20140707-SIAMAnnual.tex

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\title{Towards $\tau$ adaptivity \\ for lithosphere dynamics}
\subtitle{Non-smooth processes in heterogeneous media}
\author{{\bf Jed Brown} \texttt{jedbrown@mcs.anl.gov} (ANL and CU Boulder) \\
  \quad Mark Adams (LBL), Matt Knepley (UChicago), Dave May (ETH)
}

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

\date{SIAM Annual Meeting, Chicago, 2014-07-07 \\[1em]
This talk: \url{http://59A2.org/files/20140707-SIAMAnnual.pdf}}

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\subject{Talks}


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\lstset{language=C}
\normalem

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

\begin{frame}{Plan: ruthlessly eliminate communication}
  \begin{block}{Why?}
    \begin{itemize}
    \item Local recovery despite global coupling
    \item Tolerance for high-frequency load imbalance
      \begin{itemize}
      \item From irregular computation or hardware error correction
      \end{itemize}
    \item More scope for dynamic load balance
    \end{itemize}
  \end{block}
  \begin{block}{Requirements}
    \begin{itemize}
    \item Must retain optimal convergence with good constants
    \item Flexible, robust, and debuggable
    \end{itemize}
  \end{block}
\end{frame}

\section{$\tau$-adaptivity and multigrid compression}
\begin{frame}[fragile]{Multigrid Preliminaries}
  \begin{figure}
    \centering
    \begin{tikzpicture}
      [>=stealth,
      every node/.style={inner sep=2pt},
      restrict/.style={thick},
      prolong/.style={thick},
      mglevel/.style={rounded rectangle,draw=blue!50!black,fill=blue!20,thick,minimum size=4mm},
      ]
      \begin{scope}\scriptsize
        \newcommand\mgdx{4.0em}
        \newcommand\mgdy{4.0em}
        \newcommand\mgl[1]{(pow(2,#1+1))}
        \newcommand\mgloc[4]{(#1 + #4*\mgdx*#3,#2 + \mgdy*#3)}

        \newcommand\mghx{0.9*\mgdx}
        \newcommand\mghy{0.9*\mgdy}

        \draw[shift=\mgloc{0*\mgdx}{0}{0}{0},
        xstep=\mghy/\mgl{3},
        ystep=\mghy/\mgl{3}]
        (-0.5*\mghy,-0.5*\mghy) grid (0.5*\mghy,0.5*\mghy);

        \draw[shift=\mgloc{1*\mgdx}{0}{0}{0},
        xstep=\mghy/\mgl{2},
        ystep=\mghy/\mgl{2}]
        (-0.5*\mghy,-0.5*\mghy) grid (0.5*\mghy,0.5*\mghy);

        \draw[shift=\mgloc{2*\mgdx}{0}{0}{0},
        xstep=\mghy/\mgl{1},
        ystep=\mghy/\mgl{1}]
        (-0.5*\mghy,-0.5*\mghy) grid (0.5*\mghy,0.5*\mghy);


        \draw[shift=\mgloc{3*\mgdx}{0}{0}{0},
        xstep=\mghy/\mgl{0},
        ystep=\mghy/\mgl{0}]
        (-0.5*\mghy,-0.5*\mghy) grid (0.5*\mghy,0.5*\mghy);
      \end{scope}
    \end{tikzpicture}
    \label{fig:levels}
  \end{figure}
  \textbf{Multigrid} is an $O(n)$ method for solving algebraic problems by defining a hierarchy of scale.
  A multigrid method is constructed from:
  \begin{enumerate}
  \item a series of discretizations
    \begin{itemize}
    \item coarser approximations of the original problem
    \item constructed algebraically or geometrically
    \end{itemize}
  \item intergrid transfer operators
    \begin{itemize}
    \item residual restriction $I_h^H$ (fine to coarse)
    \item state restriction $\hat I_h^H$ (fine to coarse)
    \item partial state interpolation $I_H^h$ (coarse to fine, `prolongation')
    \item state reconstruction $\mathbb{I}_H^h$ (coarse to fine)
    \end{itemize}
  \item Smoothers ($S$)
    \begin{itemize}
    \item correct the high frequency error components
    \item Richardson, Jacobi, Gauss-Seidel, etc.
    \item Gauss-Seidel-Newton or optimization methods
    \end{itemize}
  \end{enumerate}
\end{frame}
\input{slides/MG/TauFAS.tex}

