2024-06-13-talk-CSBQ/main.tex

% vim: set foldmethod=marker foldmarker=<<<,>>>:

\input{ccmbeamer}
%<<< title, author, institute
  \title
  [Convergent Slender Body Quadrature]
  {Convergent Slender Body Quadrature}
  %\author[Dhairya Malhotra]{ \underline{Dhairya~Malhotra}, ~{Alex Barnett}}
  \author[Dhairya Malhotra]{Code: {\color{blue} \url{https://github.com/dmalhotra/CSBQ}} \\
  \phantom{.}\\
   \underline{Dhairya~Malhotra}, ~{Alex Barnett}}


  %\institute{Flatiron Institute\\ \mbox{}  \\  \pgfuseimage{FIbig} }
  %\institute{\pgfuseimage{FIbig} }

  %\date[]{Code: {\color{blue} \url{https://github.com/dmalhotra/CSBQ}} \\ June 13, 2024}
  \date[]{June 13, 2024}
%>>>
%<<< packages
  \usepackage{tikz}
  \usetikzlibrary{fit,shapes.geometric,arrows,calc,shapes,decorations.pathreplacing,patterns}
  \usepackage{pgfplots,pgfplotstable}
  \pgfplotsset{compat=1.17}

  \usepackage{mathtools}
  \usepackage{multirow}
  \usepackage{multimedia}
  \usepackage{media9}
  %\usepackage{movie15} %(obsolete)
  \usepackage{animate}
  \usepackage{fp}
  %\usepackage{enumitem}
  \usepackage{bm}

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  \usepackage{xstring}
  \usepackage{mathtools}% Loads amsmath

  \usepackage{stmaryrd}

  \usepackage{amsfonts} %% <- also included by amssymb
  \DeclareMathSymbol{\shortminus}{\mathbin}{AMSa}{"39}

  \newcommand{\vcenteredinclude}[1]{\begingroup\setbox0=\hbox{{#1}}\parbox{\wd0}{\box0}\endgroup}

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        #1.\@ }%
      {#1.\@}}}
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  \definecolor{DarkGreen}{RGB}{0,130,0}

  \usepackage{minted}
  \usemintedstyle{vs}
  %\usemintedstyle{borland}

  %\usemintedstyle{emacs}
  %\usemintedstyle{perldoc}
  %\usemintedstyle{friendly}
  %%\usemintedstyle{pastie}
  %%\usemintedstyle{vim}
%>>>

  \newcommand\vct[1]{{\ensuremath{\bm{#1}}}}
  \newcommand\convop[1]{{\ensuremath{\mathcal{#1}}}}
  \newcommand\Real{{\ensuremath{\mathbb{R}}}}

  \newcommand\StokesSL{{\ensuremath{\convop{S}}}}
  \newcommand\StokesDL{{\ensuremath{\convop{D}}}}
  \newcommand\StokesCF{{\ensuremath{\convop{K}}}}

\begin{document}
  \setbeamercovered{transparent}% Dim out "inactive" elements


  \begin{frame}%<<< Title

    \vspace{4em}
    \titlepage

    \vspace{0.5em}
    \begin{columns}
      \column{0.8\textwidth}
      \column{0.2\textwidth}
        \pgfuseimage{FIbig}
    \end{columns}
  \end{frame}%>>>


  \section{Introduction} %<<<

  \begin{FIframe}{Motivations}{} %<<<

    \vspace{-1.5em}
    \begin{columns}[t]
      \column{0.5\textwidth}

        \vspace{1em}
        Stokes simulations with fibers are key to modeling complex fluids
        (suspensions, rheology, industrial, biomedical, cellular biophysics).

        \only<2->{
        \vspace{2em}
        {\bf Slender Body Theory (SBT):}
        \begin{itemize}
          \item Asymptotic expansion in radius ($\varepsilon$) \\
                as $\varepsilon \to\ 0$ (Keller-Rubinow '76).

                \vspace{1em}
          \item Doublet correction to make velocity theta-independent (Johnson '80).

        \end{itemize}

        %\vspace{1em}
        %The force rep w/ plain Stokeslets doesn't make velocity theta-independent on the surface, so the doublet is added to do that better.
        %With doublet correction , error $\sim r^2. \log(r)$.
        }

        %\only<3->{
        %\vspace{1em}
        %SBT has only very recently been placed on rigorous footing.
        %(Koens-Lauga '18, Mori-Ohm-Spirn '19). %(error $\sim r \log^k(r)$)
        %}

      \column{0.5\textwidth}

        \begin{columns}
          \column{0.5\textwidth}
            \only<1>{\embedvideo{\includegraphics[width=0.99\textwidth]{videos/starfish}}{videos/starfish.mov}}%
            %\starttext
            %    \setupinteraction[state=start]
            %    \enabletrackers[graphics.locating]
            %    \externalfigure[sample.mov][width=10cm, height=10cm]
            %\stoptext
            \only<2->{\includegraphics[width=0.99\textwidth]{videos/starfish1}}\\
            Starfish larvae \\
            (Gilpin et al. 2016)

          \column{0.5\textwidth}
            \vspace{1em}
            \includegraphics[width=0.99\textwidth]{figs/oocyte} \\
            Drosophila oocyte (Stein et al. 2021)
        \end{columns}

        \centering
        \includegraphics[width=0.6\textwidth]{figs/mitosis} \\
        Mitotic spindle (Nazockdast et al. 2015)

    \end{columns}

  \end{FIframe} %>>>

  \begin{FIframe}{Slender Body Theory}{} %<<<

    {\bf Error estimates:} Rigorous analysis difficult (few very recent studies)
    \begin{itemize}
      \item classical asymptotics claims: $\varepsilon^2 \log(\varepsilon)$
      \item rigorous analysis: $\varepsilon \log^{3/2}(\varepsilon)$ \qquad (Mori-Ohm-Spirn '19)
      \item numerical tests: $\varepsilon^{1.7}$ \qquad (Mitchell et al. '21 -- verify close-touching breakdown)\\
        \quad close-to-touching with gap of 10$\varepsilon$,~~ only 2.5-digits in the infty-norm.\\
        \quad $\varepsilon$=1e-2 ~~only 1-2 digits achievable by SBT.\\
    \end{itemize}

    \only<1>{
      \centering
      \includegraphics[width=0.26\textwidth]{figs/cilia.jpg}

      \vspace{-2ex}
      {\tiny Source: http://remf.dartmouth.edu/imagesindex.html}
    }
    \only<2>{
      \vspace{1em}
      \begin{columns}
        \column{0.5\textwidth}

            \begin{tabular}{| r r r|}
              \hline
              $\varepsilon$ &     $\vct{u}_{exact}$ & Rel-Error \\
              \hline
              1e-1      &   6.1492138359856e-2 &   0.5e-2 \\
              1e-2      &   9.0984522324584e-2 &   0.1e-3 \\
              1e-3      &   1.2015655889904e-1 &   0.2e-5 \\
              1e-4      &   1.4931932907587e-1 &   0.2e-7 \\
              1e-5      &   1.7848191313097e-1 &   0.3e-9 \\
              \hline
            \end{tabular}


            %\begin{tabular}{r r r r | c r r r r} // these are for elipse
            %  \hline
            %  $\varepsilon$ & $\bm u_0$ & Error \\
            %  \hline
            %  $0.1$     & $0.0518$ & $0.7e-2$ \\
            %  $0.01$    & $0.0736$ & $0.2e-3$ \\
            %  $0.001$   & $0.0950$ & $0.3e-5$ \\
            %  $0.0001$  & $0.1163$ & $0.4e-7$ \\
            %  %$0.00001$ & $0.1377$ & $0.6e-9$ \\
            %  \hline
            %\end{tabular}
            % ellipse (semiaxes 2,0.5) radius eps=0.1...
            % N=480: L=8.578421775156826 drag force
            % F.     = 19.17234313264176
            % Fexact = 19.31188135187
            %
            % ellipse (semiaxes 2,0.5) radius eps=0.01...
            % N=480: L=8.578421775156826 drag force
            % F      = 13.58844162453679
            % Fexact = 13.59082284902
            %
            % ellipse (semiaxes 2,0.5) radius eps=0.001...
            % N=480: L=8.578421775156826 drag force
            % F      = 10.52899298797188
            % Fexact = 10.52902479066
            %
            % ellipse (semiaxes 2,0.5) radius eps=0.0001...
            % N=480: L=8.578421775156826 drag force
            % F.     = 8.594914613917958
            % Fexact = 8.594914990618
            %
            % ellipse (semiaxes 2,0.5) radius eps=1e-05...
            % N=480: L=8.578421775156826 drag force
            % F      = 7.261368067858561
            % Fexact = 7.2613680720
        \column{0.5\textwidth}
          \includegraphics[width=0.95\textwidth]{figs/ring-sed}
      \end{columns}
    }
    %\only<3>{
    %  \center
    %  \vspace{-0.8em}
    %  \includegraphics[width=0.78\textwidth]{figs/sbt-close-breakdown}
    %}
    \only<3>{
      \vspace{1em}
      {\bf Limitations of SBT:}
      \begin{itemize}
        \item no convergence analysis for fibers of given nonzero radius. %, you do not know errors in simulation .
        \item uncontrolled errors when fibers close $O(\varepsilon)$. %, SBT assumptions break down.
      \end{itemize}

      Efficient convergent BIE method needed, allowing adaptivity for close interactions.
    }

  \end{FIframe} %>>>

  \begin{FIframe}{Goals}{} %<<<

    Solve the slender body BVP
    \begin{itemize}
      \item in a convergent way.
      \item adaptively when fibers become close.
      \item efficiently with effort independent of radius.
    \end{itemize}
    Validate current SBT simulations.

    %\vspace{0.5em}
    %Most existing qudaratures cannot resolve high aspect ratio geometries.

    \vspace{4.5em}
    Focus on rigid fibers in this talk ~~--~~ flexible fibers for future.

    \vspace{1em}
    {\em Related work:} ~~ Mitchell et al, '21 (mixed-BVP corresponding to flexible fiber loop)


    %Only loops for now, to avoids complications with endpoint singularities.

