20141031-Structure.tex

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\newcommand\vvec{\bm v}
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\newcommand{\timeR}{t_{\mathrm{R}}}
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\newcommand{\mglevel}{\ensuremath{\ell}}
\newcommand{\mglevelcp}{\ensuremath{\mglevel_{\mathrm{cp}}}}
\newcommand{\mglevelcoarse}{\ensuremath{\mglevel_{\mathrm{coarse}}}}
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\newcommand{\vx}{\ensuremath{x}}
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%operators
\newcommand{\vA}{\ensuremath{A}}
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\newcommand{\vR}{\ensuremath{I_h^H}}
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\newcommand{\vtau}{\ensuremath{\mathbf{\tau}}}


\title{Exploiting structure with implicit methods}
\subtitle{This talk: \url{http://59A2.org/files/20141031-Structure.pdf}}
\begin{comment}
  Implicit solvers are uniquely powerful instruments for managing
  disparate scales as well as many tasks in the analysis of models, but
  performance can be subtle and challenging.  "Black-box" solvers suffer
  From limited applicability and poor efficiency on modern hardware.
  Building efficient solvers relies increasingly on pairing
  problem-specific structure with architectural capability and
  expressing the resulting algorithms in terms of existing software.
  Library design is the inverse problem: create maximally-reusable
  components to satisfy a diverse range of present and future customers
  and architectures.  This talk discusses recent examples in PETSc's
  evolution and our vision for implicit solvers.
\end{comment}

\author{{\bf Jed Brown} \texttt{jedbrown@mcs.anl.gov} (ANL and CU Boulder) \\
  Collaborators in this work: \\
  \quad Mark Adams (LBL), Peter Brune (ANL/Google), Emil Constantinescu (ANL), \\
Debojyoti Ghosh (ANL), Matt Knepley (UChicago),  \\
Dave May (ETH Z\"urich, Lois Curfman McInnes (ANL), \\
Barry Smith (ANL))
}

% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.
% \institute
% {
%   Mathematics and Computer Science Division \\ Argonne National Laboratory
% }

\date{UC Merced, 2014-10-31}

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\subject{Talks}


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\lstset{language=C}
\normalem

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

\begin{frame}{Why implicit?}
  \begin{itemize}
  \item Nature has many spatial and temporal scales
    \begin{itemize}
    \item Porous media, structures, fluids, kinetics
    \end{itemize}
  \item Science/engineering problem statement does not weak scale
    \begin{itemize}
    \item More time steps required at high resolution
    \end{itemize}
  \item Robust discretizations and implicit solvers are needed to cope
  \item Computer architecture is increasingly hierarchical
    \begin{itemize}
    \item algorithms should conform to this structure
    \end{itemize}
  \item Sparse matrices are comfortable, but outdated
    \begin{itemize}
    \item Algebraic multigrid, factorization
    \item Memory bandwidth-limited
    \end{itemize}
  \item ``black box'' solvers are not sustainable
    \begin{itemize}
    \item optimal solvers must accurately handle all scales
    \item optimality is crucial for large-scale problems
    \item hardware puts up a spirited fight to abstraction
    \end{itemize}
  \end{itemize}
\end{frame}

\input{slides/MonolithicOrSplit.tex}
\input{slides/PETSc/Coupling.tex}
\input{slides/FieldSplit.tex}
\input{slides/PETSc/LocalSpaces.tex}
\includepdf[pages=1-2]{davemay.pdf}
\input{slides/HydrostaticEigen.tex}

\begin{frame}{Plugins in PETSc}
\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}
	Third party 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}[shrink=5]{Performance of assembled versus unassembled}
  \vspace{1ex}
  \includegraphics[width=\textwidth]{figures/TensorVsAssembly} \\
  \begin{itemize}
  \item Arithmetic intensity for $\Qk p$ elements
    \begin{itemize}
    \item $< \frac 1 4$ (assembled), $\approx 10$ (unassembled), $\approx 4$ to $8$ (hardware)
    \end{itemize}
  \item store Jacobian information at Quass quadrature points, can use AD
  \end{itemize}
\end{frame}

\input{slides/Dohp/StokesScaling.tex}

\begin{frame}{\texttt{pTatin3d}: Long-term lithosphere dynamics}
  \begin{center}
    \includegraphics[width=0.8\textwidth]{figures/LaetiRifting}
  \end{center}
  \begin{itemize}
  \item Dave May (ETH Z\"urich), Laetitia Le Pourhiet (UPMC Paris)
  \item Visco-elasto-plastic rheology
  \item Material-point method for material composition, $10^{10}$ jumps
  \item Large deformation, post-failure analysis
  \item Free surface: $Q_2 - P_1^{\text{disc}}$ (non-affine)
  \end{itemize}
\end{frame}

\begin{frame}{\texttt{pTatin3d}: Long-term lithosphere dynamics}
  \begin{itemize}
  \item Assembled matrices: $9216 F/38912 B = \alert{0.235 F/B}$
    \begin{itemize}
    \item Problem size limited by memory
    \item Mediocre performance, limited by memory bandwidth
    \item Poor scalability within a node (memory contention)
    \item Lots of experimentation with different algorithms
    \item Multigrid: matrix-free on finest levels
    \end{itemize}
  \item Matrix-free: $51435 F/824 B = \alert{62.42 F/B}$
    \begin{itemize}
    \item $81\times 27$ element gradient matrix
    \item Element setup computes physical gradient matrix
    \item $1.5\times$ speedup when using all cores
    \end{itemize}
  \item Tensor-product matrix-free: $16686 F/824 B = \alert{20.25 F/B}$
    \begin{itemize}
    \item Tensor contractions with $3\times 3$ 1D matrices
    \item Tiny working set, vectorize over 4 elements within L1 cache
    \item 30\% of Haswell FMA peak, register load/store limited
    \item $7\times$ speedup ($5\times$ speedup on Sandy Bridge AVX)
    \end{itemize}
  \end{itemize}
\end{frame}