\begin{frame}{Model problem: $\pfrak$-Laplacian with slip boundary conditions}
  \begin{itemize}
  \item 2-dimensional model problem for power-law fluid cross-section
    \begin{equation*}
      -\div \big(\abs{\nabla u}^{\pfrak-2} \nabla u \big) - f = 0, \qquad 1 \le \pfrak \le \infty
    \end{equation*}
    Singular or degenerate when $\nabla u = 0$
  \item Regularized variant
    \begin{gather*}
      -\div (\eta \nabla u) - f = 0 \\
      \eta(\gamma) = (\epsilon^2 + \gamma)^{\frac{\pfrak-2}{2}} \qquad \gamma(u) = \half \abs{\nabla u}^2
    \end{gather*}
  \item Friction boundary condition on one side of domain
    \begin{gather*}
      \nabla u \cdot \bm n + A(x) \abs{u}^{q-1} u = 0
    \end{gather*}
  \end{itemize}
\end{frame}

\begin{frame}{Model problem: $\pfrak$-Laplacian with slip boundary conditions}
  \begin{itemize}
  \item $\pfrak = 1.3$ and $q = 0.2$, checkerboard coefficients $\{10^{-2},1\}$
  \item Friction coefficient $A=0$ in center, 1 at corners
  \end{itemize}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      \only<1>{\includegraphics[width=\textwidth]{figures/MG/ex15-friction/visit0010.png}}
      \only<2>{\includegraphics[width=\textwidth]{figures/MG/ex15-friction/visit0011.png}}
      \only<3>{\includegraphics[width=\textwidth]{figures/MG/ex15-friction/visit0012.png}}
      \only<4>{\includegraphics[width=\textwidth]{figures/MG/ex15-friction/visit0013.png}}
      \only<5>{\includegraphics[width=\textwidth]{figures/MG/ex15-friction/visit0014.png}}
      \only<6>{\includegraphics[width=\textwidth]{figures/MG/ex15-friction/visit0015.png}}
    \end{column}
    \begin{column}{0.5\textwidth}
      \includegraphics[width=\textwidth]{figures/MG/newton-convergence.png}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}{$\tau$ corrections}
  \begin{figure}
  \centering
  \begin{subfigure}[b]{0.18\textwidth}
    \includegraphics[width=\textwidth]{figures/MG/ElasticityCompressTrim}
    %\caption{Initial solution.}\label{fig:elast-initial}
  \end{subfigure} ~
  \begin{subfigure}[b]{0.18\textwidth}
    \includegraphics[width=\textwidth]{figures/MG/ElasticityCompressShearTrim}
    %\caption{Increment.}\label{fig:elast-increment}
  \end{subfigure} ~
  \begin{subfigure}[b]{0.28\textwidth}
    \includegraphics[width=\textwidth]{figures/MG/ElasticityCompressErrorNoTauTrim}
    %\caption{Smoothed error without $\tau$.}\label{fig:elast-error-notau}
  \end{subfigure} ~
  \begin{subfigure}[b]{0.28\textwidth}
    \includegraphics[width=\textwidth]{figures/MG/ElasticityCompressErrorTauTrim}
    %\caption{Smoothed error with $\tau$.}\label{fig:elast-error-tau}
  \end{subfigure}
  \begin{itemize}
  \item Plane strain elasticity, $E=1000,\nu=0.4$ inclusions in $E=1,\nu=0.2$ material, coarsen by $3^2$.
  \item Solve initial problem everywhere and compute $\tau_h^H = A^H \hat I_h^H u^h - I_h^H A^h u^h$
  \item Change boundary conditions and solve FAS coarse problem
    \begin{equation*}
      N^H \acute u^H = \underbrace{I_h^H \acute f^h}_{\acute f^H} + \underbrace{N^H \hat I_h^H \tilde u^h - I_h^H N^h \tilde u^h}_{\tau_h^H}
    \end{equation*}
  \item Prolong, post-smooth, compute error $e^h = \acute u^h - (N^h)^{-1} \acute f^h$
  \item<2> \alert{Coarse grid \emph{with $\tau$} is nearly $10\times$ better accuracy}
  \end{itemize}
  % \caption{Plane strain elasticity, $E=1000,\nu=0.4$ inclusions in $E=1,\nu=0.2$ material.  2-level multigrid with coarsening factor of $3^2$.
  %   Panes (a) and (b) show the deformed body colored by strain.
  %   The initial problem of compression by 0.2 from the right is solved (a) and $\tau = A^H \hat I_h^H u^h - I_h^H A^h u^h$ is computed.
  %   Then a shear increment of 0.1 in the $y$ direction is added to the boundary condition, and the coarse-level problem is resolved, interpolated to the fine-grid, and a post-smoother is applied.
  %   When the coarse problem is solved without a $\tau$ correction (c), the displacement error is nearly $10\times$ larger than when $\tau$ is included in the right hand side of the coarse problem (d).
  % }\label{fig:tau-valid}
  % ./ex49 -mx 90 -my 90 -da_refine_x 3 -da_refine_y 3 -elas_ksp_converged_reason -elas_ksp_rtol 1e-8 -no_view -c_str 3 -sponge_E0 1 -sponge_E1 1e3 -sponge_nu0 0.4 -sponge_nu1 0.2 -sponge_t 3 -sponge_w 9 -u_o vtk:ex49_sol.vts -use_nonsymbc -elas_pc_type mg -elas_pc_mg_levels 2 -elas_pc_mg_galerkin -tau1_o vtk:ex49_tau1.vts -tau2_o vtk:ex49_tau2.vts -taudiff_o vtk:ex49_taudiff.vts -u2_o vtk:ex49_sol2.vts -u2c_o vtk:ex49_sol2c.vts -u3_o vtk:ex49_sol3.vts -u4_o vtk:ex49_sol4.vts -u2err_o vtk:ex49_sol2err.vts -u3err_o vtk:ex49_sol3err.vts -u3c_o vtk:ex49_sol3c.vts -tau3_o vtk:ex49_tau3.vts
\end{figure}
\end{frame}

