.9] {};
            \node (p5_small) at (4.4,-0.05) [ultra thick, draw,circle, scale=0.9] {};
            \node (p6_small) at (5,-0.05) [ultra thick, draw,circle, scale=0.9] {};
         
            \draw[-, ultra thick] (add_mem_small.south) -- (3.5,0.4);
            \draw[-, ultra thick] (2,0.4) -- (5,0.4);
            \draw[-, ultra thick] (2,0.4) -- (p1_small);
            \draw[-, ultra thick] (2.6,0.4) -- (p2_small);
            \draw[-, ultra thick] (3.2,0.4) -- (p3_small);         
            \draw[-, ultra thick] (3.8,0.4) -- (p4_small);
            \draw[-, ultra thick] (4.4,0.4) -- (p5_small);         
            \draw[-, ultra thick] (5,0.4) -- (p6_small);
            
            
         \end{tikzpicture}
         } ;
         
         \onslide<5->
         \node (add_info) at (0,-0.1) [scale=0.9] {\scriptsize node};

% Black Arrows
         \onslide<6-> \draw[->, thick] (-2.5,-1.5) -- (-0.8,-2.2) ;
         \draw[->, thick] (2.5,-1.5) -- (0.8,-2.2) ;

% MPI Arrows
         \onslide<7-> \draw[->, ultra thick, color=yellow] (-3.5,-2.7) -- (-3.5,-1.5) ;
         \draw[->, ultra thick, color=yellow] (-2.9,-3) -- (-1.0,-3.0) ;
         \draw[->, ultra thick, color=yellow] (-3.0,-2.8) -- (1.5,-1.0) ;

% OpenMP Arrows         
         \onslide<8-> \draw[->, ultra thick, color=yellow] (3.5,-2.6) -- (3.5,-1.5) ;
         \draw[->, ultra thick, color=yellow] (2.5,-2.8) -- (-2.5,0.7) ;
         
% Network decorations
         \onslide<2-> \node (pr1) at (-4.6,0.7) [draw,circle,scale=0.5] {};
         \node (mem1) at (-4.6,0.45) [draw,rectangle,scale=0.5] {};
         \draw[-] (-4.5,0.9) -- (pr1);
         \draw[-] (mem1) -- (pr1);
         \draw[-] ([xshift=0.02cm]pr1.east) -- ([xshift=0.3cm, yshift=-0.2cm]pr1.east);
         \draw[-] ([xshift=0.02cm]mem1.east) -- ([xshift=0.3cm, yshift=-0.2cm]mem1.east);  
         \node at (-3.7,0.45) {\tiny processor};    
         \node at (-3.3,0.25) {\tiny adressable memory};   
         
         \node (pr2) at (-4.6,1.3) [draw,circle,scale=0.5] {};
         \node (mem2) at (-4.6,1.55) [draw,rectangle,scale=0.5] {};
         \draw[-] (-4.5,1.1) -- (pr2);
         \draw[-] (mem2) -- (pr2);
         \node (pr3) at (-3.9,1.5) [draw,circle,scale=0.5] {};
         \node (mem3) at (-3.9,1.75) [draw,rectangle,scale=0.5] {};
         \draw[-] (-3.8,1.3) -- (pr3);
         \draw[-] (mem3) -- (pr3);        
         \node (pr4) at (-4.9,0.95) [draw,circle,scale=0.5] {};
         \node (mem4) at (-4.9,0.7) [draw,rectangle,scale=0.5] {};
         \draw[-] (-4.55,1.0) -- (pr4);
         \draw[-] (mem4) -- (pr4);    
         \node (pr5) at (-3.0,1.5) [draw,circle,scale=0.5] {};
         \node (mem5) at (-3.0,1.75) [draw,rectangle,scale=0.5] {};
         \draw[-] (-3.08,1.3) -- (pr5);
         \draw[-] (mem5) -- (pr5);    
         \node (pr6) at (-2.2,1.1) [draw,circle,scale=0.5] {};
         \node (mem6) at (-2.2,1.35) [draw,rectangle,scale=0.5] {};
         \draw[-] (-2.45,1.0) -- (pr6);
         \draw[-] (mem6) -- (pr6);    
                           
