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

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

  \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}{} %<<<

    \only<1>{%<<<
    $\displaystyle u(x) ~= \int_{\Gamma} \mathcal{K}(x-y)~\sigma(y)~da(y)$
    }%>>>
    \only<2->{%<<<
    \vspace{-0.57em}
    $\displaystyle u(x) ~= \int_{\Gamma} \mathcal{K}(x-y)~\sigma(y)~da(y) ~= \sum_{k=1}^{N_{panel}} \int_{\Gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y)$
    }%>>>

      \only<3->{%<<<
      $\displaystyle \phantom{u(x)} ~= \underbrace{\sum_{x \notin \mathcal{N}(\Gamma_k)} \int_{\Gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y)}_{\text{far-field}}~
      + \underbrace{\sum_{x \in \mathcal{N}(\Gamma_k)} \int_{\Gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y)}_{\text{near interactions}}$
      }%>>>

      \vspace{2em}
      \only<4>{%<<<
        \begin{columns}[t]
          \column{0.5\textwidth}
          {\bf Far-field}
          \begin{itemize}
            \item Tensor product quad:\\
              Gauss-Legendre $\times$ PTR
            \item Accelerate with PVFMM \\
              $\mathcal{O}(N^2) \rightarrow \mathcal{O}(N)$
          \end{itemize}
          \column{0.5\textwidth}
          {\bf Near interactions}
          \begin{itemize}
            \item Build special quadrature rules! \\
              %Generalized Chebyshev or \\
              %Generalized Gaussian
          \end{itemize}
        \end{columns}
      }%>>>

    %\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}{Generating Special Quadrature Rules}{} %<<<

    \resizebox{.99\textwidth}{!}{\begin{tikzpicture}[thick]
      \tikzstyle{wave} = [thick, blue, domain=0:2, samples=200]
      \tikzstyle{box} = [draw, fill=blue!10, rounded corners, minimum width=2.0cm, minimum height=0.9cm, align=center, blur shadow={shadow blur steps=5}]
      \tikzstyle{arrow} = [thick,->,>=stealth, draw=black]
      \tikzstyle{input} = [align=center]

      \draw[wave] plot (\x+0.5,{0.5+0.2*sin(\x*5 r)});
      \draw[wave] plot (\x+0.6,{0.0+0.2*sin(\x*9 r)});
      \draw[wave] plot (\x+0.7,{-0.5+0.2*sin(\x*15 r)});
      \node[input] at (1.6, -1) {Integrands};

      \node[box] (magic) at (5, 0) {Magic};
      \draw[arrow] (2.9,0) -- (magic);
      \draw[arrow] (magic) -- (7.1,0);

      \draw[thick] (7.5,0) -- (9.5,0);
      \node[circle,fill=blue,inner sep=1.2pt] at (7.5,0) {};
      \node[circle,fill=blue,inner sep=1.2pt] at (7.7,0) {};
      \node[circle,fill=blue,inner sep=1.2pt] at (8.1,0) {};
      \node[circle,fill=blue,inner sep=1.2pt] at (8.9,0) {};
      \node[circle,fill=blue,inner sep=1.2pt] at (9.3,0) {};
      \node[circle,fill=blue,inner sep=1.2pt] at (9.5,0) {};
      \node at (9.2,0.25) {\small $x_i$, $w_i$};
      \node[input] at (8.5, -0.8) {Quadrature \\ nodes \& weights};

      \node at (12.8,0.4) {Generalized Gaussian Quad.};
      \node at (12.5,-0.1) {Bremer, Gimbutas and};
      \node at (12.3,-0.6) {Rokhlin - SISC 2010};
    \end{tikzpicture}}

    \vspace{1em}
    \only<2>{
      \centering
      $\displaystyle \int_{\Gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y) ~=~ \int_{s} \int_{\theta} \mathcal{K}(x-y(s,\theta))~\sigma(s,\theta)~J(s,\theta)~d\theta~ds$
      \resizebox{.7\textwidth}{!}{\begin{tikzpicture}
        \begin{scope} [rotate=-90]
          \node[anchor=south west,inner sep=0] at (3.7,0) {\includegraphics[angle=-90,width=9.25cm]{figs/slenderbody-discretization.png}};
          \node (a) at (.7, 6) {};
          \node (b) at (-.0, 6) {};

          \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);
        \node at (1.2, 6.3) {\Large $\theta$};
        \node at (-1.0, 5.5) {\Large $s$};
        \end{scope}
      \end{tikzpicture}}
    }
    \only<3-4>{%<<<
      \centering
      $\displaystyle \int_{\Gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y) ~=~ \int_{s} {\color{red} \int_{\theta}}\, \mathcal{K}(x-y(s,\theta))~\sigma(s,\theta)~J(s,\theta)~{\color{red} d\theta}~ds$

      %\begin{columns}
      %  \column{0.7\textwidth}
          \begin{center} %<<<
            \resizebox{.7\textwidth}{!}{\begin{tikzpicture}
            \path [draw=none,fill=white!0,even odd rule] (4,0) circle (1.5) (4,0) circle (0.75);
            \only<4->{\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<4->{
            \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<4->{\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;}
            %\only<3->{\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 (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{-0.5em}
      \only<4->{\small
          \!\!\!\!\!$\sim 48$ quadrature nodes for $n_0 = 8$ ~and~ 10-digits accuracy. \\
          \!\!\!\!\!$\sim 26M$ modal Green's function evaluations/sec/core (Skylake 2.4GHz)
        }

    }%>>>

    \only<5->{%<<<
      \centering
      $\displaystyle \int_{\Gamma_k} \mathcal{K}(x-y)~\sigma(y)~da(y) ~=~ {\color{red} \int_{s}} \int_{\theta} \mathcal{K}(x-y(s,\theta))~\sigma(s,\theta)~J(s,\theta)~d\theta~{\color{red} ds}$

      \vspace{1em}
      \only<5>{\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<6->{%<<<
        \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}%>>>

        \vspace{1em}
        \begin{itemize}
          \item[] {\em Instead build special quadrature rules!}
          \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}
      }%>>>
    }%>>>

  \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;}
  %        \only<3->{\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 (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} % old results
    %  \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}}
    \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
      1.0e4 &   49         &            8 &                   1e-02 &          5 &        3.5e-02 &         0.193   ~~~~~~(5.4e4) &        0.130 &          0.042 &        0.017 \\
      2.6e4 &  103         &           12 &                   1e-05 &         22 &        5.5e-05 &         0.572   ~~~~~~(4.5e4) &        4.039 &          0.045 &        0.215 \\
      4.7e4 &  157         &           20 &                   1e-07 &         33 &        6.6e-07 &         1.416   ~~~~~~(3.3e4) &       19.518 &          0.134 &        1.162 \\
      8.3e4 &  227         &           24 &                   1e-08 &         38 &        4.5e-08 &         3.623   ~~~~~~(2.3e4) &       78.907 &          0.324 &        3.689 \\
      2.2e5 &  457         &           40 &                   1e-10 &         49 &        2.9e-10 &        21.949   ~~~~~~(1.0e4) &      746.966 &          4.458 &       48.494 \\
     %4.8e5 &  893         &           48 &                   1e-11 &         54 &        2.4e-11 &        84.363   ~~~~~~(5.7e3) &     3788.948 &         15.177 &      227.747 \\
      \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} %>>>

  %>>>