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\newcommand{\dlpbar}{\ensuremath{\delta \bar{p}}}
\newcommand{\dlptwid}{\ensuremath{\delta \tilde{p}}}
\newcommand{\dlptwidi}{\ensuremath{\delta \tilde{p}_i}}
\newcommand{\dlphat}{\ensuremath{\delta \hat{p}}}
\newcommand{\dlphatj}{\ensuremath{\delta \hat{p}_j}}
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\newcommand{\uc}{\ensuremath{\mathbf{u}_c}}
\newcommand{\vc}{\ensuremath{\mathbf{v}_c}}
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\newcommand{\Vsc}{\ensuremath{\mathbf{V}_{sc}}}
\newcommand{\vgn}{\ensuremath{\mathbf{v}_{g_N}}}
\newcommand{\divdot}{\ensuremath{\mathbf{\nabla} \cdot}}
\newcommand{\Wh}{\ensuremath{W_h}}
\newcommand{\Vh}{\ensuremath{\mathbf{V}_h}}
\newcommand{\WHh}{\ensuremath{W_{H,h}}}
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\begin{document}
%\Draft
%\SetVersionDate{October 18, 2002}

\addtolength{\parskip}{0.25in}


% ----------------------------------------------------------------------
% Title Page
% ----------------------------------------------------------------------

\vfill
\begin{center}
   \frontpagetitle{Subgrid Upscaling \strut\\ Preconditioners \strut\\ for Darcy Flow \strut}
   \vspace{0.5in}
   \frontpageauthor{James M. Rath}
   \vspace{-0.5in}
   \frontpageaffil{Center for Subsurface Modeling / ICES}
   \vspace{-0.25in}
   \frontpageaffil{\parbox{5in}{
      Committee: \\
      Todd Arbogast (advisor) \\
      Steve Bryant \\
      Clint Dawson \\
      Robert van de Geijn \\
      Jack Xin}}
\end{center}
\vfill

%\frontpagelogos

\newpage

% ----------------------------------------------------------------------
% Page 0: Outline
% ----------------------------------------------------------------------

\centerline{ \textcolor{obscureweakblue}{\huge Outline:} }

\begin{firstheadlineitemize}
\item \textcolor{greyblue}{The Problem:}
  \begin{secondheadlineitemize}
  \item \textcolor{black}{Modeling subsurface flows}
  \end{secondheadlineitemize}
\item \textcolor{paleblue}{Subgrid Upscaling}
\item \textcolor{paleblue}{Two-level Multigrid-like Scheme}
\item \textcolor{paleblue}{Change the Matrix, not the RHS}
\item \textcolor{paleblue}{Applications}
\end{firstheadlineitemize}

\newpage

% ----------------------------------------------------------------------
% Page 1: statement of our small piece of the problem
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Small piece of the model:}

\begin{center}
\begin{minipage}{7in}
Darcy's law
{\LARGE \textcolor{bo}{$$\mathbf{u} = -\frac{\mathbf{\kappa}}{\mu} (\mathbf{\nabla} p - \rho \mathbf{g})$$} }

and the continuity equation
{\LARGE \textcolor{bo}{$$\divdot \mathbf{u} = q$$} }

for single-phase steady-state flow.
\end{minipage}

\vspace*{0.5in}
\parbox{7in}{Approximate using a cell-centered \\
finite difference method (CCFD) \\
for the pressures.}
\end{center}

\newpage

% ----------------------------------------------------------------------
% Page 2: motivations and goals
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Motivation:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item A single multiphase, time-dependent simulation \\
  requires \textcolor{bo}{repeated computations} of approximations like these.
\item And many more for statistical sampling and/or \\
  optimization studies.
\end{list}
}

\newpage

\textcolor{greyblue}{\LARGE Motivation:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item A single multiphase, time-dependent simulation \\
  requires \textcolor{bo}{repeated computations} of approximations like these.
\item And many more for statistical sampling and/or \\
  optimization studies.
\end{list}
}

\textcolor{greyblue}{\LARGE Goals:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Fast, accurate approximations \\
(in as much detail as possible).
\item Iterative solution methods.
\item \textcolor{bo}{Subgrid upscaling} used for \textcolor{bo}{accelerated} convergence.
\end{list}
}

\newpage

% ----------------------------------------------------------------------
% Page 3: difficulties
% ----------------------------------------------------------------------

\vspace{0.5in}
\textcolor{greyblue}{\LARGE Difficulties:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Geostatistical description of permeability has \textcolor{bo}{high-resolution}. \\
  Details necessary for correct flow prediction.
\item Permeability is \textcolor{bo}{heterogeneous}. \\
  Big jumps occur on short spatial scales.
\item Together produce an \textcolor{bo}{ill-conditioned} \\
  and \textcolor{bo}{computationally expensive} problem.
\end{list}
}

\newpage

% ----------------------------------------------------------------------
% Interlude: Outline
% ----------------------------------------------------------------------