      \end{tikzpicture}
   \end{center}

\end{frame}

% Folie 4
\begin{frame}
   \frametitle{PALM Parallelization Model}
   \scriptsize
   \onslide<2-> \textbf{General demands for a parallelized program:}
   \begin{itemize}
      \item<3-> Load balancing
      \item<4-> Small communication overhead
      \item<5-> Scalability (up to large numbers of processors)
   \end{itemize}
   \vspace{2mm}
   \onslide<6-> \textbf{The basic parallelization method used for PALM is a 2D-domain decomposition along $x$ and $y$:}\\
   \begin{columns}[T]
      \begin{column}{0.3\textwidth}
         \onslide<7-> \includegraphics[width=1.0\textwidth]{parallelization_figures/subdomain.png}
      \end{column}
      \begin{column}{0.6\textwidth}
         \ \\
         \onslide<8-> contiguous data in memory (FORTRAN):\\
         \ \\
         \ \\
         \textcolor{blue}{columns of i}\\
         \textcolor{red}{no contiguous data at all}\\
         \onslide<9-> \textcolor{blue}{columns of k}\\
         \textcolor{red}{planes of k,j (all data contiguous)}
      \end{column}
   \end{columns}   
   \vspace{2mm}
   \begin{itemize}
      \item<10-> Alternatively, a 1D-decomposition along $x$ or $y$ may be used in case of slow networks, but this generally doesn't scale for processor numbers $>$ 256.
      \item<11-> Message passing is realized using MPI.
      \item<12-> OpenMP parallelization as well as mixed usage of OpenMP and
MPI is also possible. (OpenMP tests and optimization is under way)
   \end{itemize}
\end{frame}

% Folie 5
\begin{frame}
   \frametitle{Implications of Decomposition}
   \scriptsize
   \begin{columns}[T]
      \begin{column}{0.5\textwidth}
         \begin{itemize}
            \item<2-> Central finite differences cause \textcolor{red}{local data dependencies}\\
            \ \\
            solution:  introduction of \textcolor{red}{ghost points}
            \vspace{5mm}
            
            \begin{flushright}
               \onslide<3-> $\left. \dfrac{\partial \psi}{\partial x} \right|_i = \dfrac{\psi_{i+1} - \psi_{i-1}}{2 \Delta x}$
            \end{flushright}
            \vspace{10mm}
            \item<4-> FFT and linear equation solver cause \textcolor{red}{non-local data dependencies}\\
            \ \\
            solution: transposition of 3D-arrays
         \end{itemize}
      \end{column}
      \begin{column}{0.6\textwidth}
      \begin{center}
         \vspace{-5mm}
         \onslide<3-> \includegraphics[width=0.7\textwidth]{parallelization_figures/ghost_points.png}
         \vspace{4mm}
         \onslide<5-> \includegraphics[width=0.8\textwidth]{parallelization_figures/fft.png} \end{center}
         \vspace{-4mm}
         \textbf{Example: transpositions for solving the poisson equation}
      \end{column}
   \end{columns}   
\end{frame}  

% Folie 6
\begin{frame}
   \frametitle{How to Use the Parallelized Version of PALM}   
   \scriptsize
   \begin{columns}[T]
      \begin{column}{1.12\textwidth}
         \begin{itemize} 
            \item<1-> The parallel version of PALM is switched on by \texttt{mrun}-option ''\texttt{-K parallel}''. Additionally, the number of required processors and the number of tasks per node (number of PEs to be used on one node) have to be provided:\\
                 \quad \texttt{mrun ... -K parallel -X64 -T8 ...}
                 \item<2-> From an accounting point of view, it is always most efficient to use all PEs of a node (\texttt{-T8}) (in case of a ''non-shared'' usage of nodes).
                 \item<3-> If a normal unix-kernel operating system (not a micro-kernel) is running on each CPU, then there migth be a speed-up of the code, if 1-2 PEs less than the total number of PEs on the node are used.
                 \item<4-> On machines with a comparably slow network, a 1D-decomposition (along $x$) should be used, because then only two transpositions have to be carried out by the pressure solver. A 1D-decomposition is automatically used for NEC-machines (e.g.  \texttt{-h necriam}). The virtual processor grid to be used can be set manually by d3par-parameters \texttt{npex} and \texttt{npey}.
            \item<6-> Using the Open-MP parallelization does not yield any advantage over using a pure domain decomposition with MPI (contrary to expectations, it mostly slows down the computational speed), but this may change on cluster systems for very large number of processors ($>$10000?).\\        
         \end{itemize}
         \begin{center}
         \vspace{-7mm}
         \onslide<5-> \includegraphics[width=0.13\textwidth]{parallelization_figures/folie_6.png}
         \end{center}
      \end{column}
   \end{columns}  
\end{frame}