    %\textcolor{blue}{\bf Quadratures for slender bodies}
    %\begin{itemize}
    %  \item compute interactions of filaments (eg. microtubules) in viscous fluids without asymptotic approximations.
    %  \item fully resolved boundary-integral formulation; have to deal with highly anisotropic elements.
    %\end{itemize}
  \end{FIframe} %>>>

  %\begin{FIframe}{Motivation}{} %<<<
  %  \begin{itemize}
  %    \item aspect ratios of $10^4$ or greater
  %    \item existing quadrature schemes are not efficient in this regime
  %  \end{itemize}
  %\end{FIframe} %>>>
  %\begin{FIframe}{Outline}{} %<<<
  %{\large

  %  \begin{itemize}
  %    \item Slender Body Quadrature

  %    \vspace{1em}
  %    \item Stokes Mobility Problem

  %  \end{itemize}

  %}
  %\end{FIframe} %>>>

  %>>>


  \section{Algorithms} %<<<

  \begin{FIframe}{Discretization}{} %<<<

    \vspace{-2.0em}
    \begin{columns}[t]
      \column{0.52\textwidth}

        {\bf Geometry description:}
        \begin{itemize}
          \item parameterization $s$ along fiber length
          \item coordinates $x_c(s)$ of centerline curve
          \item circular cross-section with radius $\varepsilon(s)$ %at each point along the centerline
          \item orientation vector $e_{1}(s)$
        \end{itemize}

        \vspace{1em}
        \only<2>{
          {\bf Discretization:}
          \begin{itemize}
            \item piecewise Chebyshev (order $q$) discretization in $s$ for $x_c(s)$, $\varepsilon(s)$, $e_{1}(s)$ %(either given or selected arbitrarily)
            \item Collocation nodes: tensor product of Chebyshev and Fourier discretization in angle with order $N_{\theta}$.
          \end{itemize}
        }

      \column{0.6\textwidth}

        \centering
        \only<1>{\includegraphics[width=0.99\textwidth]{figs/slender-body4}}
        \only<2>{

          \begin{tikzpicture}
            \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[height=0.99\textheight]{figs/slenderbody-discretization.png}};
            \node at (1.2, 6.3) {\Large $N_{\theta}$};
            \node (a) at (.7, 6) {};
            \node (b) at (-.0, 6) {};

            \node at (-1.0, 5.5) {\Large $q$};
            \node (d) at (.6, 8.8) {};
            \node (c) at (.1, 2.3) {};
            \draw[ultra thick, ->] (a)  to [out=60,in=120, looseness=2] (b);
            \draw[ultra thick, ->] (c)  to [out=110,in=230, looseness=1] (d);
          \end{tikzpicture}
        }

    \end{columns}

  \end{FIframe} %>>>

  \begin{FIframe}{Boundary Quadratures}{} %<<<

    \vspace{-0.7em}
    $\displaystyle u(x) ~= \int_{\Gamma} \mathcal{K}(x-y)~\sigma(y)~da(y) ~= \sum_{k=1}^{N_{panel}} \int_{\gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y)$

      $\displaystyle \phantom{u(x)} ~= \underbrace{\sum_{x \notin \mathcal{N}(\gamma_k)} \int_{\gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y)}_{\text{far-field}}~
      + \underbrace{\sum_{x \in \mathcal{N}(\gamma_k)} \int_{\gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y)}_{\text{near interactions}}$

    \only<2>{ %<<<
      \vspace{2.5em}
      {\bf Far field approximation:} %for $x \notin \mathcal{N}(\gamma_k)$
      %\vspace{0.5em}
      %$\displaystyle \qquad \int_{\gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y) ~\approx~
      %  \sum_{i,j} \frac{2 \pi w_i}{N_{\theta}} \mathcal{K}(x-y(s_i,\theta_j))~\sigma(s_i,\theta_j)~J(s_i,\theta_j) $

      \vspace{-0.5em}
      \begin{columns}
        \column{0.5\textwidth}
          \begin{itemize}
            \item Gauss-Legendre quadrature in $s$.
            \item periodic trapezoidal rule in $\theta$.
						\item determine $\mathcal{N}(\gamma_k)$ using standard \\
            error estimates.
          \end{itemize}
        \column{0.6\textwidth}
          \resizebox{.99\textwidth}{!}{\begin{tikzpicture}
            \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=10cm]{figs/bernstein1.png}};
            \node at (4.7, 3.1) {$\mathcal{N}(\gamma_i)$};

            \node at (6.0, 2.45) {\textcolor{red}{$\gamma_i$}};
            \draw [fill=orange, fill opacity=0.35] (2.20,1.70) circle (0.7cm);
            \draw [fill=orange, fill opacity=0.35] (2.73,1.97) circle (0.8cm);
            \draw [fill=orange, fill opacity=0.35] (3.45,2.17) circle (1.0cm);
          \end{tikzpicture}}%
      \end{columns}
    } %>>>

    %\only<3>{ %<<<
    %  % Ellipse:
    %  %> theta=0:0.01:2*pi;
    %  %> x=10*sin(theta);
    %  %> y=4*cos(theta);
    %  %> y1=y+((x).^2)*0.03;
    %  %> hold off; imshow(I); hold on; plot(x*25+554,-y1*20+380, '.k')

    %  \centering
    %  \vspace{1em}
    %  \begin{tikzpicture}
    %    \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.7\textwidth]{figs/bernstein1.png}};
    %    \node at (4.7, 3.1) {$\mathcal{N}(\gamma_i)$};

    %    \node at (6.0, 2.45) {\textcolor{red}{$\gamma_i$}};
    %    \draw [fill=orange, fill opacity=0.35] (2.20,1.70) circle (0.7cm);
    %    \draw [fill=orange, fill opacity=0.35] (2.73,1.97) circle (0.8cm);
    %    \draw [fill=orange, fill opacity=0.35] (3.45,2.17) circle (1.0cm);
    %  \end{tikzpicture}~~%
    %  \begin{tikzpicture}
    %    \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.31\textwidth, height=0.31\textwidth]{figs/morton.png}};

    %    \draw [fill=cyan, fill opacity=0.25] (1.655,3.80) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;
    %    \draw [fill=cyan, fill opacity=0.25] (1.655,2.73) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;
    %    \draw [fill=cyan, fill opacity=0.25] (1.655,1.65) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;
    %    \draw [fill=cyan, fill opacity=0.25] (2.730,3.80) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;
    %    \draw [fill=cyan, fill opacity=0.25] (2.730,2.73) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;
    %    \draw [fill=cyan, fill opacity=0.25] (2.730,1.65) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;
    %    \draw [fill=cyan, fill opacity=0.25] (3.805,3.80) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;
    %    \draw [fill=cyan, fill opacity=0.25] (3.805,2.73) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;
    %    \draw [fill=cyan, fill opacity=0.25] (3.805,1.65) +(-14.5pt,-14.5pt) rectangle +(14.5pt,14.5pt) ;

    %    \draw [fill=red, fill opacity=0.99] (2.5,2.5) circle (0.09cm);
    %    \draw [fill=orange, fill opacity=0.5] (2.5,2.5) circle (1.1cm);
    %  \end{tikzpicture}
    %} %>>>

    % singular, near and far field evaluation
    % finding near-neighbors, Morton ordering, Bernstein ellipse
  \end{FIframe} %>>>

  \begin{FIframe}{Boundary Quadratures}{} %<<<

    \vspace{-1em}
    {\bf Near interactions:} for $x \in \mathcal{N}(\gamma_k)$

    \vspace{0.3em}
    $\displaystyle \int_{\gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y) ~=~ \int_{s} \int_{\theta} \mathcal{K}(x-y(s,\theta))~\sigma(s,\theta)~J(s,\theta)~d\theta~ds$

    \vspace{1.5em}
    \begin{columns}
      \column{0.5\textwidth}
        {\bf Inner integral in $\theta$:}
        \begin{itemize}
          \item potential from a ring source \\
                nearly singular as $x \longrightarrow \gamma_k$.
        \end{itemize}

      \column{0.47\textwidth}
        \resizebox{.99\textwidth}{!}{\begin{tikzpicture} %<<<
          \path [draw=none,fill=white!0,even odd rule] (4,0) circle (1.5) (4,0) circle (0.75);
          \draw[color=red, ultra thick] (0,0) ellipse (4cm and 1cm);

          \draw[color=blue, ultra thick] (4.7,-0.8) circle (1pt);
          \node at (5, -0.8) {\color{blue} \Large $x$};
          %\node at (5, -0.8) {\color{blue} \Large $x_i$};
          \node at (-4.6, 0) {\color{red} \Large $y(\theta)$};
          \node at (0, -0.5) {\Large $\theta$};
          \node (c) at (-1, -0.75) {};
          \node (d) at ( 1, -0.75) {};
          \draw[ultra thick, ->] (c)  to [out=-5,in=185, looseness=1] (d);
        \end{tikzpicture}} %>>>
    \end{columns}

    \vspace{2em}
    {\bf Outer integral in $s$:}
    %\begin{itemize}
    %  \item singular if $x \in \gamma_k$ with logarithmic singularity at $s = s_0$.
    %  \item $1/|s-s_0|^{\alpha}$ decay as $|s-s_0| \longrightarrow \infty$
    %\end{itemize}

    \vspace{0.7em}
    \begin{tikzpicture}%<<<
      \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.99\textwidth]{figs/s-quad/surf.png}};

      \draw[color=red, ultra thick] (2.7,0.9) circle (1pt);
      \node at (2.5, 0.5) {\color{red} \Large $x$};

      \draw[ultra thick, ->] (4.3,0.45) to (3.1,0.5);
      \node [rotate=-6] at (5.55, 0.25) {log singularity};

      \draw[ultra thick, ->] (10.5,-0.25) to (12.1,-0.2);
      \node [rotate=-4.5] at (9.5, -0.17) {$|s-s_0|^{-\alpha}$};
    \end{tikzpicture}%>>>

  \end{FIframe} %>>>


  \begin{FIframe}{Fast Modal Green's Function Evaluation}{} %<<<

    \vspace{-1em}
    \begin{columns}
      \column{0.7\textwidth}
        \begin{center} %<<<
          \resizebox{.99\textwidth}{!}{\begin{tikzpicture}
          \path [draw=none,fill=white!0,even odd rule] (4,0) circle (1.5) (4,0) circle (0.75);
          \only<2->{\path [draw=none,fill=blue!30,even odd rule] (4,0) circle (1.5) (4,0) circle (0.75);}
          \only<3->{\path [draw=none,fill=brown!80,even odd rule] (4,0) circle (0.75) (4,0) circle (0.375);}
          \only<3->{\path [draw=none,fill=green!80,even odd rule] (4,0) circle (0.375) (4,0) circle (0.1875);}
          \draw[color=red, ultra thick] (0,0) ellipse (4cm and 1cm);
          \only<2->{
          \draw[fill=black, thick] (-4,0) circle (1pt);
          \draw[fill=black, thick] ( 4,0) circle (1pt);
          \draw[fill=black, thick] (0,-1) circle (1pt);
          \draw[fill=black, thick] (0, 1) circle (1pt);

          \draw[fill=black, thick] (-2.828,-0.7071) circle (1pt);
          \draw[fill=black, thick] (-2.828, 0.7071) circle (1pt);
          \draw[fill=black, thick] ( 2.571,-0.7660) circle (1pt);
          \draw[fill=black, thick] ( 2.571, 0.7660) circle (1pt);

          \draw[fill=black, thick] (3.464,-.5) circle (1pt);
          \draw[fill=black, thick] (3.464, .5) circle (1pt);

          \draw[fill=black, thick] (3.759,-.3420) circle (1pt);
          \draw[fill=black, thick] (3.759, .3420) circle (1pt);

          \draw[fill=black, thick] (3.939,-.1736) circle (1pt);
          \draw[fill=black, thick] (3.939, .1736) circle (1pt);

