2024-07-15-talk-SciCADE/compression.tex

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

\begin{FIframe}{Problem Setup - Stokes Mobility}{} %<<<

  \vspace{-1em}
  \begin{columns}
    \column{0.7\textwidth}

    \begin{itemize}

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

        given radius $R$, ~centers $\vct{x}^c_i$, ~forces $\vct{F}_i$, ~torques $T_i$,

        \vspace{1.8ex}
        \only<1>{
          velocity ~$\vct{V}(\vct{x}) = \vct{v}_i + \vct{\omega}_i \times (\vct{x}-\vct{x}^c_i)$.
        }%
        \only<2>{
          velocity ~{\color{red}$\vct{V}(\vct{x}) = \vct{v}_i + \vct{\omega}_i \times (\vct{x}-\vct{x}^c_i)$}.
        }

        \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{0.3\textwidth}
      \centering
      \resizebox{0.99\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[color=blue, line width=1pt, fill=gray!50] (2.19,0.975) circle (0.76cm);
        \draw[color=blue, line width=1pt, fill=gray!50] (1.5,-1.4) circle (0.76cm);
        \draw[color=blue, line width=1pt, fill=gray!50] (3.1,-2.0) circle (0.76cm);

        \draw[color=black, line width=1pt, fill=black] (2.19,0.975) circle (0.03cm);
        \node at (2.28,0.72) {$\vct{x}^c_1$};

        \draw[ultra thick, -latex] (2.3,1.07) to (3,1.5);
        \node at (3.35, 1.6) {$\vct{F}_1$};

        %\node (a) at (1.41, 0.975) {};
        %\node (b) at (2.19, 1.755) {};
        %\draw[thick, -latex] (a)  arc [out=140,in=60, looseness=3] (b);
        \draw[-latex, ultra thick] (1.3,0.975) arc
        [
            start angle=180,
            end angle=90,
            x radius=0.85cm,
            y radius=0.85cm
        ] ;
        \node at (1.35, 1.85) {$T_1$};


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

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

        %\draw[ultra thick, -latex] (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}{Boundary Integral Formulation}{} %<<<

  \only<1>{
    Represent fluid velocity: ~~$\displaystyle \vct{u}(\vct{x}) = \int_{\partial\Omega} \!\!\!\! S(\vct{x}-\vct{y}) \vct{\nu}(\vct{y}) + \int_{\partial\Omega} \!\!\!\! D(\vct{x}-\vct{y}) {\color{red}\vct{\sigma}}(\vct{y}) $
  }

  \only<2->{
    \vspace{0.25em}
    Represent fluid velocity: ~~$\displaystyle \vct{u} = \StokesSL[\vct{\nu}(\vct{F}_i, T_i)] + \StokesDL[{\color{red}\vct{\sigma}}] $
  }

  \only<3->{
  \vspace{0.3em}
  and rigid body velocity: ~~$\displaystyle \vct{V} = -\sum_{i=1}^{6n} \mathfrak{v}_i \mathfrak{v}_i^T {\color{red}\vct{\sigma}}$
  }


  \only<4->{
  \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{0.5em}
  %\only<2->{Second kind integral equation \quad {\color{red}  $\cdots$ but doesn't work for slender bodies!}
  % %, should be well-conditioned.\\
  % %What can possibly go wrong?

  % \vspace{1em}
  % \only<3->{\color{red} $\kappa(I/2 + D) ~\sim~ 1/(\varepsilon^{2} \log \varepsilon^{-1})$}
  %}

\end{FIframe} %>>>

\begin{frame}[t,fragile] \frametitle{{Nystr\"om Discretization}} \framesubtitle{{}} %<<<

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      \draw[black, line width=1pt] ({\x+0.96*\xx},{\y+0.96*\yy}) -- ({\x+1.04*\xx},{\y+1.04*\yy});
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  \vspace{-0.8em}
  \resizebox{0.62\textwidth}{!}{\begin{tikzpicture}[scale=0.8]%<<<
    \draw[color=blue, line width=2pt, fill=gray!50] (-4.2,0) circle (4cm);
    \draw[color=blue, line width=2pt, fill=gray!50] ( 4.2,0) circle (4cm);
    \only<2->{
    \draw [red, line width=2pt, domain=-30:30] plot ({ 4*cos(\x)-4.2}, {4*sin(\x)});
    \draw [red, line width=2pt, domain=-30:30] plot ({-4*cos(\x)+4.2}, {4*sin(\x)});
    \draw[rounded corners=1cm,dotted,color=black!50!green, line width=2pt] (-1.5, -2.1) rectangle (1.5, 2.1) {};
    }