% Folie 7
\begin{frame}
   \frametitle{MPI Communication}   
   \scriptsize
   \begin{columns}[T]
      \begin{column}{1.12\textwidth}
         \begin{itemize}
            \item<1-> MPI (message passing interface) is a portable interface for communication between PEs (FORTRAN or C library).
            \vspace{2mm}
            \item<2-> To make MPI available on HLRN‘s SGI-ICE, the module \texttt{mpt} must be loaded by setting the \texttt{\%modules} option  in .mrun.config appropriately: 

                 \quad \texttt{\%modules   ...:mpt:...} 
            \vspace{2mm}
                 \item<2-> The path to the MPI-library may have to be given in the compiler call, by setting an appropriate option in the configuration file .mrun.config:

                 \quad \texttt{\%lopts  -axW:-cpp:-r8:-nbs:-Vaxlib:\textcolor{blue}{-L:<replace by mpi library path>:-lmpi}}
            \vspace{2mm}
                 \item<3-> All MPI calls must be within\\
                 \quad \texttt{CALL MPI\_INIT( ierror )}\\
                 \quad $\vdots$\\
            \quad \texttt{CALL MPI\_FINALIZE( ierror )}\\
         \end{itemize}
         
      \end{column}
   \end{columns}  
\end{frame}

% Folie 8
\begin{frame}
   \frametitle{Communication Within PALM}   
   \small
   \begin{itemize}
      \item<1-> MPI calls within PALM are available when using the \texttt{mrun}-option ''\texttt{-K parallel}''.
      \item<2-> Communication is needed for
      \begin{itemize}
         \footnotesize
         \item<2-> exchange of ghost points
         \item<3-> transpositions (FFT-poisson-solver)
         \item<4-> calculating global sums (e.g. for calculating horizontal averages)
      \end{itemize}
      \item<5-> Additional MPI calls are required to define the so-called virtual processor grid and to define special data types needed for more comfortable exchange of data.
   \end{itemize}
\end{frame}

% Folie 9
\begin{frame}
   \frametitle{Virtual Processor Grid Used in PALM}   
   \footnotesize
   The processor grid and special data types are defined in file \texttt{init\_pegrid.f90}
   \begin{itemize}
      \item<2-> PALM uses a two-dimensional virtual processor grid (in case of a 1D-decomposition, it has only one element along $y$). It is defined by a so called communicator (here: \texttt{comm2d}):\\
      \scriptsize
      \quad \texttt{ndim = 2}\\
           \quad \texttt{pdims(1) = npex    ! \# of processors along x}\\
           \quad \texttt{pdims(2) = npey    ! \# of processors along y}\\
           \quad \texttt{cyclic(1) = .TRUE.}\\
           \quad \texttt{cyclic(2) = .TRUE.}\\

           \quad \texttt{CALL MPI\underline{\ }CART\underline{\ }CREATE( MPI\underline{\ }COMM\underline{\ }WORLD, ndim, pdims, cyclic,  \&}\\
           \quad \texttt{\hspace{10.5em} reorder, \textcolor{blue}{comm2d}, ierr )}  
      \item<3-> The processor number (id) with respect to this processor grid, \texttt{myid}, is given by:\\
      \scriptsize
      \quad \texttt{CALL MPI\underline{\ }COMM\underline{\ }RANK( comm2d, \textcolor{blue}{myid}, ierr )}      
      \item<4-> The ids of the neighbouring PEs are determined by:\\
      \scriptsize
      \quad \texttt{CALL MPI\underline{\ }CARD\underline{\ }SHIFT( comm2d, 0, 1, \textcolor{blue}{pleft},  \textcolor{blue}{pright}, ierr )}\\
      \quad \texttt{CALL MPI\underline{\ }CARD\underline{\ }SHIFT( comm2d, 1, 1, \textcolor{blue}{psouth}, \textcolor{blue}{pnorth}, ierr )}
   \end{itemize}
\end{frame}