          \draw[fill=black, thick] (3.985,-.0872) circle (1pt);
          \draw[fill=black, thick] (3.985, .0872) circle (1pt);
          }
          \only<2->{\draw[fill=blue!30,draw=none] (180:0.75)+(4,0) arc (180:0:0.75) -- (0:1.5)+(4,0) arc (0:180:1.5) -- cycle;}
          \only<3->{\draw[fill=brown!80,draw=none] (180:0.375)+(4,0) arc (180:0:0.375) -- (0:0.75)+(4,0) arc (0:180:0.75) -- cycle;}
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          \draw[color=blue, ultra thick] (4.7,-0.8) circle (1pt);
          \node at (5, -0.8) {\color{blue} \Large $x$};
          %\node at (5, -0.8) {\color{blue} \Large $x_i$};
          \node at (-4.6, 0) {\color{red} \Large $y(\theta)$};
          \node at (0, -0.5) {\Large $\theta$};
          \node (c) at (-1, -0.75) {};
          \node (d) at ( 1, -0.75) {};
          \draw[ultra thick, ->] (c)  to [out=-5,in=185, looseness=1] (d);

          \end{tikzpicture}}
        \end{center} %>>>
      \column{0.3\textwidth}
        $\displaystyle \int_{\theta} \mathcal{K}(x-y(\theta))~\sigma(\theta)~d\theta$
    \end{columns}

    %\vspace{-1em}
    %\begin{center} %<<<
    %\begin{tikzpicture}
    %  \draw[color=red, ultra thick] (0,0) ellipse (4cm and 1cm);
    %  \draw[color=blue, ultra thick] (4.7,-0.8) circle (1pt);
    %  \node at (5, -0.8) {\color{blue} \Large $x$};
    %  \node at (-4.6, 0) {\color{red} \Large $y(\theta)$};
    %  \node at (0, -0.5) {\Large $\theta$};
    %  \node (c) at (-1, -0.75) {};
    %  \node (d) at ( 1, -0.75) {};
    %  \draw[ultra thick, ->] (c)  to [out=-5,in=185, looseness=1] (d);
    %\end{tikzpicture}
    %\end{center} %>>>

    %$\qquad \displaystyle \int_{\theta} \mathcal{K}(x-y(\theta))~\sigma(\theta)~d\theta ~=~ \sum_n \mathcal{K}_n(x) \widehat{\sigma_n}$

    %where, $y(\theta)$ is a circular source loop, and $\mathcal{K}_n(x) = \int_{\theta} e^{-in\theta} \mathcal{K}(x-y(\theta))~d\theta$ are the modal Green's functions.

    \vspace{0.5em}
    \begin{itemize}
      \item Analytic representation in special functions - Young, Hao, Martinsson JCP-2012
        \begin{itemize}
          \item modal Green's functions -- method of choice for axisymmetric problems.
        \end{itemize}

      \only<2->{
      \vspace{1em}
      \item Build special quadrature rules!
        \begin{itemize}
          \item \eg~ generalized Gaussian quadratures: ~~Bremer, Gimbutas and Rokhlin - SISC 2010.

        \only<3->{
          \vspace{0.75em}
          \item Different rule for each nested annular region (up to $10^{-6}$ from source). %(and different accuracy tolerance $\epsilon$).

          \vspace{0.5em}
          \!\!\!\!\!$\sim 48$ quadrature nodes for $n_0 = 8$ ~and~ 10-digits accuracy. \\
          \!\!\!\!\!$\sim 26M$ modal Green's function evaluations/sec/core (Skylake 2.4GHz)
        }
        \end{itemize}
      }
    \end{itemize}


  \end{FIframe} %>>>

  %\begin{FIframe}{Fast Modal Green's Function Evaluation}{} %<<<

  %  \vspace{-1.5em}
  %  \begin{center} %<<<
  %  \begin{tikzpicture}
  %    \path [draw=none,fill=blue!30,even odd rule] (4,0) circle (1.5) (4,0) circle (0.75);
  %    \only<2->{\path [draw=none,fill=brown!80,even odd rule] (4,0) circle (0.75) (4,0) circle (0.375);}
  %    \only<2->{\path [draw=none,fill=green!80,even odd rule] (4,0) circle (0.375) (4,0) circle (0.1875);}
  %    \draw[color=red, ultra thick] (0,0) ellipse (4cm and 1cm);
  %    \draw[fill=blue!30,draw=none] (180:0.75)+(4,0) arc (180:0:0.75) -- (0:1.5)+(4,0) arc (0:180:1.5) -- cycle;
  %    \only<2->{\draw[fill=brown!80,draw=none] (180:0.375)+(4,0) arc (180:0:0.375) -- (0:0.75)+(4,0) arc (0:180:0.75) -- cycle;}
  %    \only<2->{\draw[fill=green!80,draw=none] (180:0.1875)+(4,0) arc (180:0:0.1875) -- (0:0.375)+(4,0) arc (0:180:0.375) -- cycle;}

  %    \draw[color=blue, ultra thick] (4.7,-0.8) circle (1pt);
  %    \node at (5, -0.8) {\color{blue} \Large $x$};
  %    \node at (-4.6, 0) {\color{red} \Large $y(\theta)$};
  %    \node at (0, -0.5) {\Large $\theta$};
  %    \node (c) at (-1, -0.75) {};
  %    \node (d) at ( 1, -0.75) {};
  %    \draw[ultra thick, ->] (c)  to [out=-5,in=185, looseness=1] (d);

  %  \end{tikzpicture}
  %  \end{center} %>>>

  %  \vspace{-1em}
  %  \begin{itemize}
  %    \item Build special quadrature rule ${\color{red}(w_i, \theta_i)}$ such that,

  %    \qquad\qquad $\displaystyle \int_{\theta} e^{-in\theta} \mathcal{K}(x-y(\theta))~d\theta ~\approx~ \sum_i {\color{red} w_i} e^{-in\theta_i} \mathcal{K}(x-y({\color{red}\theta_i}))$

  %    for all Fourier modes ($n \leq n_0$) and all targets $x$ in the annulus.

  %    \vspace{1em}
  %    \only<2->{
  %    \item Different rule for each nested annular region (up to $10^{-6}$ from source). %(and different accuracy tolerance $\epsilon$).

  %    \vspace{1em}
  %    \!\!\!\!\!$\sim 48$ quadrature nodes for $n_0 = 8$ ~and~ 10-digits accuracy. \\
  %    \!\!\!\!\!$\sim 26M$ modal Green's function evaluations/sec/core (Skylake 2.4GHz)
  %  }
  %  \end{itemize}

  %\end{FIframe} %>>>

  %\begin{FIframe}{Generalized Chebyshev Quadratures}{} %<<<

  %  \vspace{-2em}
  %  \begin{columns}
  %    \column{0.5\textwidth}

  %    \begin{itemize}
  %      \item Generate several integrands:
  %    \end{itemize}

  %    $\qquad\qquad f_i(\theta) = e^{-i n_{i} \theta} \mathcal{K}(x_i - y(\theta))$

  %    \column{0.5\textwidth}
  %      \begin{center} %<<<
  %        \resizebox{.99\textwidth}{!}{\begin{tikzpicture}
  %        \path [draw=none,fill=blue!30,even odd rule] (4,0) circle (1.5) (4,0) circle (0.75);
  %        \draw[color=red, ultra thick] (0,0) ellipse (4cm and 1cm);
  %        \draw[fill=blue!30,draw=none] (180:0.75)+(4,0) arc (180:0:0.75) -- (0:1.5)+(4,0) arc (0:180:1.5) -- cycle;

  %        \draw[color=blue, ultra thick] (4.7,-0.8) circle (1pt);
  %        \node at (5, -0.8) {\color{blue} \Large $x_i$};
  %        \node at (-4.6, 0) {\color{red} \Large $y(\theta)$};
  %        \node at (0, -0.5) {\Large $\theta$};
  %        \node (c) at (-1, -0.75) {};
  %        \node (d) at ( 1, -0.75) {};
  %        \draw[ultra thick, ->] (c)  to [out=-5,in=185, looseness=1] (d);

  %        \end{tikzpicture}}
  %      \end{center} %>>>
  %  \end{columns}

  %  \only<2->{
  %  \vspace{0.25em}
  %  \begin{itemize}
  %    \item Build an adaptive quadrature rule $(\theta_j, w_j)$ to integrate products $f_i f_k$.

  %    \vspace{0.75em}
  %    \only<3->{\item Set matrix $\displaystyle A_{ij} = f_{i}(\theta_j) \sqrt{w_j}$ ~~ and compute its truncated SVD: ~~$A = U \Sigma V^{*}$.}

  %    \vspace{0.75em}
  %    \only<4->{\item Compute a column pivoted QR decomposition of $V^{*}$.}

  %    \vspace{0.75em}
  %    \only<5->{\item Select nodes corresponding to pivot columns $\{\theta_{j_1}, \cdots, \theta_{j_k}\}$ and \\
  %       solve least squares problem for the quadrature weights.}
  %  \end{itemize}
  %  }

  %  \vspace{1em}
  %  \only<6>{
  %    $\approx 48$ quadrature nodes for $n_0 = 8$ ~and~ 10-digits accuracy. \\
  %    $\approx 13M$ (complex) modal Green's function evaluations/sec/core (Skylake 2.4GHz)
  %  }

  %  % Chebyshev quadrature algorithm
  %  % modal green's function evaluation rate
  %\end{FIframe} %>>>


  \begin{FIframe}{Quadratures for Outer Integral}{} %<<<
    {\bf Near Interactions:} $x$ is off-surface or adjacent panel
    \begin{itemize}
      \item panel (Gauss-Lengendre) quadrature with dyadic refienement.
    \end{itemize}

    \begin{tikzpicture}%<<<
      \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.99\textwidth]{figs/s-quad/surf.png}};
      \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.99\textwidth]{figs/s-quad/adap-quad.png}};

      \draw[color=red, ultra thick] (2.7,0.5) circle (1pt);
      \node at (2.7, 0.25) {\color{red} \Large $x$};

      \draw[ultra thick, ->] (7.1,0.1) to (5.5,0.3);
      \node [rotate=-4.5] at (9.5, -0.17) {dyadic ref. GL panel quad};
    \end{tikzpicture}%>>>

    \only<2->{
      \vspace{1em}
      {\bf Singular Interactions:} $x$ is on-surface

      \only<2>{\begin{tikzpicture}%<<<
        \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.99\textwidth]{figs/s-quad/adap-quad.png}};

        \draw[color=red, ultra thick] (2.7,0.9) circle (1pt);
        \node at (2.5, 0.5) {\color{red} \Large $x$};

        \draw[ultra thick, ->] (3.2,-0.0) to (2.8,0.65);
        \node [rotate=0] at (3.5, -0.25) {special quadrature};
        \node [rotate=0] at (3.5, -0.70) {for $p(s) \log(s) + q(s)$};

        \draw[ultra thick, ->] (7.1,0.1) to (5.5,0.3);
        \node [rotate=-4.5] at (9.5, -0.17) {dyadic ref. GL panel quad};
      \end{tikzpicture}}%>>>
      \only<3->{%<<<
        \begin{tikzpicture}%<<<
          \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.99\textwidth]{figs/s-quad/special-quad.png}};
          \draw[color=red, ultra thick] (2.49,0.89) circle (1pt);
          \node at (2.4, 0.45) {\color{red} \Large $x$};
        \end{tikzpicture}%>>>

        {\em Instead build special quadrature rules!}
        \begin{itemize}
          \item replace composite panel quadratures with a single quadrature.
          %\item integrand doesn't have closed form expression, but we can still generate quadrature rules!
          \item Separate rules for different aspect ratios ($1$ -- $10^4$ in powers of 2)
        \end{itemize}
      }%>>>
    }

    % speedup over adaptive quadrature
  \end{FIframe} %>>>

  %\begin{FIframe}{Overall Algorithm}{} %<<<
  %  %TODO: Summary

  %  \vspace{1em}
  %  {\bf Discretization:} piecewise polynomial $\times$ Fourier.

