20121114-OakRidge.tex

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\title{Composable multilevel solvers in PETSc}
\author{{\bf Jed Brown}\inst{1},\\
Mark Adams\inst{2}, Peter Brune\inst{1}, Matt Knepley\inst{3}, Dave May\inst{4}, \\ Barry Smith\inst{1}}


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% - Keep it simple, no one is interested in your street address.
\institute
{
  \inst{1}{Mathematics and Computer Science Division, Argonne National Laboratory} \\
  \inst{2}{Columbia University} \\
  \inst{3}{Computation Institute, University of Chicago} \\
  \inst{4}{ETH Z\"urich}
}

\date{Oak Ridge 2012-11-14}

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\subject{Talks}


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\input{slides/MultiphysicsExamples.tex}
\input{slides/MonolithicOrSplit.tex}
\input{slides/PETSc/Coupling.tex}
\input{slides/FieldSplit.tex}
\input{slides/PETSc/LocalSpaces.tex}
\input{slides/SNES/MultiphysicsAssemblyJacobians.tex}
\input{slides/PETSc/MatGetLocalSubMatrix.tex}
\input{slides/Stokes/OptionsTour.tex}
\input{slides/Stokes/CoupledMGOptions.tex}
\input{slides/PETSc/DistributiveSmoothing.tex}
\input{slides/PETSc/RediscretizedMultigrid.tex}
\input{slides/MG/ThreeSchools.tex}
\input{slides/MG/CoarseBasisFunctionsOverview.tex}
\input{slides/MG/BDDCCoarseBasisFunctions.tex}
\input{slides/MG/CoarseGridWhyILikeSubdomainProblems.tex}
\input{slides/MG/SmoothingNonlinearProblems.tex}

\input{slides/SNES/MonolithicFASMultistage.tex}
\input{slides/SNES/NonlinearSolversList.tex}
\input{slides/SNES/QuasiNewtonRevisited.tex}

\input{slides/DrunkenSeaman.tex}
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\input{slides/VHTEquationsSimple.tex}
\input{slides/VHTSolvers.tex}
%\input{slides/VHTSolverOptions.tex}

\input{slides/AllenCahnOptions.tex}

\input{slides/PETSc/TSARKIMEX.tex}

% \section{Implementation efficiency}
\input{slides/HardwareRoadmap.tex}
\begin{frame}{How to program this beast?}
  \begin{itemize}
  \item Decouple physics from discretization
    \begin{itemize*}
    \item Expose small, embarrassingly parallel operations to user
    \item Library schedules user threads for reuse between kernels
    \item User provides physics in kernels run at each quadrature point
    \item Continuous weak form: find $u \in \VV_D$
      \[ v^T F(u) \sim \int_\Omega v \cdot {\color{green!70!black} f_0(u,\nabla u)}
      + \nabla v \tcolon {\color{green!70!black} f_1(u,\nabla u)} = 0, \qquad \forall v \in \VV_0 \]
    \item Similar form at faces, but may involve Riemann solve
    \end{itemize*}
  \item Library manages reductions
    \begin{itemize*}
    \item Interpolation and differentiation on elements
    \item Interaction with neighbors (limiting, edge stabilization)
    \item Exploit tensor product structure to keep working set small
    \item Assembly into solution/residual vector (sum over elements)
    \end{itemize*}
  \end{itemize}
\end{frame}
\input{slides/Dohp/TensorFEM.tex}
\input{slides/Dohp/RepresentationOfJacobians.tex}
\input{slides/Dohp/TensorVsAssembly.tex}
\input{slides/HardwareArithmeticIntensity.tex}

\begin{frame}{Outlook on Solver Composition}
  \begin{itemize}
  \item Unintrusive composition of multigrid and block preconditioning
  \item Build preconditioners from literature \\
    \emph{on the command line}
  \item User code does not depend on matrix format, preconditioning method, nonlinear solution method, time integration method (implicit or IMEX), or size of coupled system (except for driver).
  \end{itemize}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      \begin{block}{In development}
        \begin{itemize}
          {\small
          \item Distributive relaxation, Vanka smoothers, coarsening for ``dual'' variables
          \item Improving operator-dependent semi-geometric multigrid
          \item More automatic spectral analysis and smoother optimization
          \item Automated support for mixing analysis into MG levels
          }
        \end{itemize}
      \end{block}
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{block}{Performance challenges}
        \begin{itemize}
          {\small
          \item Expressing NUMA distribution between libraries
          \item Cache sharing/granularity
          \item Small dense linear algebra for MG smoothers
          \item Flexibility in GPU kernels
          }
        \end{itemize}
      \end{block}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}\LARGE
  \begin{itemize}
  \item Maximize science per Watt
  \item Huge scope remains at problem formulation
  \item Raise level of abstraction at which a problem is formally specified
  \item Algorithmic optimality is crucial
  \end{itemize}
\end{frame}
\end{document}