% Folie 10
\begin{frame}
   \frametitle{Exchange of ghost points}   
   \scriptsize
         \begin{itemize}
            \item<1-> Ghost points are stored in additional array elements added at the horizontal boundaries of the subdomains, e.g.\\
            \tiny
            \quad \texttt{u(:,:,nxl\textcolor{blue}{-ngl}), u(:,:,nxr\textcolor{blue}{+ngl})    ! left and right boundary}\\
            \quad \texttt{u(:,nys\textcolor{blue}{-ngl},:), u(:,nyn\textcolor{blue}{+ngl},:)    ! south and north boundary}\\
            \hspace{3mm}
            \item<2-> \scriptsize The exchange of ghost points is done in file \texttt{exchange\underline{\ }horiz.f90}\\
            \textbf{\underline{Simplified} example:} synchroneous exchange of ghost points along $x$ ($yz$-planes, send left, receive right plane):\\
            \tiny
            \quad \texttt{CALL MPI\underline{\ }SENDRECV( ar(nzb,nys-\textcolor{blue}{ngl},nxl),   ngp\underline{\ }yz, MPI\underline{\ }REAL, pleft,  0,}\\
            \quad \texttt{\hspace{9.5em}ar(nzb,nys-\textcolor{blue}{ngl},nxr+1), ngp\underline{\ }yz, MPI\underline{\ }REAL, pright, 0,}\\
            \quad \texttt{\hspace{9.5em}comm2d, status, ierr )}\\
            \hspace{3mm}
            \item<3-> \scriptsize In the real code special MPI data types (vectors) are defined for exchange of $yz$/$xz$-planes for performance reasons and because array elements to be exchanged are not consecutively stored in memory for $xz$-planes:\\
            \tiny
            \quad \texttt{ngp\underline{\ }yz(0) = (nzt - nzb + 2) * (nyn - nys + 1 + 2 * \textcolor{blue}{ngl} )}\\
            \quad \texttt{CALL MPI\underline{\ }TYPE\underline{\ }VECTOR( \textcolor{blue}{ngl}, ngp\underline{\ }yz(0), ngp\underline{\ }yz(0), MPI\underline{\ }REAL, type\underline{\ }yz(0), ierr )}\\ 
            \quad \texttt{CALL MPI\underline{\ }TYPE\underline{\ }COMMIT( type\underline{\ }xz(0), ierr )   ! see file init\underline{\ }pegrid.f90}\\
            \ \\
            \quad \texttt{CALL MPI\underline{\ }SENDRECV( ar(nzb,nys-ngl,nxl), type\underline{\ }yz(grid\underline{\ }level), MPI\underline{\ }REAL, pleft, 0, ...}\\
         \end{itemize}       
\end{frame}

% Folie 11
\begin{frame}
   \frametitle{Transpositions}   
   \footnotesize
    \begin{columns}[T]
      \begin{column}{1.05\textwidth}
         \begin{itemize}
            \item<1-> Transpositions can be found in file \texttt{transpose.f90} (several subroutines for 1D- or 2D-decompositions; they are called mainly from the FFT pressure solver, see \texttt{poisfft.f90}.\\
            \ \\
            \item<2-> The following example is for a transposition from $x$ to $y$, i.e. for the input array all data elements along $x$ reside on the same PE, while after the transposition, all elements along $y$ are on the same PE:\\
            \ \\
            \scriptsize
            \texttt{!}\\
            \texttt{!--   in SUBROUTINE transpose\underline{\ }xy:}\\
           \texttt{CALL MPI\underline{\ }ALLTOALL( f\underline{\ }inv(nys\underline{\ }x,nzb\underline{\ }x,0), sendrecvcount\underline{\ }xy, MPI\underline{\ }REAL, \&}\\
           \texttt{\hspace{9.5em}work(1), \hspace{6.5em}sendrecvcount\underline{\ }xy, MPI\underline{\ }REAL, \&}\\
           \texttt{\hspace{9.5em}comm1dy, ierr )}\\
           \ \\
           \item<3-> The data resorting before and after the calls of MPI\_ALLTOALL is highly optimized to account for the different processor architectures.
         \end{itemize}
      \end{column}
   \end{columns}  
\end{frame}

