20130904-BOUTPETSc.tex

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\title{PETSc and BOUT++}
\author{{\bf Jed Brown} \\
Peter Brune, Emil Constantinescu, \\
Debojyoti Ghosh, Lois Curfman McInnes \\
\texttt{\{jedbrown,brune,emconsta,ghosh,curfman\}@mcs.anl.gov}
}

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

\date{BOUT++ Workshop, 2013-09-04}

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\subject{Talks}


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\begin{document}
\lstset{language=C}
\normalem

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Portable {\bf Extensible} Toolkit for Scientific computing}
\begin{block}{Philosophy: Everything has a plugin architecture}
\begin{itemize}
  \item Vectors, Matrices, Coloring/ordering/partitioning algorithms
  \item Preconditioners, Krylov accelerators
  \item Nonlinear solvers, Time integrators
  \item Spatial discretizations/topology$^*$
\end{itemize}
\end{block}
\begin{example}
	Vendor supplies matrix format and associated preconditioner, distributes
	compiled shared library.  Application user loads plugin at runtime, no source
	code in sight.
\end{example}
\end{frame}

\begin{frame}{Portable Extensible {\bf Toolkit} for Scientific computing}
Algorithms, (parallel) debugging aids, low-overhead profiling
\begin{block}{Composability}
Try new algorithms by choosing from product space and composing
existing algorithms (multilevel, domain decomposition, splitting).
\end{block}
\begin{block}{Experimentation}
\begin{itemize}
  \item It is not possible to pick the solver \emph{a priori}. \\
  What will deliver best/competitive performance for a given physics, discretization, architecture, and problem size?
  \item PETSc's response: expose an algebra of composition so new solvers can be created at runtime.
  \item Important to keep solvers decoupled from physics and discretization because we also experiment with those. 
\end{itemize}
\end{block}
\end{frame}

\section{Time Integration}
\begin{frame}{Trade-offs in time integration}
  \begin{itemize}
  \item Properties
    \begin{itemize}
    \item Nonlinear stability (e.g., positivity preservation)
    \item Stability along imaginary axis
    \item $L$-stability (damping at infinity)
    \item Implicitness and reuse
    \end{itemize}
  \item What is expensive?
    \begin{itemize}
    \item Function evaluation
    \item Operator assembly/preconditioner setup
      \begin{itemize}
      \item How much can be reused for how long?
      \end{itemize}
    \item Implicit solves
      \begin{itemize}
      \item Can we find better solver algorithm?
      \item More effort in setup?
      \end{itemize}
    \end{itemize}
  \item What is ``convergence''?
    \begin{itemize}
    \item Wave propagation: implicitness useless for convergence \emph{in a norm}
    \item Non-norm functionals could be robust
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Reusing implicit solver setup}
  \begin{itemize}
  \item Linearization
  \item MG interpolants
  \item Lagged preconditioner
  \item Modified Newton
  \item Quasi-Newton
  \item IMEX with linear implicit part
  \item Rosenbrock/W
  \end{itemize}
\end{frame}

\input{slides/PETSc/TSARKIMEX.tex}
% \input{slides/PETSc/TSUsage.tex}
%\input{slides/TS/}

\begin{frame}[fragile]{Time integration method design}
  \begin{figure}
    \centering
    \includegraphics[width=.8\textwidth]{figures/TS/EmilMethodDesignFeatures.png}
  \end{figure}
  \begin{itemize}
  \item Select order, number of stages, required properties
  \item Optimize properties like SSP coefficient, accuracy, or linear stability
  \item \cverb|TSARKIMEXRegister("my-method", ...coefficients...)|
  \item \cverb|-ts_type arkimex -ts_arkimex_type my-method|
  \end{itemize}
\end{frame}

\begin{frame}{Example: Additive Runge-Kutta design}
  \begin{itemize}
  \item 3-stage, second order, $L$-stable implicit part
  \item one-parameter family of solutions
  \end{itemize}
  \begin{description}
  \item[ARK2c] Maximize SSP coefficient
  \item[ARK2E] Minimize leading error coefficient
  \end{description}
  \begin{figure}
    \centering
    \includegraphics[width=0.55\textwidth]{figures/TS/ssp_ark_poster.png}
    \includegraphics[width=0.49\textwidth]{figures/TS/Stability_ARK2E_ARK2C.pdf}
  \end{figure}
\end{frame}

\input{slides/PETSc/TSMethods.tex}

\begin{frame}{Adaptive controllers}
  \begin{itemize}
  \item ``Stiff'' waves are not stiff if one wants to converge \emph{in a norm}
  \item PETSc integrators provide embedded methods to estimate errors
  \item Automatic controllers optimize local truncation error and nonlinear solve cost
  \item User can register custom controllers
  \item Use a priori knowledge of the physics, robust functionals
  \item Choose from list of methods, choose next step size
  \end{itemize}
\end{frame}

\section{Nonlinear solvers}
\begin{frame}{Which nonlinear solver?}
  \begin{itemize}
  \item Global linearization (NewtonLS, NewtonTR)
    \begin{itemize}
    \item Preconditioning libraries for assembled matrices
    \item Low arithmetic intensity
    \end{itemize}
  \item Quasi-Newton
    \begin{itemize}
    \item Build low-rank updates to Jacobian inverse
    \item Brown and Brune, ``Low-rank quasi-Newton updates for robust Jacobian lagging in Newton-type methods'', ANS MC13.
    \end{itemize}
  \item Nonlinear multigrid and domain decomposition
    \begin{itemize}
    \item ASPIN (left-preconditioned nonlinear Schwarz), also right-preconditioned
    \item Full Approximation Scheme with linear or nonlinear smoothers
    \item More intrusive, but freakishly efficient for difficult problems
    \end{itemize}
  \item Nonlinear GMRES, Anderson mixing, nonlinear CG
    \begin{itemize}
    \item Accelerator for nonlinear preconditioning
    \item Good alternative to matrix-free finite differencing
    \item More robust line search possible: operates in reduced basis
    \end{itemize}
  \end{itemize}
\end{frame}
\begin{frame}
  \includegraphics[width=\textwidth]{figures/BruneNGMRESFAS2.png}
\end{frame}

\input{slides/MonolithicOrSplit.tex}
\input{slides/PETSc/LocalSpaces.tex}

\section{Comments on performance}
\input{slides/JFNKBottlenecks.tex}
\input{slides/ScalabilityWarning.tex}
\input{slides/Dohp/TensorVsAssembly.tex}
\input{slides/HardwareArithmeticIntensity.tex}

\end{document}