  %  \vspace{1em}
  %  {\bf Far-field interactions:} standard quadratures (GL $\times$ PTR) + FMM

  %  \vspace{1em}
  %  {\bf Near interactions:}
  %  \begin{itemize}
  %    \item special quadratures for modal Green's function and singular integral in $s$.
  %    \item dyadic refined Gauss-Legendre quadrature in $s$ for non-singular case.
  %    \item build local correction matrix instead of computing on-the-fly.
  %  \end{itemize}

  %\end{FIframe} %>>>


  %\begin{frame} %<<<
  %  \centering
  %  \huge Numerical Results
  %\end{frame} %>>>

  %\begin{FIframe}{Numerical Results - comparison with BIEST}{} %<<<
  %  \begin{columns}
  %    \column{0.5\textwidth}

  %    {\bf Green's identity (Laplace):}

  %    $\Delta u = 0$, ~~ then for ~~ $x \in \Gamma$,

  %    \vspace{-1em}
  %    \[  u(x) = \frac{u(x)}{2} + \StokesSL[\partial_{n} u](x) - \StokesDL[u](x) \]

  %    \column{0.5\textwidth}
  %      \begin{tikzpicture}
  %        \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.99\textwidth]{figs/biest-conv}}; % R0 = 2, r = 0.5
  %        \draw [red, ultra thick, ->|](1.15,3.35) -- (1.30,2.99);
  %        \node at (1.65, 2.50) {\color{red} $0.5$};
  %        \draw [red, ultra thick, ->|](2.13,1.65) -- (1.98,2.01);

  %        \draw [red, ultra thick, ->](3.5,1.9) -- (6.1,1.9);
  %        \node at (4.25, 2.12) {\color{red} $1.0$};
  %      \end{tikzpicture}
  %  \end{columns}

  %  \only<1>{
  %    \vspace{1em}
  %    {\bf Boundary Integral Equation Solver for Taylor States (BIEST)\footnotemark}
  %    \begin{itemize}
  %      \item quadrature for general toroidal surfaces with uniform grid.
  %      \item partition-of-unity to separate singular part of boundary integral.
  %      \item polar coordinate transform for singular integral.
  %    \end{itemize}
  %  }
  %  \only<2>{
  %  \centering
  %  \vspace{1em}
  %  \begin{tabular}{r r r r | c r r r r}
  %    \hline
  %    \multicolumn{4}{c|}{Slender-body Quadrature} & \multicolumn{5}{c}{BIEST\footnotemark} \\
  %    $N$  & $\left\|e\right\|_{\infty}$ & $T_{setup}$ & $T_{eval}$    & $~$ &   $N$ &    $\left\|e\right\|_{\infty}$ & $T_{setup}$ & $T_{eval}$ \\
  %    \hline
  %    320  & 1.5e-04 &       0.032 &        0.0004 & $~$ &   507 &    2.0e-03 &      0.1319 &     0.0017 \\
  %    720  & 3.5e-06 &       0.094 &        0.0013 & $~$ &  1323 &    4.0e-06 &      1.4884 &     0.0042 \\
  %    1280 & 5.4e-09 &       0.228 &        0.0033 & $~$ &  2523 &    4.3e-09 &      6.6825 &     0.0313 \\
  %    2000 & 2.5e-10 &       0.501 &        0.0079 & $~$ &  4107 &    3.5e-10 &     15.4711 &     0.0862 \\
  %    \hline
  %  \end{tabular}
  %  }

  %  \footnotetext[1]{JCP 2019 - Malhotra, Cerfon, Imbert-Gérard, O'Neil ({\href{https://github.com/dmalhotra/BIEST}{\textcolor{blue}{https://github.com/dmalhotra/BIEST}}})}

  %   % Slenderbody - Green's identity test
  %   %       N             Setup         Eval          Error    Nelem    FourierOrder
  %   %      80        3.8000e-03   2.0000e-04    1.55153e-01        2               4
  %   %      80        5.8000e-03   2.0000e-04    5.71073e-03        2               4
  %   %     320        2.4900e-02   4.0000e-04    6.80325e-03        4               8
  %   %     320        3.1500e-02   4.0000e-04    1.57426e-04        4               8
  %   %     720        8.0700e-02   1.3000e-03    2.77260e-05        6              12
  %   %     720        9.3700e-02   1.3000e-03    3.52629e-06        6              12
  %   %     720        1.1270e-01   1.3000e-03    3.25405e-07        6              12
  %   %    1280        2.2890e-01   3.3000e-03    5.48754e-09        8              16
  %   %    2000        4.3300e-01   7.8000e-03    3.79014e-10       10              20
  %   %    2000        5.0100e-01   7.9000e-03    2.57239e-10       10              20
  %   %    2880        8.5870e-01   1.6100e-02    2.73956e-10       12              24
  %   %    3920        1.3213e+00   2.9200e-02    3.88062e-10       14              28
  %   %    5120        2.0569e+00   4.8900e-02    6.05052e-10       16              32
  %   %    5120        2.5004e+00   5.0600e-02    6.57478e-10       16              32

  %   % BIEST - Green's identity test
  %   %    N    T_setup    T_eval      Error   M    q   N1   N2
  %   %  507     0.0677    0.0017    6.7e-03   6    4   39   13
  %   %  507     0.0669    0.0017    6.7e-03   6    4   39   13
  %   %  507     0.1319    0.0017    2.0e-03   6    6   39   13
  %   %  507     0.3398    0.0016    5.1e-05   6   10   39   13
  %   %  867     0.4813    0.0017    4.1e-05   6   12   51   12
  %   % 1323     1.4884    0.0042    4.0e-06   8   16   63   17
  %   % 1875     4.0895    0.0177    8.7e-08  12   18   75   25
  %   % 2523     6.6825    0.0313    4.3e-09  14   20   87   29
  %   % 3267    10.4136    0.0581    1.1e-09  16   22   99   33
  %   % 4107    15.4711    0.0862    3.5e-10  18   24  111   37
  %   % 5043    22.0902    0.1253    1.0e-10  20   26  123   41
  %   % 6075    30.8523    0.1972    4.1e-11  22   28  135   45

  %\end{FIframe} %>>>


  %%\begin{FIframe}{Numerical Results}{} %<<<

  %%  \vspace{-1.5em}
  %%  \begin{columns}[t]
  %%    \column{0.66\textwidth}

  %%      \embedvideo{\includegraphics[width=0.99\textwidth]{videos/tangle}}{videos/tangle.mov}

  %%      \vspace{-1ex}
  %%      \begin{columns}
  %%        \column{0.59\textwidth}
  %%      Exterior Laplace BVP:
  %%
  %%      \quad $\displaystyle \Delta u = 0, \quad u |_{\Gamma} = 1,$

  %%      \quad $\displaystyle u(x) \rightarrow 0 ~\text{as}~ |x|\rightarrow 0$

  %%        \column{0.39\textwidth}

  %%        wire radius = \\
  %%        ~~1.5e-3~to~4e-3

  %%        \vspace{1ex}
  %%        wire length = 16

  %%      \end{columns}

  %%    \column{0.33\textwidth}

  %%      \includegraphics[width=0.99\textwidth]{figs/tangle-cross-section-potential-laplace.png}

  %%      \vspace{1ex}
  %%      \includegraphics[width=0.99\textwidth]{figs/tangle-cross-section-error-laplace.png}

  %%  \end{columns}

  %%  % Geometry = Tangle
  %%  % points / s / core
  %%  % with fourier order
  %%  % Stokes and Laplace
  %%  % with different accuracy
  %%  % BVP-solve
  %%\end{FIframe} %>>>

  %%\begin{FIframe}{Numerical Results - Laplace BVP}{} %<<<

  %%  \vspace{-1em}
  %%  \begin{columns}
  %%    \column{0.23\textwidth}
  %%      \quad$\displaystyle \Delta u = 0$

  %%      \quad$\displaystyle u |_{\Gamma} = 1$

  %%      \vspace{1ex}
  %%      \quad $\displaystyle u(x) \rightarrow 0$ \\
  %%      \quad as~ $\displaystyle |x|\rightarrow \infty$

  %%    \column{0.76\textwidth}
  %%      %\includegraphics[width=0.56\textwidth]{figs/tangle}
  %%      \includegraphics[width=0.49\textwidth]{figs/tangle-cross-section-potential-laplace.png}
  %%      \includegraphics[width=0.49\textwidth]{figs/tangle-cross-section-error-laplace.png}
  %%  \end{columns}


  %%  \resizebox{1.05\textwidth}{!}{\begin{tabular}{r r r r | r r | r r | r r}
  %%    \hline
  %%          &              &              &                         &            &                &   \multicolumn{2}{c |}{1-core} & \multicolumn{2}{c }{40-cores} \\
  %%    $N$   &  $N_{panel}$ & $N_{\theta}$ &   $\epsilon_{_{GMRES}}$ & $N_{iter}$ &  $\left\|e\right\|_{\infty}$ &   $T_{setup}~~(N/T_{setup})$ &  $T_{solve}$ &   $T_{setup}$ & $T_{solve}$ \\
  %%    \hline
  %%    2.8e3 &   70         &            4 &                   1e-02 &          4 &        4.2e-02 &         0.13   ~~~~~~(2.1e4) &         0.03 &         0.020 &       0.013 \\
  %%   %4.9e3 &  122         &            4 &                   1e-03 &          7 &        4.9e-03 &         0.23   ~~~~~~(2.1e4) &         0.16 &         0.020 &       0.027 \\
  %%    1.4e4 &  172         &            8 &                   1e-04 &         10 &        1.0e-03 &         0.72   ~~~~~~(1.9e4) &         1.81 &         0.051 &       0.094 \\
  %%    3.0e4 &  252         &           12 &                   1e-05 &         14 &        3.1e-05 &         1.82   ~~~~~~(1.6e4) &        12.25 &         0.091 &       2.527 \\
  %%    3.1e4 &  262         &           12 &                   1e-07 &         20 &        2.4e-07 &         2.47   ~~~~~~(1.2e4) &        18.97 &         0.213 &       4.239 \\
  %%    6.5e4 &  272         &           24 &                   1e-09 &         28 &        1.1e-09 &         7.74   ~~~~~~(8.4e3) &       114.05 &         0.325 &       7.136 \\
  %%   %7.7e4 &  276         &           28 &                   1e-11 &         35 &        6.6e-11 &        11.75   ~~~~~~(6.5e3) &       200.05 &         0.539 &      10.690 \\
  %%    \hline
  %%  \end{tabular}}