% Folie 12
\begin{frame}
   \frametitle{Parallel I/O} 
   \scriptsize
   \begin{columns}[T]
      \begin{column}{1.1\textwidth}
         \begin{itemize}
            \item<1-> PALM writes and reads some of the input/output files in parallel, i.e. each processor writes/reads his own file. \textbf{Each file then has a different name!}\\
            \ \\
            \textbf{Example:} binary files for restart are written into a subdirectory of the PALM working directory:\\
            \quad \texttt{BINOUT/\_0000}\\
                 \quad \texttt{BINOUT/\_0001}\\
                 \quad $\vdots$
                 \item<2-> These files can be handled (copied) by \texttt{mrun} using the file attribute \texttt{pe} in the configuration file:\\
                 \texttt{BINOUT  out:loc:pe restart \~{}/palm/current\underline{\ }version/JOBS/\$fname/RESTART  \underline{\ }d3d}\\
                 \ \\
                 \onslide<3->In this case, filenames are interpreted as directory names. 
                 An \texttt{mrun} call using option\\
                 ''\texttt{-d example\underline{\ }cbl -r restart}'' will copy the local \textbf{\underline{directory}} \texttt{BINOUT} to the \textbf{\underline{directory}} \texttt{.../RESTART/example\underline{\ }cbl\underline{\ }d3d}  .
         \end{itemize}
         \onslide<4-> \textbf{General comment:}
         \begin{itemize}
            \item Parallel I/O on a large number of files ($>$1000) currently may cause severe file system problems (e.g. on Lustre file systems). A workaround for this problem will\\ be available soon. 
         \end{itemize}
      \end{column}
   \end{columns}  
\end{frame}



% Folie 13
\begin{frame}
   \frametitle{PALM Parallel I/O for 2D/3D Data} 
   \footnotesize
   \begin{itemize}
      \item<1-> 2D- and 3D-data output is also written in parallel by the processors (2D: by default, 3D: generally).
      \item<2-> Because the graphics software (\texttt{ncview}, \texttt{ncl}, \texttt{ferret}, etc.) expect the data to be in one file, these output files have to be merged to one single file after PALM has finished.\\
      \ \\
      This is done within the job by calling the utility program \texttt{combine\underline{\ }plot\underline{\ }fields.x} after PALM has successfully finished.
      \item<3-> \texttt{combine\underline{\ }plot\underline{\ }fields.x} is automatically executed by \texttt{mrun}.
      \item<4-> The executable \texttt{combine\underline{\ }plot\underline{\ }fields.x} is created during the installation process by the command\\
      \ \\
      \quad \texttt{mbuild -u -h <host identifier>}
   \end{itemize}
\end{frame}

% Folie 14
\begin{frame}
   \frametitle{Performance Examples (I)} 
   \begin{itemize}
      \item Simulation using 1536 * 768 * 242 grid points  ($\sim$ 60 GByte)
   \end{itemize}
   \includegraphics[scale=0.28]{parallelization_figures/perf_left.png} 
   \includegraphics[scale=0.28]{parallelization_figures/perf_right.png}
   \includegraphics[scale=0.28]{parallelization_figures/legende.png}  \\
   \scriptsize
      \begin{columns}[T]
         \begin{column}{0.18\textwidth}
         \end{column}
         \begin{column}{0.4\textwidth}
            IBM-Regatta, HLRN, Hannover\\
            (1D domain decomposition)
         \end{column}
         \begin{column}{0.4\textwidth}
            Sun Fire X4600, Tokyo Institute of Technology\\
(2D domain decomposition)
         \end{column} 
         \begin{column}{0.2\textwidth}
         \end{column}           
         
      \end{columns}

\end{frame}

% Folie 15
\begin{frame}
   \frametitle{Performance Examples (II)} 
   \begin{itemize}
      \item Simulation with $2048^3$ grid points  ($\sim$ 2 TByte memory)
   \end{itemize}
      \begin{columns}[T]
         \begin{column}{0.5\textwidth}
            \includegraphics[scale=0.25]{parallelization_figures/perf_3.png} \\
            \scriptsize
            \quad SGI-ICE2, HLRN-II, Hannover\\
            \quad (2D-domain decomposition)
         \end{column}
         \begin{column}{0.5\textwidth}
            \vspace{35mm}
            \onslide<2-> currently largest simulation feasible on that system:\\
            \ \\
            $4096^3$ grid points
         \end{column} 
      \end{columns}
\end{frame}

\end{document}