20110906-ThesisDefense.tex

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\title{Computational methods for ice flow simulation \\
with applications to Jakobshavn Isbr{\ae}}
\subtitle{Diss. ETH No. 19948}

\author{Jed Brown}


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\institute
{VAW, ETH Z\"urich}

\date{2011-09-06}

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\subject{Talks}


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

\section{Perspective}
\begin{frame}{Grand Challenge Problems}
  \begin{block}{Global Prosperity Optimization}
    Maximize the prosperity functional by adjusting energy policy, urban and groundwater planning, geoengineering projects, etc.
  \end{block}
  \begin{block}{Ice Sheet Stability Questions (short-term connection via Ocean)}
    \begin{itemize}
    \item Where is the stability manifold at Pine Island, Thwaites, and other outlet glaciers relative to current trajectories?
    \item In the event of entering an unstable regime, how rapid would the resulting sea level rise be?
    \item What possible anthropogenic sources have the most influence on stability and related sea level rise?
    \item What are the errors associated with these computations?
    \item Which field work and remote sensing studies would provide the most cost-effect way to reduce uncertainty?
    \end{itemize}
  \end{block}
\end{frame}

\begin{frame}{Where does computation fit in?}
  \begin{itemize}
  \item Scientific applications: use computation to investigate scientific questions
    \begin{itemize}
    \item non-model problems: messy, coupled, highly nonlinear
    \item do not want to spend career focusing on methods
    \item slow to respond to developments in mathematics
    \item difficult to evaluate performance/robustness without implementing
    \end{itemize}
  \item Applied mathematics: create and analyze new methods
    \begin{itemize}
    \item identify key ingredients needed for robustness and accuracy
    \item well-posed formulations and (quadratically) convergent methods to provide quantitative answers to scientific questions (``why'', ``how'')
    \item demonstrate on simplified models (geometry, boundary conditions, materials)
    \item efficient implementation not important: focus on asymptotics
    \end{itemize}
  \item Scientific software: make methods available and performant
    \begin{itemize}
    \item modify methods for robustness on wider variety of problems
    \item make interfaces usable by non-experts
    \item decouple from problem specification
    \item implementation efficiency is important, follow new hardware
    \item solved problems should not need to be revisited
    \item reusable components: experiment in minutes instead of months
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Ingredients for ice sheets}
  \begin{itemize}
  \item Implicit solution algorithms and composability
    \begin{itemize}
    \item \alert<2>{Stokes and hydrostatic solvers}
    \item \alert<2>{Coupling strategies: software maintainability vs. algorithms}
    \item \alert<2>{Exposing the right capability for stability, sensitivity, optimization/inversion, uncertainty quantification}
    \end{itemize}
  \item Discretizations well-suited to the problem
    \begin{itemize}
    \item \alert<2>{technical issues: aspect ratio, inf-sup stability}
    \item \alert<2>{surface-volume coupling, identifying stiffness,} topology change
    \item transport-dominated regions ($\Peclet_h > 10^4$) \uncover<2>{\alert{[badly]}}
    \item contact problems of many flavors
    \item \alert<2>{efficiency on modern hardware, memory bandwidth bottleneck}
    \end{itemize}
  \item Verification and Validation
    \begin{itemize}
    \item \alert<2>{maintaining capability with new material models and additional coupled processes}
    \item sparse observations: geometry, initial/boundary conditions, materials; cannot validate forward models, only inverse models
    \end{itemize}
  \item Conservative model for polythermal ice
    \begin{itemize}
    \item \alert<2>{discretization, unstructured meshing, implementation}
    \item \alert<2>{modularity and implicit solvers}
    \end{itemize}
  \end{itemize}
\end{frame}

\section[Hydrostatic]{A robust multigrid solver for the hydrostatic equations}
\input{slides/THI/Equations.tex}
% \begin{frame}
%   \includegraphics[width=\textwidth]{figures/THI/x-shear}
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\begin{frame}
  \includegraphics[width=\textwidth]{figures/THI/x-80km-m16p2l6-ew} \\
  Grid-sequenced Newton-Krylov solution of test $X$. \\
  \begin{itemize}
  \item solid lines denote nonlinear iterations
  \item dotted lines with $\times$ denote linear residuals.
  \end{itemize}
\end{frame}
\input{slides/THI/Y5kmClip.tex}
\begin{frame}
  \begin{figure}
    \includegraphics[width=\textwidth]{figures/THI/y-10km-m10p6l5-ew}
    \centering\caption{Grid sequenced Newton-Krylov convergence for test $Y$.
    The ``cliff'' has \SI{58}{\degree} angle in the red line ($12\times 125$ meter elements), \SI{73}{\degree} for the cyan line ($6\times 62$ meter elements).}\label{fig:testy}
  \end{figure}
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  \begin{figure}
    \includegraphics[width=\textwidth]{figures/THI/linear4}
    \centering\caption{Average number of Krylov iterations per nonlinear iteration.  Each nonlinear system was solved to a relative tolerance of $10^{-2}$.}\label{fig:linear}
  \end{figure}
\end{frame}
\begin{frame}
  \includegraphics[width=\textwidth]{figures/KeyesStrongWeak} \\
  \uncover<2>{\large \alert{The easiest way to make software scalable is to make it sequentially inefficient. -- Gropp 1999}}
\end{frame}
\input{slides/THI/Shaheen.tex}
\begin{frame}
  \begin{center}
    \alert{\huge One high-accuracy solve \\[0.2em]
      costs 30 times as much \\[0.5em]
      as a residual evaluation}
  \end{center}
  \begin{center}
    about 15 to reach truncation error

