20110218-Hutter70.tex

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\title{Computational methods for several models of\\ice stream flow}

\author{Jed Brown}


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% - Keep it simple, no one is interested in your street address.
\institute[ETH Z\"urich]
{
  Laboratory of Hydrology, Hydraulics, and Glaciology \\
  ETH Zürich
}

\date[2011-02-18]{2011-02-18}

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\subject{Talks}


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\titlepage
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%   \frametitle{Outline}
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%   % You might wish to add the option [pausesections]
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\section{Practicalities}
\input{slides/GroundingLine/Steepness.tex}
\begin{frame}
  \includegraphics[width=1.1\textwidth]{figures/KeyesAllAboutAlgorithms}
\end{frame}

\section[Hydrostatic]{A robust multigrid solver for the hydrostatic equations}
\input{slides/THI/Equations.tex}
%\input{slides/THI/X5kmClip.tex}
\begin{frame}
  \includegraphics[width=\textwidth]{figures/THI/x-shear}
\end{frame}
\begin{frame}
  \includegraphics[width=\textwidth]{figures/THI/x-80km-m16p2l6-ew} \\
  Grid-sequenced Newton-Krylov solution of test $X$.  The solid lines denote nonlinear iterations, and the dotted lines with $\times$ denote linear residuals.
\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}
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\begin{frame}
  \includegraphics[width=\textwidth]{figures/KeyesStrongWeak}
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\input{slides/THI/Shaheen.tex}
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  \begin{center}
    \alert{\Huge One high-accuracy solve \\

      costs 30 times as much

      \medskip

      as a residual evaluation}
  \end{center}
  \begin{center}
    about 15 to reach truncation error

    \bigskip

    \uncover<2>{\alert{\Large 1000 times faster than existing methods}}
    
    \bigskip

    \uncover<2>{(Brown, Smith, Ahmadia 2011; submitted to JGR)}
  \end{center}
\end{frame}

\section{Other models}
\input{slides/GroundingLine/SSA.tex}
\begin{frame}{Non-Newtonian Stokes system: velocity $\bm u$, pressure $p$}
\begin{columns}
\begin{column}{0.5\textwidth}
  \alert{\begin{align*}
    -\nabla \cdot(\eta D\uu) + \nabla p - \ff &= 0 \\
    \nabla \cdot \uu &= 0
  \end{align*}}
\end{column}
\begin{column}{0.5\textwidth}
    \begin{align*}
      D\uu &= \tfrac 1 2 \left(\nabla \uu + (\nabla \uu)^T \right) \\
      \gamma(D\uu) &= \tfrac 1 2 D\uu \tcolon D\uu \\
      \eta(\gamma) &= B(\theta,\dotsc)\big(\gamma_0 + \gamma \big)^{\frac{\mathfrak{p}-2}{2}} \\
      \mathfrak{p} &= 1 + \tfrac{1}{\mathfrak{n}} \approx \tfrac 4 3 \\
      T &= \bm 1 - \bm n \otimes \bm n \\
    \end{align*}
\end{column}
\end{columns}
\vspace{-1.5em}
    with boundary conditions
    \begin{align*}
      (\eta D\bm u - p\bm 1)\cdot\bm n =
      \begin{cases}\bm 0 & \text{free surface} \\
        -\rho_w z \bm n & \text{ice-ocean interface}\end{cases} \\
      \bm u = \bm 0\qquad\qquad \text{frozen bed}, \theta < \theta_0 \\
      \left. \begin{aligned}
          \bm u \cdot \bm n &= \bm g_{\text{melt}}(T\uu,\dotsc) \\
          T (\eta D\bm u - p\bm 1)\cdot\bm n &= \bm g_{\text{slip}}(T \bm u,\dotsc) \end{aligned}\right\}
      \text{nonlinear slip}, \theta \ge \theta_0 \\
    \end{align*}
    \vspace{-3em}
    \[ \bm g_{\text{slip}}(T\uu) = \beta_{\mathfrak{m}}(\dotsc) \lvert T\bm u \rvert^{\mathfrak{m}-1} T \bm u \]
    Navier $\mathfrak{m}=1$, \quad Weertman $\mathfrak{m}\approx \frac 1 3$, \quad Coulomb $\mathfrak{m}=0$.
\end{frame}

\begin{frame}{Stokes challenges}
  \begin{block}{Mass conservation is critical}
    \begin{itemize}
    \item Staggered grid finite difference (hard to deal with geometry)
    \item Stabilized methods (conservation artifacts when non-smooth)
    \item Inf-sup stable mixed finite element method
      \begin{itemize}
      \item Use discontinuous pressure to enforce local mass conservation
      \item Inf-sup constant decays like $\sqrt{\epsilon}$ for $Q_k-P_{k-1}^{\text{disc}}$
      \item Sub-optimal order of accuracy for $Q_k-Q_{k-2}^{\text{disc}}$
      \end{itemize}
    \end{itemize}      
  \end{block}
  \begin{block}{Solving saddle-point problems}
    \begin{itemize}
    \item Not uniformly elliptic: solvers are much less robust
    \item Standard preconditioners do not work
    \item Coupled multigrid with Vanka smoothers offer best performance,\\
      \quad not robust for stretched grids or anisotropic viscosity
    \item Block preconditioners require approximate commutators, \\
      \quad fragile for strong anisotropy and non-smooth viscosity
    \end{itemize}
  \end{block}
\end{frame}

\begin{frame}{Outlook}
  \begin{block}{}
    \begin{itemize}
    \item We have textbook multigrid efficiency for hydrostatic equations
    \item All other models are currently slower at high resolution \\ \quad because there are no scalable implementations
    \item Daniel Goldberg's model could be as much as 4 times faster, \\ \quad probably closer to 2 times
    \item Bueler \& Brown's model (or SSA) could be up to 10 times faster
    \item Technical challenges for Stokes
    \item Bathymetry is rough enough that we should solve Stokes
    \item Singularities: reentrant corners, transition from frozen \\ \quad 
      to slip bounadry conditions, grounded margins, grounding lines
    \item Implicit time integration
    \end{itemize}    
  \end{block}
  % \begin{block}{Tools}
  %   \begin{itemize}
  %   \item PETSc\ \url{http://mcs.anl.gov/petsc}
  %     \begin{itemize}\item ML, Hypre, MUMPS
  %     \end{itemize}
  %   \item ITAPS \url{http://itaps.org}
  %     \begin{itemize}\item MOAB, CGM, Lasso
  %     \end{itemize}
  %   \end{itemize}
  % \end{block}
\end{frame}

\begin{frame}
  \includegraphics[width=0.5\textwidth]{figures/THI/c-steady-crop}
  \includegraphics[width=0.5\textwidth]{figures/THI/erosion300k}
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\end{document}