  %%  % Tangle BVP - Laplace
  %%  %      geom   gmres_tol      tol       N     Nelem     FourierOrder         iter    MaxError       L2-error    T_setup   setup-rate    T_solve    T_setup    T_solve
  %%  % tangle50         1e-2     1e-3    2800        70                4            4      4.2e-2         9.2e-4     0.1302        21505     0.0314     0.0200     0.0131
  %%  % tangle100        1e-3     1e-4    4880       122                4            7      4.9e-3         6.9e-5     0.2338        20873     0.1617     0.0195     0.0272
  %%  % tangle150        1e-4     1e-5   13760       172                8           10      1.0e-3         8.5e-6     0.7216        19069     1.8098     0.0514     0.0940
  %%  % tangle230        1e-5     1e-6   30240       252               12           14      3.1e-5         8.1e-7     1.8162        16650    12.2452     0.0905     2.5270
  %%  % tangle240        1e-7     1e-8   31440       262               12           20      2.4e-7         8.2e-9     2.4693        12732    18.9716     0.2125     4.2385
  %%  % tangle250        1e-9    1e-10   65280       272               24           28      1.1e-9        4.5e-11     7.7427         8431   114.0527     0.3250     7.1356
  %%  % tangle254       1e-11    1e-12   77280       276               28           35     6.6e-11        5.5e-13    11.7547         6574   200.0480     0.5391    10.6896

  %%\end{FIframe} %>>>


  \begin{FIframe}{Numerical Results - Stokes BVP}{} %<<<

    \vspace{-1.5em}
    \embedvideo{\includegraphics[width=0.6\textwidth]{videos/tangle}}{videos/tangle.mov}
    \includegraphics[width=0.39\textwidth]{figs/tangle-stokes-streamlines.png}

    \vspace{-1.0em}
    \begin{columns}[t]
      \column{0.25\textwidth}

        {\bf Exterior Stokes Dirichlet BVP:}

        \quad $\displaystyle \Delta {\bm u} - \nabla p = 0,$

        \quad $\displaystyle \nabla \cdot {\bm u} = 0,$


      \column{0.4\textwidth}

        \vspace{1.2em}
        \quad $\displaystyle {\bm u} |_{\Gamma} = {\bm u_0},$

        \quad $\displaystyle u(x) \rightarrow 0 ~\text{as}~ |x|\rightarrow 0 ,$

      \column{0.33\textwidth}

        \vspace{1.2em}
        wire radius =
        ~1.5e-3~to~4e-3

        \vspace{0.2ex}
        wire length = 16

        %\includegraphics[width=0.99\textwidth]{figs/tangle-stokes-streamlines.png}

        %\vspace{1ex}
        %\includegraphics[width=0.99\textwidth]{figs/tangle-cross-section-error-stokes.png}

    \end{columns}


    \vspace{1.0em}
    {\bf BIE formulation:}\quad
    $
    \displaystyle (I/2 + D + S~/~({\color{red}2 \varepsilon \log \varepsilon^{-1}}) ) \, {\bm \sigma} = {\bm u_0}
    $


    % Geometry = Tangle
    % points / s / core
    % with fourier order
    % Stokes and Laplace
    % with different accuracy
    % BVP-solve
  \end{FIframe} %>>>

  \begin{FIframe}{Numerical Results - Stokes BVP}{} %<<<

    \centering
    \vspace{-1.5em}
    \includegraphics[width=0.35\textwidth]{figs/tangle-stokes-streamlines.png}
    \hspace{5em}
    \includegraphics[width=0.40\textwidth]{figs/tangle-cross-section-error-stokes.png}

    \resizebox{1.05\textwidth}{!}{\begin{tabular}{r r r r | r r | r r | r r}
      \hline
            &              &              &                         &            &                &   \multicolumn{2}{c |}{1-core} & \multicolumn{2}{c }{40-cores} \\
      $N$   &  $N_{panel}$ & $N_{\theta}$ &   $\epsilon_{_{GMRES}}$ & $N_{iter}$ &  $\left\|e\right\|_{\infty}$ &   $T_{setup}~~(N/T_{setup})$ &  $T_{solve}$ &   $T_{setup}$ & $T_{solve}$ \\
      \hline
     %8.4e3 &  70          &            4 &                   1e-02 &          6 &        2.1e-01 &         0.18   ~~~~~~(4.5e4) &          0.1 &         0.024 &       0.02 \\
      1.5e4 &  122         &            4 &                   1e-03 &         10 &        1.9e-02 &         0.33   ~~~~~~(4.4e4) &          0.7 &         0.024 &       0.05 \\
     %4.1e4 &  172         &            8 &                   1e-04 &         16 &        1.7e-02 &         1.22   ~~~~~~(3.3e4) &          9.8 &         0.077 &       1.84 \\
      9.1e4 &  252         &           12 &                   1e-05 &         21 &        1.7e-04 &         3.31   ~~~~~~(2.7e4) &         61.2 &         0.197 &       5.25 \\
      9.4e4 &  262         &           12 &                   1e-07 &         33 &        4.1e-06 &         4.43   ~~~~~~(2.1e4) &        104.3 &         0.224 &       7.69 \\
      2.0e5 &  272         &           24 &                   1e-09 &         43 &        1.4e-08 &        17.70   ~~~~~~(1.1e4) &        586.0 &         0.796 &      22.94 \\
      2.3e5 &  276         &           28 &                   1e-11 &         54 &        4.1e-09 &        27.67   ~~~~~~(8.4e3) &       1034.2 &         1.229 &      38.85 \\
      \hline
    \end{tabular}}

    % Tangle BVP - Stokes
    %      geom   gmres_tol      tol       N     Nelem     FourierOrder         iter    MaxError       L2-error    T_setup   setup-rate    T_solve    T_setup    T_solve
    % tangle50         1e-2     1e-3    8400        70                4            6     2.1e-01        2.2e-03     0.1856        45259     0.1589     0.0248     0.0234
    % tangle100        1e-3     1e-4   14640       122                4           10     1.9e-02        1.0e-04     0.3313        44190     0.7745     0.0243     0.0565
    % tangle150        1e-4     1e-5   41280       172                8           16     1.7e-02        1.6e-05     1.2295        33575     9.8059     0.0770     1.8448
    % tangle230        1e-5     1e-6   90720       252               12           21     1.7e-04        9.0e-07     3.3138        27376    61.2092     0.1975     5.2584
    % tangle240        1e-7     1e-8   94320       262               12           33     4.1e-06        7.6e-09     4.4355        21265   104.3853     0.2241     7.6990
    % tangle250        1e-9    1e-10  195840       272               24           43     1.4e-08        1.1e-10    17.7085        11059   586.0695     0.7960    22.9405
    % tangle254       1e-11    1e-12  231840       276               28           54     4.1e-09        6.9e-12    27.6771         8377  1034.2305     1.2298    38.8589

  \end{FIframe} %>>>

  \begin{FIframe}{Numerical Results - close-to-touching}{} %<<<

    \centering

    \only<1>{
    \begin{tikzpicture}
        \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=0.9\textwidth]{figs/touching.png}};
        \node[anchor=south west,inner sep=0] at (10,-1.7) {\includegraphics[width=0.4\textwidth]{figs/touching-zoom.png}};
        \draw[red,ultra thick,rounded corners] (5.75,2.55) rectangle (6.65,3.65);

        \draw[red,ultra thick,rounded corners] (10,-1.7) rectangle (14.98,2.95);

        \draw [red, ultra thick, ->|](0.7,0.7) -- (1.03,1.03);
        \draw [red, ultra thick, ->|](1.57,1.57) -- (1.24,1.24);
        \node at (1.75, 1.85) {\color{red} $0.125$};

        \draw [red, ultra thick, ->](3.4,2.9) -- (3.4,0.18);
        \node at (3.8, 1.7) {\color{red}  $1.0$};

        \node at (7.95, 3.3) {\color{red}  gap $= 0.003$};
        \node at (7.7, 2.8) {\color{red}  $N_\theta = 88$};

    \end{tikzpicture}
    }
    \only<2>{
      \includegraphics[width=0.8\textwidth]{figs/close-to-touching-streamlines}
    }

  \end{FIframe} %>>>

  \begin{FIframe}{Numerical Results - close-to-touching}{} %<<<

    \centering

    \includegraphics[width=0.55\textwidth]{figs/touching.png}
    \includegraphics[width=0.4\textwidth]{figs/close-to-touching-streamlines}

    \begin{tabular}{r r | r r | r r | r r}
      \hline
            &                         &            &                &   \multicolumn{2}{c |}{1-core} & \multicolumn{2}{c }{40-cores} \\
      $N$   &   $\epsilon_{_{GMRES}}$ & $N_{iter}$ &  $\left\|e\right\|_{\infty}$ &   $T_{setup}~~(N/T_{setup})$ &  $T_{solve}$ &   $T_{setup}$ & $T_{solve}$ \\
      \hline
     %6.5e4 &                   1e-01 &          2 &        1.3e-01 &          5.6        (1.1e+4) &          3.2 &          0.85 &         0.5 \\
      6.5e4 &                   1e-02 &          4 &        2.1e-02 &          8.1        (8.0e+3) &          6.5 &          1.28 &         1.4 \\
     %6.5e4 &                   1e-03 &          7 &        1.6e-02 &         10.8        (6.0e+3) &         11.8 &          1.73 &         2.3 \\
     %6.5e4 &                   1e-04 &         13 &        9.3e-03 &         13.6        (4.7e+3) &         22.6 &          2.13 &         4.8 \\
      6.5e4 &                   1e-05 &         24 &        2.4e-03 &         16.8        (3.8e+3) &         42.9 &          2.50 &         7.7 \\
     %6.5e4 &                   1e-06 &         34 &        3.4e-05 &         19.9        (3.2e+3) &         62.5 &          2.80 &        10.9 \\
      6.5e4 &                   1e-07 &         43 &        2.8e-06 &         23.5        (2.7e+3) &         81.6 &          3.31 &        12.8 \\
     %6.5e4 &                   1e-08 &         49 &        2.6e-07 &         27.4        (2.3e+3) &         96.2 &          3.72 &        14.8 \\
     %6.5e4 &                   1e-09 &         54 &        9.3e-08 &         31.4        (2.1e+3) &        109.3 &          3.91 &        16.3 \\
      6.5e4 &                   1e-10 &         59 &        5.4e-08 &         35.6        (1.8e+3) &        122.9 &          4.06 &        19.2 \\
     %6.5e4 &                   1e-11 &         64 &        5.0e-09 &         40.5        (1.6e+3) &        137.1 &          4.56 &        20.2 \\
     %6.5e4 &                   1e-12 &         69 &        5.0e-10 &         45.6        (1.4e+3) &        152.2 &          5.00 &        22.3 \\
      6.5e4 &                   1e-13 &         72 &        1.3e-10 &         49.9        (1.3e+3) &        162.6 &          5.27 &        23.2 \\
      \hline
    \end{tabular}