    \node at (-5.5, 1.5) {\LARGE $\Omega_k$};
    %\node at (-7.0, 3.6) {\Large $\partial\Omega_k$};


    \drawpanels{-4.2}{0}{4}{30}{330}{8};
    \drawpanels{ 4.2}{0}{4}{-150}{150}{8};

    \drawpanels{-4.2}{0}{4}{-15}{15}{2};
    \drawpanels{-4.2}{0}{4}{-7.5}{7.5}{2};
    \drawpanels{-4.2}{0}{4}{-3.75}{3.75}{2};
    \drawpanels{-4.2}{0}{4}{-1.875}{1.875}{3};
    %\drawpanels{-4.2}{0}{4}{-0.9375}{0.9375}{3};

    \drawpanels{ 4.2}{0}{4}{165}{195}{2};
    \drawpanels{ 4.2}{0}{4}{172.5}{187.5}{2};
    \drawpanels{ 4.2}{0}{4}{176.25}{183.75}{2};
    \drawpanels{ 4.2}{0}{4}{178.125}{181.875}{3};
    %\drawpanels{ 4.2}{0}{4}{179.0625}{180.9375}{3};

    %\draw [dashed, line width=1pt] (-4.2,0) -- (-0.2,0);
    \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-4.2,0) -- (-1.371,2.828);
    %\draw [line width=2pt, domain=0:45] plot ({1*cos(\x)-4.2}, {1*sin(\x)});
    \node at (-3.1, 1.6) {\Large $\radius$};
    \node at (-4.55,0) {\Large $\vct{x}^c_{k}$};

    \draw [dashed, line width=1pt] (-0.2,0) -- (-0.2,-3.55);
    \draw [dashed, line width=1pt] ( 0.2,0) -- ( 0.2,-3.55);
    \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-1.2,-3.45) -- (-0.2,-3.45);
    \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] ( 1.2,-3.45) -- ( 0.2,-3.45);
    \node at (0, -4.0) {\Large $d$};

    %\node at (0, 1.75) {\color{red} \Large $\Gamma_{2}$};
    %\node at (0, 1.25) {\color{red} \Large $\sigma_{2}$};

    %\node at (4.0, 3.5) {\color{blue} \huge $\Gamma_{1}$};
    %\node at (5.0, 3.5) {\color{blue} \Large $\sigma_{1}$};
  \end{tikzpicture}}%>>>
  \resizebox{0.38\textwidth}{!}{\begin{tikzpicture}%<<<
    \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=5cm]{figs/plot-mobility-density_.png}};
    \node at (-0.1, 2.1) {$\vct{\sigma}$};
  \end{tikzpicture}}%>>>

  \begin{columns}
    \begin{column}[T]{0.61\textwidth}
      \begin{itemize}
        \setlength\itemsep{1.5ex}
        \item Discretize $\partial\Omega$ into panels.
        \item Layer-potential operators:
          \begin{itemize}
            \item adaptive quadrature for near integrals
            \item special quadrature for singular integrals
          \end{itemize}
        \item Solve BIE: ~~~\scalebox{1.3}{$K \sigma = g$}
      \end{itemize}
    \end{column}

    \begin{column}[T]{0.39\textwidth}
      \only<2->{
        {\bf
          \color{red}
          \vspace{2em}

          \begin{center}
          Compress close-interactions,

          and interpolate in $d$.
          \end{center}
        }
      }
    \end{column}
  \end{columns}

\end{frame}
%>>>

\begin{frame}[t,fragile] \frametitle{{Compressing Close Interactions}} \framesubtitle{{}} %<<<
  %\resizebox{0.34\textwidth}{!}{\input{figs/tikz/disc-suspension}}

  \newcommand*\drawpanels[6]{% coord, radius, start, end, count
    \pgfmathsetmacro{\x}{{#1}}
    \pgfmathsetmacro{\y}{{#2}}
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    \pgfmathsetmacro{\N}{{#6-1}}

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      \draw[black, line width=1pt] ({\x+0.96*\xx},{\y+0.96*\yy}) -- ({\x+1.04*\xx},{\y+1.04*\yy});
    }
  }

