20121219-FCP.tex

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\title{Multilevel solvers with adaptive coarse space construction for lithosphere dynamics}
\author{{\bf Jed Brown}\inst{1}, Mark Adams\inst{2}, Matt Knepley\inst{3}, Barry Smith\inst{1}}

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\institute
{
  \inst{1}{Mathematics and Computer Science Division, Argonne National Laboratory} \\
  \inst{2}{Columbia University} \\
  \inst{3}{University of Chicago} \\
}

\date{Frontiers in Computational Physics, 2012-12-19}

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\subject{Talks}


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\begin{frame}
  \titlepage
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\input{slides/MonolithicOrSplit.tex}
\begin{frame}{Status quo for implicit solves in lithosphere dynamics}
  \begin{itemize}
  \item global linearization using Newton or Picard
  \item assembly of a sparse matrix
  \item ``block'' factorization preconditioner with approximate Schur complement
  \item algebraic or geometric multigrid on positive-definite systems
  \end{itemize}
  \begin{block}{Why is this bad?}
    \vspace{-1em}
    \begin{itemize}
    \item nonlinearities (e.g., plastic yield) are mostly local
      \begin{itemize}
      \item feed back through nearly linear large scales
      \item frequent visits to fine-scales even in nearly-linear regions
      \item no way to locally update coarse grid operator
      \item Newton linearization introduces anisotropy
      \end{itemize}
    \item assembled sparse matrices are terrible for performance on modern hardware
      \begin{itemize}
      \item memory bandwidth is very expensive compared to flops
      \item fine-scale assembly costs a lot of memory
      \item assembled matrices are good for algorithmic experimentation
      \end{itemize}
    \item block preconditioners require more parallel communication
    \end{itemize}
  \end{block}
\end{frame}

\input{slides/HardwareArithmeticIntensity.tex}
\input{slides/Dohp/TensorVsAssembly.tex}

\input{slides/MG/TauFAS.tex}
\input{slides/MG/FMGRecovery.tex}
\input{slides/MG/FourSchools.tex}

% Coarsening
\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/SubdomainInterfaces.tex}
\input{slides/MG/CoarseGridComplicationForSaddlePoint.tex}

% 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}

\begin{frame}{Outlook}
  \begin{itemize}
  \item smoothing with point-block Jacobi Chebyshev and scaled diagonal for pressure
  \item needs only (subdomain ``Neumann'') nonlinear function evaluations and assembly of point-block diagonal matrices
  \item convergence rates similar to smoothed aggregation, but without fine-grid assembly
  \item allows local updates of coarse operator, but currently slower due to naive implementation
  \item Development in progress within PETSc
    \begin{itemize}
    \item parallel implementation of dual-support problems without duplicating lots of work
    \item homogenization-based nonlinear coarsening
    \item true $\tau$ formulation with adaptive fine-grid visits and partial coarse operator updates
    \item microstructure-compatible pressure interpolation
    \item ``spectrally-correct'' nonlinear saddle-point smoothers
    \item locally-computable spectral estimates for guaranteed-stable additive smoothers
    \end{itemize}
  \end{itemize}
\end{frame}

\end{document}