    %      N  gmres_tol      tol      iter    MaxError       L2-error    T_setup   setup-rate    T_solve    T_setup    T_solve
    %  64560      1e-01     1e-2         2     1.3e-01        3.2e-02     5.6700                  3.1944     0.8531     0.4806
    %  64560      1e-02     1e-3         4     2.1e-02        2.5e-03     8.1061                  6.5360     1.2818     1.3614
    %  64560      1e-03     1e-4         7     1.6e-02        3.1e-04    10.8099                 11.8118     1.7274     2.2869
    %  64560      1e-04     1e-5        13     9.3e-03        2.4e-05    13.6997                 22.5707     2.1291     4.8351
    %  64560      1e-05     1e-6        24     2.4e-03        3.7e-06    16.8026                 42.8992     2.5001     7.6538
    %  64560      1e-06     1e-7        34     3.4e-05        2.2e-07    19.9488                 62.5492     2.8044    10.8931
    %  64560      1e-07     1e-8        43     2.8e-06        1.4e-08    23.5213                 81.6355     3.3077    12.7662
    %  64560      1e-08     1e-9        49     2.6e-07        1.9e-09    27.4751                 96.2095     3.7236    14.7706
    %  64560      1e-09    1e-10        54     9.3e-08        5.5e-10    31.4113                109.2922     3.9118    16.2876
    %  64560      1e-10    1e-11        59     5.4e-08        2.3e-10    35.6971                122.8530     4.0588    19.2035
    %  64560      1e-11    1e-12        64     5.0e-09        2.2e-11    40.5914                137.0600     4.5563    20.2282
    %  64560      1e-12    1e-13        69     5.0e-10        2.5e-12    45.6508                152.2238     4.9972    22.3425
    %  64560      1e-13    1e-14        72     1.3e-10        1.5e-12    49.9494                162.6172     5.2653    23.2362

  \end{FIframe} %>>>

  %>>>


  \section{Mobility problem} %<<<


    \begin{FIframe}{Mobility problem}{} %<<<

      \vspace{-1em}
      \begin{columns}
        \column{0.6\textwidth}

        \begin{itemize}

          \item $n$ rigid bodies ~~$\Omega = \sum\limits_{i=1}^{n} \Omega_i$

            \only<1>{
            with velocities ~$\vct{V}(\vct{x}) = \vct{v}_i + \vct{\omega}_i \times (\vct{x}-\vct{x}^c_i)$,
            }%
            \only<2>{
              with velocities ~{\color{red}$\vct{V}(\vct{x}) = \vct{v}_i + \vct{\omega}_i \times (\vct{x}-\vct{x}^c_i)$},
            }

            \vspace{0.8ex}
            and given forces $\vct{F}_i$, ~torques $\vct{T}_i$ abount $\vct{x}^c_i$.

            \vspace{1.4em}
          \item Stokesian fluid in  $\Real^3 \setminus \Omega$

            \vspace{0.7ex}
            \qquad $\displaystyle \Delta \vct{u} - \nabla p = 0, ~~\nabla \cdot \vct{u} = 0,$ \\

            \vspace{0.6ex}
            \qquad $\displaystyle \vct{u} \rightarrow 0$ ~as~ $\vct{x} \rightarrow \infty$.

            \vspace{1.3em}
          \item Boundary conditions on $\partial\Omega$,

            \vspace{0.6ex}
            \only<1>{\qquad $\displaystyle \vct{u} = \vct{V} + \vct{u}_s$.}
            \only<2>{\qquad $\displaystyle \vct{u} = {\color{red}\vct{V}} + \vct{u}_s$.}

        \end{itemize}

        \vspace{1em}
        \qquad\quad
        \only<1>{\phantom{\color{red} unknown: $\vct{V}(\vct{u}_i, \vct{\omega}_i)$}}
        \only<2>{\color{red} unknown: $\vct{V}(\vct{u}_i, \vct{\omega}_i)$}

        \column{0.4\textwidth}
          \resizebox{0.98\textwidth}{!}{\begin{tikzpicture}
            \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[angle=90,origin=c,width=4cm]{figs/rigid-bodies.png}};

            \draw[ultra thick, ->] (2.19,0.95) to (3,1.5);
            \node at (3.25, 1.5) {$\vct{F}_1$};

            \node (a) at (2.0, 1.3) {};
            \node (b) at (2.08, 1.3) {};
            \draw[thick, ->] (a)  to [out=140,in=60, looseness=3] (b);
            \draw[ultra thick, ->] (2.1,1) to (1.85,2.15);
            \node at (1.85, 2.3) {$\vct{T}_1$};

            %\draw[color=red, ultra thick] (2.7,0.9) circle (1pt);
            %\node at (2.5, 0.5) {\color{red} \Large $x$};

            %\draw[ultra thick, ->] (4.3,0.45) to (3.1,0.5);
            %\node [rotate=-6] at (5.55, 0.25) {log singularity};

            %\draw[ultra thick, ->] (10.5,-0.25) to (12.1,-0.2);
            %\node [rotate=-4.5] at (9.5, -0.17) {$|s-s_0|^{-\alpha}$};
          \end{tikzpicture}}

      \end{columns}

    \end{FIframe} %>>>


    \begin{FIframe}{Mobility problem - double-layer formulation}{} %<<<

      Represent fluid velocity: ~~$\displaystyle \vct{u} = \StokesSL[\vct{\nu}(\vct{F}_i, \vct{T}_i)] + \StokesDL[{\color{red}\vct{\sigma}}] $

      \vspace{0.3em}
      and rigid body velocity: ~~$\displaystyle \vct{V} = -\sum_{i=1}^{6n} \mathfrak{v}_i \mathfrak{v}_i^T {\color{red}\vct{\sigma}}$


      \vspace{1.5em}
      Applying boundary conditions ~ ($\displaystyle \vct{u} = \vct{V} + \vct{u}_s$ ~on~ $\partial\Omega$),

      \vspace{0.3em}
      \qquad$\qquad\displaystyle
      (I/2 + D) \, {\color{red}\vct{\sigma}} + \sum_{i=1}^{6n} \mathfrak{v}_i \mathfrak{v}_i^T {\color{red}\vct{\sigma}} = \vct{u}_s - S \, \vct{\nu}
      $

      \vspace{0.5em}{\em(Pozrikidis - Boundary Integral and Singularity Methods for Linearized Viscous Flow)}


      \vspace{2em}
      \only<2>{
      Second kind integral equation, should be well-conditioned.\\
      What can possibly go wrong?
      }


    \end{FIframe} %>>>

    \begin{FIframe}{Conditioning of layer-potential operators}{} %<<<

      \vspace{1em}
      \begin{columns}
        \column{0.05\textwidth}

        \column{0.35\textwidth}
          %\quad$\displaystyle \nabla^2 u - \nabla p = 0$ ~~in~~ $\mathbb{R}^3 \setminus \Omega$

          %\quad$\displaystyle u |_{\Gamma} = u_0$

          %\quad $\displaystyle u(x) \rightarrow 0 ~\text{as}~ |x|\rightarrow \infty$

          \vspace{1em}
          $\kappa(S)$ \hfill $\sim 2.6e6$

          \vspace{1em}
          $\kappa(I/2 + D)$ \hfill $\sim 4.3e6$

          \vspace{1em}
          $\kappa(I/2 + D + 16 S)$ \hfill $\sim 80$

        \column{0.1\textwidth}

        \column{0.50\textwidth}
            \includegraphics[width=0.99\textwidth]{figs/slender-torus}
      \end{columns}

      \vspace{3em}
      \begin{itemize}
        \item For infinite cylinder (Laplace case): ~~ $\kappa(I/2 + D) ~\sim~ 1/(\varepsilon^{2} \log \varepsilon^{-1})$

          \vspace{0.5em}
        \item Combined field operator well-conditioned: ~~ $I/2 + D + S ~/~ (2\varepsilon \log \varepsilon^{-1})$
      \end{itemize}

    \end{FIframe} %>>>

    %\begin{FIframe}{Boundary Integral Formulation}{Dirichlet BVP} %<<<

    %  \begin{columns}
    %    \column{0.5\textwidth}
    %      \quad$\displaystyle \nabla^2 u - \nabla p = 0$ ~~in~~ $\mathbb{R}^3 \setminus \Omega$

    %      \quad$\displaystyle u |_{\Gamma} = u_0$

    %      \quad $\displaystyle u(x) \rightarrow 0 ~\text{as}~ |x|\rightarrow \infty$

    %    \column{0.5\textwidth}
    %        \includegraphics[width=0.99\textwidth]{figs/biest-conv}
    %  \end{columns}

    %  {\bf Integral equation formulation:}

    %  \begin{columns}
    %    \column{0.75\textwidth}

    %      \vspace{1em}
    %      $u = \frac{\sigma}{2} + \StokesDL[\sigma] \text{~~~~on~~~~} \Gamma$ \hfill $\kappa \sim 324$

    %      \vspace{1em}
    %      $u = \StokesSL[\sigma] \text{~~~~on~~~~} \Gamma$ \hfill $\kappa \sim 651$

    %      \vspace{1em}
    %      $u = \frac{\sigma}{2} + \StokesDL[\sigma] + \StokesSL[\sigma] \text{~~~~on~~~~} \Gamma$ \hfill $\kappa \sim 9$

    %  \end{columns}

    %\end{FIframe} %>>>

    \begin{FIframe}{Mobility problem - combined field formulation}{} %<<<

      \only<1>{
      Represent fluid velocity: ~~$\displaystyle \vct{u} = \StokesSL[\vct{\nu}(\vct{F}_i, \vct{T}_i)] + \StokesCF[{\color{red}\vct{\sigma}} ] $
      }
      \only<2->{
        \vspace{-0.7em}
      Represent fluid velocity: ~~$\displaystyle \vct{u} = \StokesSL[\vct{\nu}(\vct{F}_i, \vct{T}_i)] + \StokesCF[{\color{red}\vct{\sigma}} - \sum_{i=1}^{6n} \mathfrak{v}_i \mathfrak{v}_i^T {\color{red}\vct{\sigma}} ] $
      }

      \only<1>{\vspace{0.5em}}%
      \only<2->{\vspace{-0.65em}}%
      and rigid body velocity: ~~$\displaystyle \vct{V} = -\sum_{i=1}^{6n} \mathfrak{v}_i \mathfrak{v}_i^T {\color{red}\vct{\sigma}}$

      \vspace{0.3em}
      where, ~$\displaystyle \StokesCF =  \StokesDL + \StokesSL / (2 \varepsilon \log \varepsilon^{-1}) $.


      \only<3>{
      \vspace{2.5em}
      Applying boundary conditions,

      \vspace{0.3em}
      \qquad$\qquad\displaystyle
      (I/2 + K) \, \left( {\color{red}\vct{\sigma}} - \sum_{i=1}^{6n} \mathfrak{v}_i \mathfrak{v}_i^T {\color{red}\vct{\sigma}} \right) + \sum_{i=1}^{6n} \mathfrak{v}_i \mathfrak{v}_i^T {\color{red}\vct{\sigma}} = \vct{u}_s - S \, \vct{\nu}
      $

      \vspace{1em}
      Second kind integral equation and  well-conditioned!
      }

    \end{FIframe} %>>>

    \begin{FIframe}{Numerical Results - Sedimentation Flow}{} %<<<

      \vspace{-1.5em}
      \begin{columns}[t]
        \column{0.7\textwidth}

          \vspace{1ex}
          {\bf Time-stepping:} 5-th order adaptive SDC

          \vspace{1ex}
          {\bf 8-digits accuracy} in quadratures, GMRES solve, \\
          and time-stepping.