  \vspace{-1.6em}
  \begin{columns}
    \begin{column}[T]{0.63\textwidth}
      \hfill
      \resizebox{0.99\textwidth}{!}{\begin{tikzpicture}[scale=0.8]%<<<
        \draw[color=blue, line width=2pt, fill=gray!50] (-4.2,0) circle (4cm);
        \draw[color=blue, line width=2pt, fill=gray!50] ( 4.2,0) circle (4cm);
        \draw [red, line width=2pt, domain=-30:30] plot ({ 4*cos(\x)-4.2}, {4*sin(\x)});
        \draw [red, line width=2pt, domain=-30:30] plot ({-4*cos(\x)+4.2}, {4*sin(\x)});

        %\draw [orange, line width=2pt, domain=-30:30] plot ({ 3.85*cos(\x)-4.2}, {3.85*sin(\x)});
        %\draw [orange, line width=2pt, domain=-30:30] plot ({-3.85*cos(\x)+4.2}, {3.85*sin(\x)});
        %\node at (-0.73,-0.6) {\color{orange} \Large $\overline{\sigma}_{2}$};

        %\draw[dotted,color=black!50!green, line width=2pt] (0,0) circle (2.09cm);
        \draw[rounded corners=1cm,dotted,color=black!50!green, line width=2pt] (-1.5, -2.1) rectangle (1.5, 2.1) {};


        %\draw[fill=red, opacity=0.1] (0,0) circle (2.09cm);
        %\node at (2.25, -1) {\color{black!50!green} \Large $\Gamma_{3}$};

        %\node at (-5.5, 1.5) {\huge $\Omega_k$};
        %\node at (-7.0, 3.6) {\Large $\partial\Omega_k$};


        \drawpanels{-4.2}{0}{4}{30}{330}{8};
        \drawpanels{ 4.2}{0}{4}{-150}{150}{8};

        \drawpanels{-4.2}{0}{4}{-15}{15}{2};
        \drawpanels{-4.2}{0}{4}{-7.5}{7.5}{2};
        \drawpanels{-4.2}{0}{4}{-3.75}{3.75}{2};
        \drawpanels{-4.2}{0}{4}{-1.875}{1.875}{3};
        %\drawpanels{-4.2}{0}{4}{-0.9375}{0.9375}{3};

        \drawpanels{ 4.2}{0}{4}{165}{195}{2};
        \drawpanels{ 4.2}{0}{4}{172.5}{187.5}{2};
        \drawpanels{ 4.2}{0}{4}{176.25}{183.75}{2};
        \drawpanels{ 4.2}{0}{4}{178.125}{181.875}{3};
        %\drawpanels{ 4.2}{0}{4}{179.0625}{180.9375}{3};

        %\draw [dashed, line width=1pt] (-4.2,0) -- (-0.2,0);
        \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-4.2,0) -- (-1.371,2.828);
        %\draw [line width=2pt, domain=0:45] plot ({1*cos(\x)-4.2}, {1*sin(\x)});
        %\node at (-3.0, 0.5) {\Large $\theta$};
        \node at (-3.1, 1.6) {\Large $\radius$};
        \node at (-4.55,0) {\Large $\vct{x}^c_{k}$};

        \draw [dashed, line width=1pt] (-0.2,0) -- (-0.2,-3.55);
        \draw [dashed, line width=1pt] ( 0.2,0) -- ( 0.2,-3.55);
        \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-1.2,-3.45) -- (-0.2,-3.45);
        \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] ( 1.2,-3.45) -- ( 0.2,-3.45);
        \node at (0, -4.0) {\Large $d$};

        \node at (0.85,  0.35) {\color{red} \Large $\Gamma_{2}$};
        \node at (0.85, -0.40) {\color{red} \Large $\sigma_{2}$};

        \node at (4.0, 3.4) {\color{blue} \LARGE $\Gamma_{1}$};
        \node at (5.0, 3.4) {\color{blue} \Large $\sigma_{1}$};
      \end{tikzpicture}}%>>>
    \end{column}
    \begin{column}[T]{0.36\textwidth}

