20140325-HPGMG.tex

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\title{Finite-element FAS GMG}
\author{{\bf Jed Brown} \texttt{jedbrown@mcs.anl.gov}}

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{ \small
  Mathematics and Computer Science Division, Argonne National Laboratory \\
  {\small Department of Computer Science, University of Colorado Boulder}
}

\date{HPCG \\ 2014-03-25}

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\subject{Talks}


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\begin{frame}{Objectives}
  \begin{itemize}
  \item Scale-free problem definition
    \begin{itemize}
    \item Do not adjudicate ``cores'', ``processors''
    \end{itemize}
  \item Scale-free communication
    \begin{itemize}
    \item Globally-coupled problems need long-range messages
    \end{itemize}
  \item Quantify dynamic range
    \begin{itemize}
    \item $N_{\text{max}}$ to fill memory, find $N_{1/2}$ at half efficiency
    \item ``dynamic range'' $N_{\text{max}}/N_{1/2}$
    \end{itemize}
  \item Stress conflict between vectorization and cache reuse
  \item 
  \end{itemize}
\end{frame}

\begin{frame}{Finite-element FAS}
  \begin{itemize}
  \item 
  \end{itemize}
\end{frame}

\begin{frame}{Newton linearization of $p$-Laplacian using $Q_2$ elements}
  \begin{equation*}
    -\nabla\cdot \Big[ (\eta + \eta' \nabla u \otimes \nabla u) \nabla w \Big] = f
  \end{equation*}
  \begin{itemize}
  \item Asymptotic data motion
    \begin{itemize}
    \item Coordinates: 24 \texttt{double}s per element (81 without reuse)
    \item State+Residual: 16 \texttt{double}s per element (54 without reuse)
    \item Coefficients: $4\cdot 27$ \texttt{double}s per element (no reuse)
    \end{itemize}
  \item Floating point (counting FMA)
    \begin{itemize}
    \item Evaluate metric terms to quadrature points: $3^4 3^2 3 = 2187$
    \item Inversion/determinants at quadrature points: $3^3 22 = 594$
    \item State/residual evaluation: $2 \cdot 3^4 3^2 = 1458$
    \item Pointwise ``physics'': $\approx 30 \cdot 3^3 = 810$
    \end{itemize}
  \item Arithmetic intensity: $5049 FMA/1184 B = 4.26 FMA/B$
  \end{itemize}
\end{frame}

\begin{frame}{Absorb metric terms into coefficients}
  \begin{equation*}
    -\nabla_\xi\cdot \Big[ (\nabla_x\xi)^T(\eta + \eta' \nabla u \otimes \nabla u)(\nabla_x\xi) \nabla_\xi w \Big] = f
  \end{equation*}
  \begin{itemize}
  \item Asymptotic data motion
    \begin{itemize}
    \item State+Residual: 16 \texttt{double}s per element (54 without reuse)
    \item Coefficients: $6\cdot 27$ \texttt{double}s per element (no reuse)
    \end{itemize}
  \item Floating point (counting FMA)
    \begin{itemize}
    \item State/residual evaluation: $2 \cdot 3^4 3^2 = 1458$
    \item Pointwise ``physics'': $9 \cdot 3^3 = 243$
    \end{itemize}
  \item Arithmetic intensity: $1701 FMA/1424 B = 1.2 FMA/B$
  \end{itemize}
\end{frame}


\begin{frame}{Proposed changes to HPCG}
  \begin{itemize}
  \item F-cycle multigrid
    \begin{itemize}
    \item Multiscale communication, actually solves a problem (optimally)
    \item Mathematically clean, less discretionary decisions needed
    \end{itemize}
  \item Chebyshev(2) smoother
    \begin{itemize}
    \item Avoids fixing a scale for multiplicativeness in Red-Black GS
    \item Enables pipelining/fusion, stresses cache
    \end{itemize}
  \item Unassembled matrix representation
    \begin{itemize}
    \item Increase arithmetic intensity (1/6 to $\approx 10$) and integer issue
    \item FE vs FD/FV?  Scalar or vector problem?
    \item $Q_2$ FE and/or elasticity puts more pressure on cache and registers
    \end{itemize}
  \item Quantify dynamic range
    \begin{itemize}
    \item Ratio between filling memory and $50\%$ efficiency
    \item Helps understand strong scaling/turn-around time
    \end{itemize}
  \end{itemize}
\end{frame}

\end{document}