20120730-CIG.tex

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\title{Multilevel Stokes flow solvers}
\subtitle{Adapting to heterogeneity and rheology}
\author{{\bf Jed Brown}}

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{
  {Mathematics and Computer Science Division, Argonne National Laboratory} \\
}

\date{CIG Mantle/Lithosphere 2012-07-30}

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\subject{Talks}


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\begin{frame}
  \titlepage
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\begin{frame}{Intent of this talk}
  \begin{itemize}
  \item observation: solver scalability is the bottleneck at scale
  \item ``black box'' solvers are not sustainable
    \begin{itemize}
    \item optimal solvers must accurately handle all scales
    \item optimality is crucial for large-scale problems
    \item hardware puts up a spirited fight to abstraction
    \end{itemize}
  \item introduce multilevel solver concepts
  \item outline ingredients that discretizations can provide to solvers
  \item discuss algorithmic trade-offs
  \item current state of solver software and what we are working on
  \end{itemize}
\end{frame}

\begin{frame}{Outline}
  \tableofcontents
\end{frame}

\section{Introduction}
\begin{frame}
  \includegraphics[width=1.05\textwidth]{figures/KeyesAllAboutAlgorithms}
\end{frame}
\begin{frame}{Challenges for elliptic solvers}
  \begin{itemize}
  \item multiscale material coefficients
    \begin{itemize}
    \item long, thin high viscosity: transmit stresses long distances
    \item ``jelly sandwich'': release long-range stresses locally
    \end{itemize}
  \item nonlinearity
    \begin{itemize}
    \item plasticity: creates ``jelly sandwich''
    \item Newton linearization produces local anisotropy
    \item heating: localization
    \item coupling to other physical processes
    \end{itemize}
  \item multilevel methods
    \begin{itemize}
    \item need accurate coarse grids
    \item need effective smoothers
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Multigrid separates scales, feedback between scales}
  \includegraphics[width=\textwidth]{figures/YangAMGSummary} \\
\end{frame}

\input{slides/MonolithicOrSplit.tex}

\section{Multiscale Toolbox}
\input{slides/MG/ThreeSchools.tex}

\subsection{Coarse grids}
\input{slides/MG/CoarseGridComputableConvergenceMeasures.tex}
\input{slides/MG/CompatibleRelaxation.tex}
\input{slides/MG/CoarseBasisFunctionsOverview.tex}
\input{slides/MG/BDDCCoarseBasisFunctions.tex}
\input{slides/MG/CoarseGridWhyILikeSubdomainProblems.tex}
\input{slides/MG/CoarseGridComplicationForSaddlePoint.tex}

\subsection{Smoothing}
\input{slides/MG/SmoothingNonlinearProblems.tex}
\input{slides/MG/SmoothingSaddlePoint.tex}
\input{slides/MG/VankaBlockSmoothers.tex}
\input{slides/PETSc/DistributiveSmoothing.tex}
\input{slides/Stokes/CoupledMGOptions.tex}

\section{Software and performance}
\subsection{Coupling software}
\input{slides/PETSc/Coupling.tex}
\input{slides/FieldSplit.tex}
\input{slides/PETSc/LocalSpaces.tex}
\input{slides/SNES/MultiphysicsAssemblyJacobians.tex}
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\subsection{Performance considerations}
\input{slides/Dohp/TensorVsAssembly.tex}
\begin{frame}{Coarse levels may not be cheaper than fine levels}
  \begin{center}
    \includegraphics[width=0.6\textwidth]{figures/YangAMGLevelCost} \\
    {\scriptsize [Gahvari, Schulz, Yang, Jordan, Gropp 2011]}
  \end{center}
  \begin{itemize}
  \item latency for longer-range communication outweighs smaller data
  \item very aggressive coarsening important to limit number of levels
  \item alternatives: additive multigrid, redundant coarse grids
  \end{itemize}
\end{frame}

\begin{frame}{Multilevel Solvers are a \emph{Way of Life}}
  \begin{itemize}
  \item ingredients that discretizations can provide
    \begin{itemize}
    \item identify ``fields''
    \item topological coarsening, possibly for fields
    \item near-null space information
    \item ``natural'' subdomains
    \item subdomain integration, face integration
    \item element or subdomain assembly/matrix-free smoothing
    \end{itemize}
  \item solver composition
    \begin{itemize}
    \item most splitting methods accessible from command line
    \item energy optimization for tentative coarse basis functions
    \item algebraic form of distributive relaxation
    \item generic assembly for large systems and components
    \item working on flexibile ``library-assisted'' nonlinear multigrid
    \item adding support for interactive eigenanalysis
    \end{itemize}
  \end{itemize}
\end{frame}

\end{document}