\centerline{ \textcolor{obscureweakblue}{\huge Outline:} }

\begin{firstheadlineitemize}
\item \textcolor{paleblue}{The Problem}
\item \textcolor{greyblue}{Subgrid Upscaling}
  \begin{secondheadlineitemize}
  \item \textcolor{black}{Mixed Variational Form and Its Approximation}
  \item \textcolor{black}{Subgrid Upscaling and Its Approximation}
  \item \textcolor{black}{Examples}
  \end{secondheadlineitemize}
\item \textcolor{paleblue}{Two-level Multigrid-like Scheme}
\item \textcolor{paleblue}{Change the Matrix, not the RHS}
\item \textcolor{paleblue}{Applications}
\end{firstheadlineitemize}

\newpage

% ----------------------------------------------------------------------
% Page 6: mixed variational form
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Mixed Variational Form:}

{\Large
\hspace{0.25in} Let
\begin{align*}
W &= L^2(\Omega)/\mathbb{R}, \\
\mathbf{V} &= \{ \mathbf{v} \in H(\mathrm{div},\Omega) \; | \;
\mathbf{v} \cdot \nu = 0 \;\mathrm{on}\; \partial \Omega \}.
\end{align*}

\hspace{0.25in} Find $p \in W$ and $\mathbf{u} \in \mathbf{V}$ such that
\begin{align*}
(\divdot \mathbf{u},w) &= (q,w)  &  \forall w \in W, \\
(\mu \mathbf{\kappa}^{-1} \mathbf{u}, \mathbf{v}) - (p,\divdot \mathbf{v}) &=
(\rho \mathbf{g}, \mathbf{v})  &  \forall \mathbf{v} \in \mathbf{V}.
\end{align*}
}

\newpage

\textcolor{greyblue}{\LARGE Finite Element Approximation:}

Fine-scale problem:
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{\setlength{\topsep}{-0.5ex}}
\item Grid with spacing $h$,
\item $\mathbf{\kappa}$ piecewise constant,
\item and lowest-order Raviart-Thomas-Nedelec \\
  (RT0) elements $\Wh\times\Vh\subset W\times\mathbf{V}$.
\end{list}
%
Find $p_h \in \Wh$ and $\mathbf{u}_h \in \Vh$ such that
\begin{align*}
(\divdot \mathbf{u}_h,w_h) &= (q,w_h)  &  \forall w_h \in \Wh, \\
(\mu \mathbf{\kappa}^{-1} \mathbf{u}_h, \mathbf{v}_h) - (p_h,\divdot \mathbf{v}_h) &=
(\rho \mathbf{g}, \mathbf{v}_h)  &  \forall \mathbf{v}_h \in \Vh.
\end{align*}

Using quadrature --- the trapezoidal rule --- to compute
$(\mu \mathbf{\kappa}^{-1} \mathbf{u}_h, \mathbf{v}_h)$ is equivalent to
solving the original problem using CCFD.

\newpage

\begin{center}
Degrees of freedom for 2-D RT0 elements on \\
a {\large\textcolor{orangered}{fine}} rectangular grid:
\end{center}

\begin{center}
{\setlength\unitlength{7pt}
\begin{picture}(72,48)(0,0)
\multiput(0,0)(0,4){13}{ \line(1,0){72} }
\multiput(0,0)(6,0){13}{ \line(0,1){48} }
\textcolor{seafoamgreen}{
%  \multiput(3,0)(6,0){12}{\multiput(0,2)(0,4){12}{\makebox(0,0){\circle*{1}}}}
%  \multiput(3,0)(6,0){12}{\multiput(0,2)(0,4){12}{\circle*{1}}}
%  \multiput(6,0)(6,0){12}{\multiput(0,2)(0,4){12}{\circle*{1}}}
  \multiput(4.25,0)(6,0){12}{\multiput(0,2)(0,4){12}{\circle*{1}}}
}
\textcolor{orangered}{
  \multiput(0,0)(6,0){13}{\multiput(0,2)(0,4){12}{\makebox(0,0){$\times$}}}
  \multiput(0,0)(0,4){13}{\multiput(3,0)(6,0){12}{\makebox(0,0){$\times$}}}
}
\end{picture}}
\end{center}

\bigskip

\centerline{ \textcolor{orangered}{$\times$} Velocity }

\centerline{
\textcolor{seafoamgreen}{{\setlength\unitlength{7pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} Pressure}

\newpage

\textcolor{greyblue}{\LARGE Coarse-Scale Approximation:}

Coarse-scale problem:
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{\setlength{\topsep}{-0.5ex}}
\item Grid with spacing $H = C h$,
\item $\mathbf{\kappa}$ piecewise constant (with $h$ resolution),
\item and RT0 or lowest-order Brezzi-Douglas-Dur\`{a}n-Fortin \\
  (BDDF1) elements $\WH\times\VH\subset W\times\mathbf{V}$.
\end{list}

Find $p_H \in \WH$ and $\mathbf{u}_H \in \VH$ such that
\begin{align*}
(\divdot \mathbf{u}_H,w_H) &= (q,w_H)  &  \forall w_H \in \WH, \\
(\mu \mathbf{\kappa}^{-1} \mathbf{u}_H, \mathbf{v}_H) - (p_H,\divdot \mathbf{v}_H) &=
(\rho \mathbf{g}, \mathbf{v}_H)  &  \forall \mathbf{v}_H \in \VH.
\end{align*}