      \vspace{2em}
      \begin{align*}
        \begin{pmatrix}
        {\color{blue} \mathcal{K}_{11}} & {\color{black!50!green} \mathcal{K}_{12}} \\
        {\color{black!50!green} \mathcal{K}_{21}} & {\color{red} \mathcal{K}_{22}}
        \end{pmatrix}
        \begin{pmatrix}
        {\color{blue} \sigma_1 } \\
        {\color{red} \sigma_2 }
        \end{pmatrix}
        =
        \begin{pmatrix}
        {\color{blue} g_1 } \\
        {\color{red} g_2 }
        \end{pmatrix}
      \end{align*}

      \only<2->{
      Build compression using \\
      RCIP method of Helsing
      }

    \end{column}

  \end{columns}


  \only<3->{
  \vspace{0.5em}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      Right precondition with $\mathcal{K}_{22}^{-1}$:
      \begin{align*}
        \begin{pmatrix}
          \mathcal{K}_{11} & \mathcal{K}_{12} \mathcal{K}_{22}^{-1} \\
          \mathcal{K}_{21} & I
        \end{pmatrix}
        \begin{pmatrix}
          \sigma_1 \\
          \overline{\sigma}_2
        \end{pmatrix}
        =
        \begin{pmatrix}
          g_1 \\
          g_2
        \end{pmatrix}
      \end{align*}
      where $\overline{\sigma}_2 = \mathcal{K}_{22} \sigma_2$
    \end{column}

    \begin{column}{0.2\textwidth}

      \only<4->{
      \begin{center}

        $\implies$

        coarsen

      \end{center}
      }
    \end{column}

    \begin{column}{0.4\textwidth}
      \only<4->{
      \begin{align*}
        \begin{pmatrix}
          K_{11} & K^{c}_{12} R \\
          K^{c}_{21} & I
        \end{pmatrix}
        \begin{pmatrix}
          \sigma_1 \\
          \overline{\sigma}^{c}_2
        \end{pmatrix}
        =
        \begin{pmatrix}
          g_1 \\
          g^{c}_2
        \end{pmatrix}
      \end{align*}
      where $R = W_c^{-1} P^{T} W_f K_{22}^{-1} P$.
      }
    \end{column}

  \end{columns}
  }

\end{frame}
%>>>

\begin{FIframe}{Computing ~$R_d$~ On-the-Fly}{} %<<<

  {\bf Cost of computing $R_d$:}

  \vspace{0.4em}
  {\renewcommand{\arraystretch}{1.6}
  \begin{tabular}{ l l l }
    Direct: & $\mathcal{O}((q \log d)^3)$ & \\

    RCIP: & $\mathcal{O}(q^3 \log d)$ & $\quad \left[~ \mathcal{O}(q^6 \log d) \text{ ~in~ 3D} ~\right]$ \\

  \end{tabular}}

  \vspace{4em}
  \only<2->{
  {\bf Interpolating $R_d$:} ~~~~Interpolated Compressed Inverse Preconditioning (ICIP)

  \vspace{1em}
  \begin{columns}
    \column{0.35\textwidth}
      $\displaystyle R_{ij}(d) = \sum\limits_{k=0}^{p-1} \alpha_k T_k(\log d)$
    \column{0.64\textwidth}
      \begin{tikzpicture}%<<<
        % Draw the base line
        \draw[thick] (0,0) -- (8,0);

        % Draw the panel divisions
        \foreach \x in {0,2,4,6,8} {
            \draw[thick] (\x,0.2) -- (\x,-0.2);
        }

        % Add panel labels
        \node at (0,-0.5) {$10^{0}$};
        \node at (2,-0.5) {$10^{-2}$};
        \node at (4,-0.5) {$10^{-4}$};
        \node at (6,-0.5) {$10^{-6}$};
        \node at (8,-0.5) {$10^{-8}$};

        \node at (4,-1.0) {$\log d \longrightarrow$};

        % Compute and draw Chebyshev nodes for each panel
        \foreach \i in {0, 2, 4, 6} {
            \foreach \j in {1, 2, 3, 4, 5, 6, 7, 8} {
                \pgfmathsetmacro{\theta}{(2*\j-1)*180/16}
                \pgfmathsetmacro{\x}{\i + 1 + cos(\theta)}
                \filldraw[blue] (\x,0) circle (1.5pt);
            }
        }
        %\begin{axis}[
        %  xmode=log,
        %  log basis x=10,
        %  axis x line=bottom,
        %  axis y line=none,
        %  xmin=1e-8, xmax=1,
        %  xtick={1,1e-1,1e-2,1e-3,1e-4,1e-5,1e-6,1e-7,1e-8},
        %  xticklabels={$10^0$, $$, $10^{-2}$, $$, $10^{-4}$, $$, $10^{-6}$, $$, $10^{-8}$},
        %  tick align=outside,
        %  enlargelimits=false,
        %  width=12cm,
        %  height=2cm
        %]
        %\end{axis}
      \end{tikzpicture}%>>>
  \end{columns}