          \vspace{1ex}
          {\bf 40 CPU cores}

          \only<2>{
          \embedvideo{\includegraphics[width=0.99\textwidth]{videos/sed2-top-zoom1}}{videos/sed2-top-zoom1.avi}
          }

        \column{0.4\textwidth}
          \vspace{-0.5em}
          \only<1>{
          \embedvideo{\includegraphics[width=0.99\textwidth]{videos/sed2-side-color}}{videos/sed2-side-color.mov}
          }
          \only<2>{
          \includegraphics[width=0.99\textwidth]{videos/sed2-side-color}
          }

      \end{columns}

      %\embedvideo{\includegraphics[width=0.25\textwidth]{videos/mesh}}{videos/mesh.avi}
      % 5-th order SDC time-stepping
      % 7 - digits accuracy
      % adaptively refined mesh

      % block diagonal preconditioner - plot of GMRES iterations
    \end{FIframe} %>>>

    \begin{FIframe}{Numerical Results - Sedimentation Flow}{} %<<<

      \vspace{-2.0em}
      \begin{center}
      \resizebox{0.85\textwidth}{!}{\begin{tikzpicture}
        \pgfplotsset{
          xmin=0, xmax=250,
          width=12cm, height=7cm,
          xlabel={$T$}, xtick distance=50,
        }
        \begin{axis}[ymin=0, ymax=110, ylabel={$N_{iter}$}, legend style={draw=none,at={(0,1)},anchor=north west}]
          \addplot [thick,color=blue] table [x={t},y={noprecond}] {data/sed2}; \addlegendentry{no-preconditioner};
          \addplot [thick,color=red] table [x={t},y={precond}] {data/sed2}; \addlegendentry{block-preconditioner};
          %\addplot [thick,color=green] table [x={t},y={KSPprecond}] {data/sed2}; \addlegendentry{Krylov-preconditioner};
        \end{axis}

        \begin{axis}[axis y line*=right, ymin=0, ymax=65000, ylabel={$N$}, legend style={draw=none,at={(0.97,0.97)},anchor=north east}]
          \addplot [thick,dashed,color=black] table [x={t},y={N}] {data/sed2}; \addlegendentry{$N$};
        \end{axis}
      \end{tikzpicture}}
      \end{center}

      \vspace{-1em}
      {\bf Close-to-touching:} ~~smaller time-steps, ~~more unknowns ($N$), \\
      high GMRES iteration count (one-body preconditioner doesn't help). \\
      {\color{red} $\sim 125 \times$ more expensive!}
    \end{FIframe} %>>>



    \begin{FIframe}{Accelerating GMRES Solves}{} %<<<

      \vspace{0.5em}
      \resizebox{.9\textwidth}{!}{\begin{tikzpicture}%<<<
        % draw horizontal line
        \draw[ultra thick, ->] (0,0) -- (14,0);

        % draw vertical lines
        \foreach \x in {2,4,6,8,10,12}
        \draw[ultra thick] (\x cm,3pt) -- (\x cm,-3pt);

        % draw node
        \draw[ultra thick] ( 4,0) node[below=3pt] {$t_{n-2}$};
        \draw[ultra thick] ( 6,0) node[below=3pt] {$t_{n-1}$};
        \draw[ultra thick] ( 8,0) node[below=3pt] {$t_{n}$};
        \draw[ultra thick] (10,0) node[below=3pt] {$t_{n+1}$};
      \end{tikzpicture}}%>>>

      \vspace{0.5em}
      \begin{itemize}
        \setlength\itemsep{0.5em}
        \item Forward Euler: ~$n\text{-}{th}$~ time step
          \begin{itemize}
            \item solve BIE using GMRES:~~ $A_{y_n} \sigma_{\!n} = b_{y_n}$
            \item advance to $t_{n+1}$:~~~ $y_{n+1} = y_{n} + h\, v(\sigma_{\!n})$
          \end{itemize}
        \only<2->{\item Use ~$\sigma_{\!n-1}$~ as initial guess to GMRES}
        \only<3->{: {\color{red} doesn't work well}}
      \end{itemize}

      \only<4->{
      \vspace{0.5em}
      \begin{itemize}
        \item Re-use Krylov subspace from previous time step?

        \only<5->{
        \begin{itemize}
          \item Krylov subspace: ~~$X \leftarrow [b, ~A b, ~\cdots, ~A^{k\shortminus\!1} b]$

          \item Compute QR decomposition: ~ $QR \leftarrow AX$

          \item Preconditioner: ~ $P \coloneq I - Q Q^{T} + X R^{-1} Q^{T}$

            \vspace{0.5em}
          \item[] \qquad $P \, Ax = x$ \quad for all~~ $x \in span(X)$
          \item[] \qquad ~\, $P \, y = y$  \quad for all~~ $y \perp span(X)$

        \end{itemize}
        }
      \end{itemize}}

    \end{FIframe} %>>>


    \begin{FIframe}{Krylov Preconditioning with SDC}{} %<<<

      \vspace{-1em}
      \resizebox{.9\textwidth}{!}{\begin{tikzpicture}%<<<
        % draw horizontal line
        \draw[ultra thick, ->] (0,0) -- (14,0);

        % draw vertical lines
        \foreach \x in {1.7, 12.3}
        \draw[ultra thick] (\x cm,6pt) -- (\x cm,-6pt);
        \draw[ultra thick] (1.7000 ,0) node[above=5pt] {$a$};
        \draw[ultra thick] (12.3000,0) node[above=5pt] {$b$};

        \foreach \x in {1.9000, 2.6699, 4.5000, 7.0000, 9.5000, 11.3301, 12.1000}
        \draw[ultra thick, blue] (\x cm,3pt) -- (\x cm,-3pt);

        % draw node
        \draw[ultra thick, blue] (1.9000 ,0) node[below=3pt] {$t_1$};
        \draw[ultra thick, blue] (2.6699 ,0) node[below=3pt] {$t_2$};
        \draw[ultra thick, blue] (4.5000 ,0) node[below=3pt] {$t_3$};
        %\draw[ultra thick, blue] (7.0000 ,0) node[below=3pt] {$t_4$};
        %\draw[ultra thick, blue] (9.5000 ,0) node[below=3pt] {$t_5$};
        \draw[ultra thick, blue] (8.5,0) node[below=3pt] {$\cdots$};
        %\draw[ultra thick, blue] (11.3301,0) node[below=3pt] {$t_6$};
        \draw[ultra thick, blue] (12.1000,0) node[below=3pt] {$t_m$};

        \only<2->{%<<<
          \draw[ultra thick, DarkGreen] (1.9000 ,-1) node {$P_1$};
          \draw[ultra thick, DarkGreen] (2.6699 ,-1) node {$P_2$};
          \draw[ultra thick, DarkGreen] (4.5000 ,-1) node {$P_3$};
          \draw[ultra thick, DarkGreen] (8.5000 ,-1) node {$\cdots$};
          \draw[ultra thick, DarkGreen] (12.1000,-1) node {$P_m$};

          \draw[ultra thick, DarkGreen, ->] (2.10 ,-1) -- (2.40 ,-1);
          \draw[ultra thick, DarkGreen, ->] (3.00 ,-1) -- (3.50 ,-1);
          \draw[ultra thick, DarkGreen, ->] (4.86 ,-1) -- (5.40 ,-1);
        }%>>>
        \only<3->{%<<<
          \draw[ultra thick, DarkGreen, ->] (1.9000 ,-1.25) -- (1.9000 ,-1.8);
          \draw[ultra thick, DarkGreen, ->] (2.6699 ,-1.25) -- (2.6699 ,-1.8);
          \draw[ultra thick, DarkGreen, ->] (4.5000 ,-1.25) -- (4.5000 ,-1.8);
          \draw[ultra thick, DarkGreen, ->] (12.100 ,-1.25) -- (12.100 ,-1.8);

          \draw[ultra thick, DarkGreen] (1.9000 ,-2.1) node {~~\,$P^{(1)}_1$};
          \draw[ultra thick, DarkGreen] (2.6699 ,-2.1) node {~~\,$P^{(1)}_2$};
          \draw[ultra thick, DarkGreen] (4.5000 ,-2.1) node {~~\,$P^{(1)}_3$};
          \draw[ultra thick, DarkGreen] (8.5000 ,-2.1) node {$\cdots$};
          \draw[ultra thick, DarkGreen] (12.100 ,-2.1) node {~~\,$P^{(1)}_m$};

          \draw[ultra thick, black, ->] (0.75 ,-0.5) -- (0.75 ,-3.5);
          \draw[ultra thick, black] (0.5,-2) node[rotate=-90] {corrections};
        }%>>>
        \only<4->{%<<<
          \draw[ultra thick, DarkGreen, ->] (1.9000 ,-2.5) -- (1.9000 ,-3.0);
          \draw[ultra thick, DarkGreen, ->] (2.6699 ,-2.5) -- (2.6699 ,-3.0);
          \draw[ultra thick, DarkGreen, ->] (4.5000 ,-2.5) -- (4.5000 ,-3.0);
          \draw[ultra thick, DarkGreen, ->] (12.100 ,-2.5) -- (12.100 ,-3.0);

          \draw[ultra thick, DarkGreen] (1.9000 ,-3.3) node {~~\,$P^{(2)}_1$};
          \draw[ultra thick, DarkGreen] (2.6699 ,-3.3) node {~~\,$P^{(2)}_2$};
          \draw[ultra thick, DarkGreen] (4.5000 ,-3.3) node {~~\,$P^{(2)}_3$};
          \draw[ultra thick, DarkGreen] (8.5000 ,-3.3) node {$\cdots$};
          \draw[ultra thick, DarkGreen] (12.100 ,-3.3) node {~~\,$P^{(2)}_m$};

          %\draw[ultra thick, DarkGreen, ->] (1.9000 ,-3.7) -- (1.9000 ,-4.2);
          %\draw[ultra thick, DarkGreen, ->] (2.6699 ,-3.7) -- (2.6699 ,-4.2);
          %\draw[ultra thick, DarkGreen, ->] (4.5000 ,-3.7) -- (4.5000 ,-4.2);
          %\draw[ultra thick, DarkGreen, ->] (12.100 ,-3.7) -- (12.100 ,-4.2);
        }%>>>

      \end{tikzpicture}}%>>>

      \only<5->{
        \vspace{1.4em}

        \begin{columns}
          \column{0.05\textwidth}
          \column{0.45\textwidth}
            GMRES iter without preconditioner:

            \vspace{0.5em}
            \begin{tabular}{r | r r r r r}
              \parbox[t]{2mm}{\multirow{5}{*}{\rotatebox[origin=c]{-90}{correction $\rightarrow$}}}
                & \multicolumn{5}{c}{sub-step $\rightarrow$}  \\
              \hline
                                      & 66 & 66 & 66 & 66 & 66    \\
                                      & 66 & 66 & 66 & 66 & 66    \\
                                      & 66 & 66 & 66 & 66 & 66    \\
                                      & 66 & 66 & 66 & 66 & 66    \\
            \end{tabular}

          \column{0.05\textwidth}
          \column{0.44\textwidth}
            \only<6->{
            GMRES iter with preconditioner:

            \vspace{0.5em}
            \begin{tabular}{r | r r r r r}
              \parbox[t]{2mm}{\multirow{5}{*}{\rotatebox[origin=c]{-90}{correction $\rightarrow$}}}
                & \multicolumn{5}{c}{sub-step $\rightarrow$}  \\
              \hline
                                      & 66 & 30 & 22 & 45 & 30    \\
                                      & 35 & 17 & 33 & 28 & 24    \\
                                      &  8 &  4 & 14 &  5 & 12    \\
                                      &  1 &  1 &  2 &  2 &  4    \\
            \end{tabular}}

        \end{columns}
      }

    \end{FIframe} %>>>


    \begin{FIframe}{Numerical Results - Sedimentation Flow}{} %<<<

      \vspace{-2.0em}
      \begin{center}
      \resizebox{0.95\textwidth}{!}{\begin{tikzpicture}
        \pgfplotsset{
          xmin=0, xmax=250,
          width=12cm, height=7cm,
          xlabel={$T$}, xtick distance=50,
        }
        \begin{axis}[ymin=0, ymax=110, ylabel={$N_{iter}$}, legend style={draw=none,at={(0,1)},anchor=north west}]
          \addplot [thick,color=blue] table [x={t},y={noprecond}] {data/sed2}; \addlegendentry{no-preconditioner};
          \addplot [thick,color=red] table [x={t},y={precond}] {data/sed2}; \addlegendentry{block-preconditioner};
          \addplot [thick,color=DarkGreen] table [x={t},y={KSPprecond}] {data/sed2}; \addlegendentry{Krylov-preconditioner};
        \end{axis}

        \begin{axis}[axis y line*=right, ymin=0, ymax=65000, ylabel={$N$}, legend style={draw=none,at={(0.97,0.97)},anchor=north east}]
          \addplot [thick,dashed,color=black] table [x={t},y={N}] {data/sed2}; \addlegendentry{$N$};
        \end{axis}
      \end{tikzpicture}}
      \end{center}

      %\vspace{-1em}
      %{\bf Close-to-touching:} ~~smaller time-steps, ~~more unknowns ($N$), \\
      %high GMRES iteration count (block preconditioner doesn't help). \\
      %{\color{red} $\sim 125 \times$ more expensive!}
    \end{FIframe} %>>>
  %>>>


  %<<< Sedimentation flow
    \begin{FIframe}{Numerical Results - Sedimentation Flow}{} %<<<

      \begin{columns}

        \column{0.35\textwidth}

          \vspace{1ex}
          {\bf 5-th order adaptive SDC}

          \vspace{1ex}
          {\bf 8-digits accuracy} in quadratures, GMRES solve, \\
          and time-stepping.

          \vspace{1ex}
          {\bf 0.5 million unknowns} \\
          64 rings.

          \vspace{1ex}
          {\bf 160 CPU cores}

        \column{0.65\textwidth}

        \embedvideo{\includegraphics[width=0.99\textwidth]{videos/sed64}}{videos/sed64.mov}

      \end{columns}

    \end{FIframe} %>>>

    \begin{FIframe}{Numerical Results - Sedimentation Flow}{} %<<<

      \vspace{-1.9em}
      \centering
      \only<1>{ \embedvideo{\includegraphics[width=0.94\textwidth]{videos/bacteria2_.png}}{videos/bacteria2_.mov} }%
      \only<2>{ \embedvideo{\includegraphics[width=0.47\textwidth]{videos/bacteria64-density.png}}{videos/bacteria64-density.mov} }

    \end{FIframe} %>>>
  %>>>

  \section{Software} %<<<

  \begin{frame}[t,fragile] \frametitle{CSBQ library}{} %<<<

    \vspace{1em}
    {\bf Code:} ~~ {\color{blue} \url{https://github.com/dmalhotra/CSBQ}}

    \vspace{1em}
    {\bf Requirements:} ~~ C++11 compiler ~~with~~ OpenMP 4.0

    \vspace{1em}
    {\bf Build system:} ~~ none (header only)

    \vspace{0.5em}
    \begin{minted}[
        %frame=lines,
        fontsize=\footnotesize,
        %linenos,
        gobble=0,
        mathescape
      ]{C++}
               #include <csbq.hpp>
    \end{minted}


    \vspace{3em}
    {\bf Optional dependencies:} ~~ BLAS, LAPACK, MPI, and PVFMM

      %\begin{itemize}
      %\item includes simple `Makefile' for example codes.
      %\end{itemize}

  \end{frame}
  %>>>

  \begin{frame} \frametitle{Library classes}{} %<<<

    \resizebox{1.05\linewidth}{!}{\begin{tikzpicture}

      \node[anchor=north west, draw=none, rounded corners=.45cm, minimum height=7cm, minimum width=15.2cm] at (-0.1,1.6) {};

      \only<3->{
        \node[anchor=north west, draw=blue,thick, rounded corners=.45cm] at (1.0,1.5) {\begin{tabular}{c} \bf Taylor states  \end{tabular}};
        \node[anchor=north west, draw=blue,thick, rounded corners=.45cm] at (5.3,1.5) {\begin{tabular}{c} \bf Vacuum fields  \end{tabular}};
        \node[anchor=north west, draw=blue,thick, rounded corners=.45cm] at (9.8,1.5) {\begin{tabular}{c} \bf Virtual casing     \end{tabular}};

        \draw [line width=1mm] (0,0.25) -- (15.0,0.25);
      }


      \only<2->{
        \node[anchor=north west, draw=red,thick, rounded corners=.55cm] at (0,0) {\small
        \begin{tabular}{c}
          {\bf Surface} \\
          W7X, LHD, \\
          QAS3, etc, \\
          $(\mathbf{X}, \mathbf{Y}, \mathbf{Z})$
        \end{tabular}};

        \node[anchor=north west, draw=red,thick, rounded corners=.55cm] at (2.7,0) {\small
        \begin{tabular}{c}
          {\bf SurfaceOp} \\
          resample, $\mathbf{n}$, \\
          $\nabla_{\Gamma}$,~
          $\nabla_{\Gamma} \cdot$, \\
          $\nabla_{\Gamma} \times$,~
          $\Delta_{\Gamma}$,\\
          $\Delta^{-1}_{\Gamma}$
        \end{tabular}};

        \node[anchor=north west, draw=red,thick, rounded corners=.55cm] at (5.8,0) {\small
        \begin{tabular}{c}
          {\bf BoundaryIntegralOp}\\
          $\int\limits_\Gamma \sigma(\mathbf{r}') K(\mathbf{r}-\mathbf{r}') d a(\mathbf{r}')$
        \end{tabular}};

        \node[anchor=north west, draw=red,thick, rounded corners=.55cm] at (10.9,0) {\small
        \begin{tabular}{c}
          {\bf KernelFunction}\\
            Laplace, Helmholtz, \\
            Biot-Savart, etc.
        \end{tabular}};
      }

      \node[anchor=north west, draw=black,thick, rounded corners=.55cm] at (1,-3.3) {\small
      \begin{tabular}{c}
        {\bf SCTL:} Scientific Computing Template Library\\
        \hspace{30em} \\
        \\
      \end{tabular}};

      \node[anchor=north west, draw=red,thick, rounded corners=.45cm] at  (2,-4.1) {\begin{tabular}{c} \bf Vector \end{tabular}};
      \node[anchor=north west, draw=red,thick, rounded corners=.45cm] at  (5,-4.1) {\begin{tabular}{c} \bf Matrix \end{tabular}};
      \node[anchor=north west, draw=red,thick, rounded corners=.45cm] at  (8,-4.1) {\begin{tabular}{c} \bf GMRES \end{tabular}};
      \node[anchor=north west, draw=red,thick, rounded corners=.45cm] at (11,-4.1) {\begin{tabular}{c} \bf FFT \end{tabular}};

    \end{tikzpicture}}

  \end{frame}%>>>

  %>>>

  \section{Conclusions} %<<<

  \begin{FIframe}{Conclusions}{} %<<<

    %\vspace{1.2em}
    \begin{itemize}
      \item Convergent boundary integral formulation for slender bodies.
        \begin{itemize}
          \item unlike SBT, boundary conditions are actually enforced to high accuracy.
        \end{itemize}
        \vspace{0.5ex}
      \item Special quadratures - efficient for aspect ratios as large as $10^5$.
        \begin{itemize}
          %\item fast computation of modal/toroidal Green's function.
          %\item special (Chebyshev) quadratures for singular integrals along length of fibers.
          \item quadrature setup rates up to $20,\!000$ unknowns/s/core (comparable to FMM speeds).
        \end{itemize}
        \vspace{0.5ex}
      \item Stokes mobility problem - combined field BIE formulation.
        \begin{itemize}
          \item well-conditioned formulation for slender-body geometries.
          \item high-order time stepping (SDC), Krylov subspace preconditioner.
        \end{itemize}
    \end{itemize}

    \only<2>{
    \vspace{1.5em}
    {\bf Limitations and ongoing work:}
    \begin{itemize}
      \item Flexible fibers -- applications in biological fluids.
      \item Open problems: collision handling.

      %\item Open fibers (singularities at ends). %Special elements (and quadratures) for fiber endpoints (non-loop geometries).
      %%\item Replace Chebyshev quadratures with generalized Gaussian quadratures% of Bremer, Gimbutas and Rokhlin - SISC 2010.
      %%\item Parallelisation with proper load balancing.
      %%\item FMM acceleration of far-field computation.
      %%\item Apply to problems in biological fluids.
      %\item Mobility problem and flexible fibers.
      %\item Comparison w/ SBT efficiency when SBT is sufficiently accurate.
    \end{itemize}
    }



    % - end-caps so that we can have non-loop geometries
    % - replace Chebyshev quadratures with Generalized Gaussian Quadratures
    %%%%%%%% - develop preconditioners for close to touching geometries?
    % direct comparison with slender-body theory
    % develop applications ...
    % parallelization



    %\vspace{1em}
    %\textcolor{blue}{\bf Future directions}

    %\vspace{0.5em}
    %\begin{columns}
    %  \column{0.9\textwidth}
    %    \begin{itemize}
    %      \item apply quadratures to numerical simulations of biological processes (collaboration with CCB).
    %    \end{itemize}
    %  \column{0.1\textwidth}
    %\end{columns}

    %\vspace{0.5em}
    %\begin{columns}
    %  \column{0.65\textwidth}
    %    \begin{itemize}
    %      \item study convergence in close-to-touching setups; ~~ require adaptivity in length as well as $\theta$-dimensions.
    %    \end{itemize}
    %  \column{0.35\textwidth}
    %    \includegraphics[width=0.9\textwidth]{figs/close-touching}

    %    {\small (fig from Morse et al.)}
    %\end{columns}

    % - end-caps so that we can have non-loop geometries
    % - replace Chebyshev quadratures with Generalized Gaussian Quadratures
    %%%%%%%% - develop preconditioners for close to touching geometries?
    % direct comparison with slender-body theory
    % develop applications ...
    % parallelization

    %\vspace{0.75em}
    %\begin{columns}
    %  \column{0.9\textwidth}
    %    \begin{itemize}
    %      \item develop similar ideas for other special cases and more generally for high aspect ratio panels

    %      \begin{center}
    %        \includegraphics[width=0.6\textwidth]{slender-body/high-aspect-panels.png}
    %      \end{center}
    %    \end{itemize}
    %  \column{0.1\textwidth}
    %\end{columns}

  \end{FIframe} %>>>

  %>>>


\end{document}