    \bigskip

    \uncover<2>{\alert{\Large 1000 times faster than methods currently in use} \\
      e.g. Lemieux, Price, Evans, Knoll, Salinger, Holland, Payne 2011 \\ (J. Computational Physics)}
    
    \bigskip

    {(Brown, Smith, Ahmadia 2011; submitted to SIAM J. Scientific Computing)}
  \end{center}
\end{frame}

\section{High throughput for high order finite elements}
\begin{frame}{What hardware is coming down the pike? (2018)}
  \begin{itemize}
  \item DOE Exascale program forecasts
    \begin{itemize}
    \item $10^6$ nodes, $10^3$ threads/node, $10$ GB/node, $10$ MB/thread
    \item 1--10 TF/node, \alert<2->{``balanced'' arithmetic intensity: 10--40 flops/byte}
    \item deep memory hierarchy, locality crucial, ``free'' flops
    \end{itemize}
  \item Arithmetic intensity: flops/byte
  \begin{itemize}
  \item sparse matrix kernels: 0.17--0.25
  \item matrix-free kernels: $\ge 5$
  \end{itemize}
\item Mitigation
  \begin{itemize}
  \item hierarchical methods with fewer/no global reductions
  \item reduce memory/bandwidth usage in exchange for more flops
    \begin{itemize}
    \item \alert<3>{higher order methods, recompute coefficients instead of storing}
    \end{itemize}
  \item ``communication avoiding'': apply operators multiple times while coefficients are in cache
    \begin{itemize}
    \item mathematics disagrees, sacrifice algorithmic scalability
    \end{itemize}
  \end{itemize}
\end{itemize}
\end{frame}

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

% \input{slides/VerificationAndValidation.tex}
% \input{slides/Dohp/ElasticityCodeVerification.tex}

\section{Algebraic coupling algorithms and interfaces}
\input{slides/MonolithicOrSplit.tex}
\input{slides/PETSc/Coupling.tex}
\input{slides/PETSc/LocalSpaces.tex}
\input{slides/PETSc/MatGetLocalSubMatrix.tex}

\section{Conservative polythermal ice flow}
\input{slides/VHTEquations.tex}
\input{slides/VHTNondimensional.tex}
\input{slides/VHTSolvers.tex}
%\input{slides/VHTSmoother.tex}
\input{slides/VHTJako.tex}

\begin{frame}{Summary}
  \begin{itemize}
  \item textbook multigrid efficiency for hydrostatic ice
  \item dual-order $h$ and $p$-version finite element library
    \begin{itemize}
    \item alleviate memory bandwidth bottleneck by factor of 50
    \item run-time choice of order, compatibility for mixed spaces
    \item $>70\%$ of FPU peak for core library kernels
    \item geometric model is available to physics/solvers
    \item automation of code verification
    \end{itemize}
  \item bridging the Great Solver Schism
    \begin{itemize}
    \item commuting multilevel methods with field-splitting
    \item data structure independence for assembly
    \item decouple software packaging from available solution algorithms
    \end{itemize}
  \item technical requirements for robust discretizations of ice flow
  \item 3D implicit polythermal ice flow
    \begin{itemize}
    \item new conservative formulation
    \item quadratically convergent steady-state solver
    \item run in Jakobshavn geometry
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Outlook}
  \begin{itemize}
  \item Need better transport algorithm: discontinuous Galerkin
  \item Many flavors of contact: large-deformation, self, frictional, thermal
  \item Unstructured geometric multigrid/full-space DD
  \item Non-interactive remeshing after topology change/large deformation
  \item Move from symbolics to AD for manufactured solutions/codegen
  \item IMEX methods now implemented $G(t,x,\dot x) = F(t,x)$
  \item General/nonsymmetric pivoting for full-space optimization
  \item Uncertainty quantification for semi-smooth systems
  \item Multi-parameter continuation for stability analysis
  \end{itemize}
\end{frame}

\end{document}