\input{slides/HardwareArithmeticIntensity.tex}

\begin{frame}{This is a dead end}
  \begin{itemize}
  \item Arithmetic intensity $< 1/4$
  \item Idea: multiple right hand sides
    \begin{equation*}
      \frac{(2 k \text{ flops})(\text{bandwidth})}{\texttt{sizeof(Scalar)} + \texttt{sizeof(Int)}}, \quad k \ll \text{avg. nz/row}
    \end{equation*}
  \item Problem: popular algorithms have nested data dependencies
    \begin{itemize}
    \item Time step \\
      \qquad Nonlinear solve \\
      \qquad \qquad Krylov solve \\
      \qquad \qquad \qquad Preconditioner/sparse matrix
    \end{itemize}
  \item Cannot parallelize/vectorize these nested loops
  \item<2> \alert{Can we create new algorithms to reorder/fuse loops?}
    \begin{itemize}
    \item Reduce latency-sensitivity for communication
    \item Reduce memory bandwidth (reuse matrix)
    \item Implicit Runge-Kutta, creates tensor product structure
    \item Full space/one-shot methods for PDE-constrained optimization
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Beyond global linearization: FAS multigrid}
  \includegraphics[width=\textwidth]{figures/BruneNGMRESFAS.png}
  \begin{itemize}
  \item Geometric coarse grids and rediscretization
  \end{itemize}
\end{frame}

\begin{frame}{Lagged quasi-Newton for nonlinear elasticity}
  \begin{tabular}{llll llll}
  \toprule
  Method & Lag & LS & Linear Solve & Its. & $F(u)$ & Jacobian & $P^{-1}$ \\
  \midrule
  LBFGS & 3 & cp & \texttt{preonly} & 18 & 37 & 5 & 18 \\
  LBFGS & 3 & cp & \num{1e-05} & 21 & 43 & 6 & 173 \\
  LBFGS & 6 & cp & \texttt{preonly} & 24 & 49 & 4 & 24 \\
  LBFGS & 6 & cp & \num{1e-05} & 30 & 61 & 5 & 266 \\[1ex]
  JFNK & 0 & cp & \texttt{preonly} & 11 & 23 & 11 & 11 \\
  JFNK & 0 & cp & \num{1e-05} & 8 & 69 & 8 & 60 \\
  JFNK & 1 & cp & \texttt{preonly} & 15 & 31 & 8 & 15 \\
  JFNK & 1 & cp & \num{1e-05} & 7 & 2835 & 4 & 2827 \\
  JFNK & 3 & cp & \texttt{preonly} & 23 & 47 & 6 & 23 \\
  JFNK & 3 & cp & \num{1e-05} & 7 & 3143 & 2 & 3135
\end{tabular}
\begin{itemize}
\item B and Brune, MC2013
\end{itemize}
\end{frame}

\input{slides/PETSc/TSARKIMEX.tex}

% \input{slides/MG/TauFAS.tex}

\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}

\input{slides/MG/LowComm.tex}
\begin{frame}{Segmental refinement: no horizontal communication}
  \begin{itemize}
  \item 27-point second-order stencil, manufactured analytic solution
  \item 5 SR levels: $16^3$ cells/process local coarse grid
  \item $\text{Overlap} = \text{Base} + (L-\ell) \text{Increment}$
    \begin{itemize}
    \item Implementation requires even number of cells---round down.
    \end{itemize}
  \item FMG with $V(2,2)$ cycles
  \end{itemize}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      \begin{table}\small
        \centering\caption{$\norm{e_{SR}}_\infty / \norm{e_{FMG}}_\infty$}\label{tab:sr-error}
        \begin{tabular}{l rrr}
          \toprule
          & \multicolumn{3}{c}{Base} \\
          Increment & 1 & 2 & 3 \\
          \midrule
          1 & {\color{red} 1.59} & {\color{red} 2.34} & 1.00 \\
          2 & 1.00 & 1.00 & 1.00 \\
          3 & 1.00 & 1.00 & 1.00 \\
          \bottomrule
        \end{tabular}
      \end{table}
    \end{column}
    \begin{column}{0.6\textwidth}
      \includegraphics[width=\textwidth]{figures/MG/weak_scaling_edison-eps-converted-to.pdf}
    \end{column}
  \end{columns}
\end{frame}
\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}

\input{slides/MG/HPGMGFE.tex}

\begin{frame}\large
  \begin{itemize}
  \item Maximize science per Watt
  \item Huge scope remains at problem formulation
  \item Raise level of abstraction at which a problem is formally specified
  \item Algorithmic optimality is crucial
  \item Real problems are messy
  \item Performance is always messy at scale
  \item Improve matrix-free abstractions, robustness, diagnostics
  \item Ideas are easy, implementation and practical issues are hard
  \item Better language/library support for aggregating
  \end{itemize}
\end{frame}

\end{document}