\begin{frame}{$\tau$ adaptivity: an idea for heterogeneous media}
  \begin{itemize}
  \item Applications with localized nonlinearities
    \begin{itemize}
    \item Subduction, rifting, rupture/fault dynamics
    \item Carbon fiber, biological tissues, fracture
    \end{itemize}
  \item Adaptive methods fail for heterogeneous media
    \begin{itemize}
    \item Rocks are rough, solutions are not ``smooth''
    \item Cannot build accurate coarse space without scale separation
    \end{itemize}
  \item $\tau$ adaptivity
    \begin{itemize}
    \item Fine-grid work needed everywhere at first
    \item Then $\tau$ becomes accurate in nearly-linear regions
    \item Only visit fine grids in ``interesting'' places: active nonlinearity, drastic change of solution
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Comparison to nonlinear domain decomposition}
  \begin{itemize}
  \item ASPIN (Additive Schwarz preconditioned inexact Newton) \\
    \begin{itemize}
    \item Cai and Keyes (2003)
    \item More local iterations in strongly nonlinear regions
    \item Each nonlinear iteration only propagates information locally
    \item Many real nonlinearities are activated by long-range forces
      \begin{itemize}
      \item locking in granular media (gravel, granola)
      \item binding in steel fittings, crack propagation
      \end{itemize}
    \item Two-stage algorithm has different load balancing
      \begin{itemize}
      \item Nonlinear subdomain solves
      \item Global linear solve
      \end{itemize}
    \end{itemize}
  \item $\tau$ adaptivity
    \begin{itemize}
    \item Minimum effort to communicate long-range information
    \item Nonlinearity sees effects as accurate as with global fine-grid feedback
    \item Fine-grid work always proportional to ``interesting'' changes
    \end{itemize}
  \end{itemize}
\end{frame}