  \vspace{1em}
  Interpolation cost: \quad $\mathcal{O}(q^2 p)$ \quad  $\quad \left[~ \mathcal{O}(q^4 p) \text{ ~in~ 3D} ~\right]$
  }

\end{FIframe}%>>>

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

  \begin{columns}
  \column{0.5\textwidth}
    Errors (Stokes mobility with 2 discs):
  \column{0.5\textwidth}
    \resizebox{0.4\textwidth}{!}{\begin{tikzpicture}[scale=0.8]%<<<
      \draw[color=blue, line width=2pt, fill=gray!50] (-3.2,0) circle (3cm);
      \draw[color=blue, line width=2pt, fill=gray!50] ( 3.2,0) circle (3cm);

      \draw [dashed, line width=1pt] (-0.2,0) -- (-0.2,-3.55);
      \draw [dashed, line width=1pt] ( 0.2,0) -- ( 0.2,-3.55);
      \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-1.2,-3.45) -- (-0.2,-3.45);
      \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] ( 1.2,-3.45) -- ( 0.2,-3.45);
      \node at (0,-4.0) {\Huge $d$};
    \end{tikzpicture}}%>>>
  \end{columns}

  \vspace{1em}
  \begin{tabular}{r r c r r r r r }
    \hline
           & ~ & Adaptive       & ~ &    \multicolumn{4}{c}{Interpolating $R_d$} \\
     $d$   & ~ & Discretization & ~ &    $p=8$ & ~~$p=16$ & ~~$p=24$ & ~~$p=32$ \\ % & ~~$p=40$
    \hline
    1e-1   & ~ &  7.6e-15       & ~ & 1.0e-4 & 2.9e-07 & 2.1e-09 & 9.1e-12 \\ % 3.419e-14 8.966e-15
   %1e-2   & ~ &  1.8e-13       & ~ & 2.8e-3 & 3.6e-06 & 3.6e-08 & 6.5e-10 \\ % 2.635e-12 1.552e-14
    1e-3   & ~ &  4.4e-13       & ~ & 3.4e-5 & 5.6e-10 & 4.8e-14 &         \\ % 6.656e-14 8.237e-14
   %1e-4   & ~ &  3.8e-11       & ~ & 1.5e-3 & 1.4e-09 & 1.4e-13 &         \\ % 1.244e-13 8.435e-14
    1e-5   & ~ &  9.0e-09       & ~ & 1.5e-5 & 2.1e-12 &         &         \\ % 1.170e-12 1.012e-12
   %1e-6   & ~ &  2.0e-07       & ~ & 6.0e-4 & 1.4e-11 &         &         \\ % 1.487e-11 1.936e-11
    1e-7   & ~ &  4.3e-07       & ~ & 1.7e-5 & 4.1e-11 &         &         \\ % 2.812e-11
    1e-8   & ~ &  5.3e-08       & ~ & 6.3e-4 & 3.9e-09 &         &         \\ % 1.220e-09
    \hline
  \end{tabular}

  \vspace{1em}
  $p$: interpolation order

\end{FIframe}%>>>

\begin{FIframe}{GMRES Iterations}{} %<<<
  %Iteration counts for 2-discs, and disc-chain

  \vspace{-0.8em}
  \begin{columns}[T]
    \column{0.25\textwidth}
      \centering

      Iteration counts \\
      for $\epsilon_{\text{GMRES}}$=1e-8

      \vspace{1em}
      \resizebox{0.4\textwidth}{!}{\begin{tikzpicture}[scale=0.8]%<<<
        \draw[color=blue, line width=2pt, fill=gray!50] (0,-3.2) circle (3cm);
        \draw[color=blue, line width=2pt, fill=gray!50] (0, 3.2) circle (3cm);
        \only<2->{
        \draw[color=blue, line width=2pt, fill=gray!50] (0, -9.6) circle (3cm);
        \draw[color=blue, line width=2pt, fill=gray!50] (0,-16.0) circle (3cm);
        }
        \only<3->{
        \draw[color=black, line width=2pt, fill=black] (0,-19.5) circle (0.15cm);
        \draw[color=black, line width=2pt, fill=black] (0,-20.4) circle (0.15cm);
        \draw[color=black, line width=2pt, fill=black] (0,-21.3) circle (0.15cm);
        \draw[color=blue, line width=2pt, fill=gray!50] (0,-24.8) circle (3cm);
        }