\input{slides/MG/LowComm.tex}
\input{slides/MG/SmoothingNonlinearProblems.tex}

\begin{frame}[fragile]{Multiscale compression and recovery using $\tau$ form}
   \begin{tikzpicture}
    [scale=0.7,every node/.style={scale=0.7},
    >=stealth,
    restrict/.style={thick,double},
    prolong/.style={thick,double},
    cprestrict/.style={green!50!black,thick,double,dashed},
    control/.style={rectangle,red!40!black,draw=red!40!black,thick},
    mglevel/.style={rounded rectangle,draw=blue!50!black,fill=blue!20,thick,minimum size=6mm},
    checkpoint/.style={rectangle,draw=green!50!black,fill=green!20,thick,minimum size=6mm},
    mglevelhide/.style={rounded rectangle,draw=gray!50!black,fill=gray!20,thick,minimum size=6mm},
    tau/.style={text=red!50!black,draw=red!50!black,fill=red!10,inner sep=1pt},
    crelax/.style={text=green!50!black,fill=green!10,inner sep=0pt}
    ]
    \begin{scope}
      \newcommand\mgdx{1.9em}
      \newcommand\mgdy{2.5em}
      \newcommand\mgloc[4]{(#1 + #4*\mgdx*#3,#2 + \mgdy*#3)}
      \node[mglevel] (fine0) at \mgloc{0}{0}{4}{-1} {\mglevelfine};
      \node[mglevel] (finem1down0) at \mgloc{0}{0}{3}{-1} {};
      \node[mglevel] (cp1down0) at \mgloc{0}{0}{2}{-1} {$\mglevelcp+1$};
      \node[mglevel] (cpdown0) at \mgloc{0}{0}{1}{-1} {\mglevelcp};
      \node[mglevel] (coarser0) at \mgloc{0}{0}{0}{0} {\ldots};

      \node[mglevelhide] (cpup0) at \mgloc{0}{0}{1}{1} {};
      \node (cp1up0) at \mgloc{0}{0}{2}{1} {};

      \node (cpdown1) at \mgloc{4em}{0}{1}{-1} {};
      \node[mglevelhide] (coarser1) at \mgloc{4em}{0}{0}{1} {\ldots};
      \node[mglevel] (cpup1) at \mgloc{4em}{0}{1}{1} {\mglevelcp};
      \node[mglevel] (cp1up1) at \mgloc{4em}{0}{2}{1} {$\mglevelcp+1$};
      \node[mglevel] (finem1up1) at \mgloc{4em}{0}{3}{1} {};
      \node[mglevel] (fine1) at \mgloc{4em}{0}{4}{1} {\mglevelfine};

      \draw[->,restrict,dashed] (fine0) -- (finem1down0);
      \draw[->,restrict] (finem1down0) -- (cp1down0);
      \draw[->,restrict] (cp1down0) -- (cpdown0);
      \draw[->,restrict,dashed] (cpdown0) -- (coarser0);
      \draw[->,prolong,dashed] (coarser0) -- (cpup0);
      \draw[->,prolong,dashed] (cpup0) -- (cp1up0);

      \draw[->,restrict,dashed] (cpdown1) -- (coarser1);
      \draw[->,prolong,dashed] (coarser1) -- (cpup1);
      \draw[->,prolong] (cpup1) -- (cp1up1);
      \draw[->,prolong] (cp1up1) -- (finem1up1);
      \draw[->,prolong,dashed] (finem1up1) -- (fine1);

      \node[checkpoint] at (4em + \mgdx*4,\mgdy) (cp) {CP};
      \draw[>->,cprestrict] (fine1) -- node[below,sloped] {Restrict} (cp);

      \node[left=\mgdx of fine0] (bnanchor) {};
      \node[control,fill=red!20] at (1.1*\mgdx,3*\mgdy) {Solve $F(u^n;b^n) = 0$};
      \node[mglevel,right=of fine1] (finedt) {next solve};
      \draw[->, >=stealth, control] (fine1) to[out=20,in=170] node[above] {$b^{n+1}(u^n,b^n)$} (finedt);
      \draw[->, >=stealth, control] (bnanchor) to[out=45,in=155] node[above] {$b^n$} (fine0);

      % Recovery process
      \begin{scope}[xshift=8*\mgdx]
        \node[checkpoint] (rcp) at \mgloc{0}{0}{0}{0} {CP};
        \node[mglevel] (r0a) at \mgloc{0}{\mgdy}{0}{0} {CR};
        \node[mglevel] (r1a) at \mgloc{0}{\mgdy}{1}{1} {};
        \node[mglevel] (r0b) at \mgloc{2*\mgdx}{\mgdy}{0}{0} {CR};
        \node[mglevel] (r1b) at \mgloc{2*\mgdx}{\mgdy}{1}{1} {};
        \node[mglevel] (r2b) at \mgloc{2*\mgdx}{\mgdy}{2}{1} {\mglevelfine};
        \node[mglevel] (r1c) at \mgloc{6*\mgdx}{\mgdy}{1}{-1} {};
        \node[mglevel] (r0d) at \mgloc{6*\mgdx}{\mgdy}{0}{0} {CR};
        \node[mglevel] (r1d) at \mgloc{6*\mgdx}{\mgdy}{1}{1} {};
        \node[mglevel] (r2d) at \mgloc{6*\mgdx}{\mgdy}{2}{1} {\mglevelfine};

        \draw[-,prolong,green!50!black] (rcp) -- (r0a);
        \draw[->,prolong] (r0a) -- (r1a);
        \draw[->,restrict] (r1a) -- (r0b);
        \draw[->,restrict] (r0b) -- (r1b);
        \draw[->,restrict,dashed] (r1b) -- (r2b);
        \draw[->,restrict,dashed] (r2b) -- (r1c);
        \draw[->,restrict] (r1c) -- (r0d);
        \draw[->,restrict] (r0d) -- (r1d);
        \draw[->,restrict,dashed] (r1d) -- (r2d);

        \foreach \smooth in {finem1down0, cp1down0, cpdown0, coarser0,
          cpup1, cp1up1, finem1up1,
          r0b,r1c,r0d,r1d} {
          \node[above left=-5pt of \smooth.west,tau] {$\tau$};
        }
        \node[rectangle,fill=none,draw=green!50!black,thick,fit=(rcp)(r2d)] (recoverbox) {};
        \node[rectangle,draw=green!50!black,fill=green!20,thick,minimum size=6mm,above={0cm of recoverbox.south east},anchor=south east] (recover) {FMG Recovery};
      \end{scope}
      \node (notation) at (\mgdx,5*\mgdy) {
        \begin{minipage}{18em}\small\sf
          \begin{itemize}\addtolength{\itemsep}{-5pt}
          \item checkpoint converged coarse state
          \item recover using FMG anchored at $\mglevelcp+1$
          \item needs only $\mglevelcp$ neighbor points
          \item $\tau$ correction is local
          \end{itemize}
        \end{minipage}
      };
    \end{scope}
  \end{tikzpicture}
  \begin{itemize}
  \item Normal multigrid cycles visit all levels moving from $n \to n+1$
  \item FMG recovery only accesses levels finer than $\ell_{CP}$
  % \item Only failed processes and neighbors participate in recovery
  \item Lightweight checkpointing for transient adjoint computation
  \item Postprocessing applications, e.g., in-situ visualization at high temporal resolution in part of the domain
  \end{itemize}
\end{frame}

\begin{frame}{Outlook on $\tau$-FAS adaptivity and compression}
  \begin{itemize}
  \item Benefits of AMR without fine-scale smoothness
  \item Coarse-centric restructuring is a major interface change
  \item Nonlinear smoothers (and discretizations)
    \begin{itemize}
    \item Smooth in neighborhood of ``interesting'' fine-scale features
    \item Which discretizations can provide efficient matrix-free smoothers?
    \item Does there exist an efficient smoother based on element Neumann problems?
    \end{itemize}
  \item Dynamic load balancing
  \item Reliability of error estimates for refreshing $\tau$
    \begin{itemize}
    \item We want a coarse indicator for whether $\tau$ needs to change
    \end{itemize}
  \item Worthwhile for resilience and to better use hardware
  \end{itemize}
\end{frame}

\end{document}