        \draw [dashed, line width=1pt] (0,-0.2) -- (-3.55,-0.2);
        \draw [dashed, line width=1pt] (0, 0.2) -- (-3.55, 0.2);
        \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-3.45,-1.2) -- (-3.45,-0.2);
        \draw [line width=2pt,-{Latex[length=10pt,width=10pt]}] (-3.45, 1.2) -- (-3.45, 0.2);
        \node at (-4.0,0) {\Huge $d$};
      \end{tikzpicture}}%>>>

    \column{0.75\textwidth}
      {\bf Adaptive discretization:}
      \only<1>{\begin{tabular}{r r r r r r r r}%<<<
        \hline
          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
        \hline
            2               &       15 &  37  &  104 &  337 & 1283 & 1848 & 2344 \\
            &&&&&&&\\
            &&&&&&&\\
            &&&&&&&\\
            &&&&&&&\\
        %\hline
      \end{tabular}
      }%>>>
      \only<2>{\begin{tabular}{r r r r r r r r}%<<<
        \hline
          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
        \hline
            2               &       15 &  37  &  104 &  337 & 1283 & 1848 & 2344 \\
            4               &       25 &  75  &  271 & 1134 & 3770 & 5301 & 6620 \\
            &&&&&&&\\
            &&&&&&&\\
            &&&&&&&\\
        %\hline
      \end{tabular}
      }%>>>
      \only<3>{\begin{tabular}{r r r r r r r r}%<<<
        \hline
          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & ~1e-4 & ~~~1e-5 & 1e-6 & 1e-7 \\
        \hline
            2               &       15 &   37 &  104 &  337 & 1283 & 1848 & 2344 \\
            4               &       25 &   75 &  271 & 1134 & 3770 & 5301 & 6620 \\
         %  8               &       32 &  124 &  494 & 1939 & 7488 &>8000 &>8000 \\
           16               &       35 &  147 &  629 & 2754 &>8000 &      &      \\
         % 32               &       36 &  148 &  682 & 3092 &      &      &      \\
           64               &       36 &  148 &  683 & 3094 &      &      &      \\
         %128               &       37 &  149 &  683 & 3094 &      &      &      \\
          256               &       37 &  149 &  683 & 3094 &      &      &      \\
        %\hline
      \end{tabular}}%>>>

      \vspace{1.5em}
      {\bf Interpolated Compressed Inverse Preconditioning (ICIP):}
      \only<1>{\begin{tabular}{r r r r r r r r}%<<<
        \hline
          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
        \hline
            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
        %\hline
      \end{tabular}
      }%>>>
      \only<2>{\begin{tabular}{r r r r r r r r}%<<<
        \hline
          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
        \hline
            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
            4               &       28 &  34  &   36 &   37 &   37 &   37 &   37 \\
        %\hline
      \end{tabular}
      }%>>>
      \only<3>{\begin{tabular}{r r r r r r r r}%<<<
        \hline
          $N_{\text{disc}}$ & $d=$1e-1 & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6 & 1e-7 \\
        \hline
            2               &       18 &  20  &   21 &   21 &   21 &   21 &   21 \\
            4               &       28 &  34  &   36 &   37 &   37 &   37 &   37 \\
         %  8               &       38 &  52  &   54 &   53 &   57 &   57 &   57 \\
           16               &       46 &  71  &   74 &   80 &   86 &   87 &   88 \\
         % 32               &       48 &  90  &   98 &  113 &  139 &  146 &  150 \\
           64               &       49 &  96  &  108 &  131 &  186 &  237 &  251 \\
         %128               &       49 &  98  &  110 &  134 &  215 &  326 &  431 \\
          256               &       49 &  98  &  110 &  134 &  220 &  371 &  608 \\
        %\hline
      \end{tabular}}%>>>

  \end{columns}

\end{FIframe}%>>>