\newpage

\begin{center}
Degrees of freedom for 2-D BDM1 elements on \\
a {\large\textcolor{orangered}{coarse}} rectangular grid:
\end{center}

\begin{center}
{\setlength\unitlength{7pt}
\begin{picture}(72,48)(0,0)
\multiput(0,0)(0,16){4}{ \line(1,0){72} }
\multiput(0,0)(24,0){4}{ \line(0,1){48} }
\textcolor{seafoamgreen}{
  \multiput(13.5,0)(24,0){3}{\multiput(0,8)(0,16){3}{\makebox(0,0){\circle*{1}}}}
}
\textcolor{orangered}{
%  \multiput(0,0)(24,0){4}{\multiput(0,8)(0,16){3}{\makebox(0,0){$\otimes$}}}
%  \multiput(0,0)(0,16){4}{\multiput(12,0)(24,0){3}{\makebox(0,0){$\otimes$}}}
  \multiput(0,0)(24,0){4}{\multiput(0,4)(0,16){3}{\makebox(0,0){$\otimes$}}}
  \multiput(0,0)(24,0){4}{\multiput(0,12)(0,16){3}{\makebox(0,0){$\otimes$}}}
  \multiput(0,0)(0,16){4}{\multiput(6,0)(24,0){3}{\makebox(0,0){$\otimes$}}}
  \multiput(0,0)(0,16){4}{\multiput(18,0)(24,0){3}{\makebox(0,0){$\otimes$}}}
}
\end{picture}}
\end{center}

\bigskip

\centerline{ \textcolor{orangered}{$\otimes$} Velocity (linear) }

\centerline{
\textcolor{seafoamgreen}{{\setlength\unitlength{7pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} Pressure}

\newpage

% ----------------------------------------------------------------------
% Pages 2-?: Description of the upscaling technique
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE The Upscaling Technique:}

{\small
\textcolor{bo}{Decompose} $W$ and $\mathbf{V}$ (using a chosen coarse grid)
so that:
\vspace{-0.25in}
\begin{enumerate}
\item[(1)] no information is lost, $W = \Wc \oplus \dlW$, $\mathbf{V} = \Vc \oplus \dlV$;
\item[(2)] mass is conserved, $\divdot \Vc = \Wc$, $\divdot \dlV = \dlW$;
\item[(3)] fine-scale velocities are locally supported over the coarse grid, \\
  $\dlV \cdot \nu = 0$ on the boundary of each coarse cell.
\end{enumerate}
}

%\vspace*{0.25in}
{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item \textcolor{bo}{Disconnect} the subgrid problems from one another --- \\
  but not from the coarse grid problem.
\item Use Green's or \textcolor{bo}{influence function} approach for \\
  \textcolor{bo}{forward elimination} of subgrid problems into coarse problem.
\item Follow with \textcolor{bo}{back substitution} to recover the subgrid
  solutions.
\item \textcolor{bo}{Recombine} to obtain fine-scale solution.
\end{list}
}


\newpage

\textcolor{greyblue}{\LARGE Upscaling Approximation:}

{\small
Approximate $\dlW\times\dlV$ by RT0, $\dlWh\times\dlVh$, \\
and for $\Wc\times\Vc$ use BDDF1, $\WH\times\VH$.

Let $\WHh=\WH\oplus\dlWh$ and $\VHh=\VH\oplus\dlVh$.

\begin{description}
\item[\textcolor{bo}{$\WH$}:] piecewise constant on coarse cells
\item[\textcolor{bo}{$\dlWh$}:] piecewise constant on fine cells, \\
  with zero average over each coarse cell
\item[\textcolor{bo}{$\VH$}:] normal components continuous across coarse faces, \\
  and linear variation along coarse faces
\item[\textcolor{bo}{$\dlVh$}:] normal components continuous across fine faces, \\
  and constant along those faces; \\ normal components across coarse faces are zero
\end{description}

\newpage

\begin{center}
Degrees of freedom for 2-D RT0 elements on \\
a {\large\textcolor{orangered}{fine}} rectangular grid:
\end{center}

\begin{center}
{\setlength\unitlength{7pt}
\begin{picture}(72,48)(0,0)
\multiput(0,0)(0,4){13}{ \line(1,0){72} }
\multiput(0,0)(6,0){13}{ \line(0,1){48} }
\textcolor{seafoamgreen}{
  \multiput(4.25,0)(6,0){12}{\multiput(0,2)(0,4){12}{\circle*{1}}}
}
\textcolor{orangered}{
  \multiput(0,0)(6,0){13}{\multiput(0,2)(0,4){12}{\makebox(0,0){$\times$}}}
  \multiput(0,0)(0,4){13}{\multiput(3,0)(6,0){12}{\makebox(0,0){$\times$}}}
}
\end{picture}}
\end{center}

\bigskip

\centerline{ \textcolor{orangered}{$\times$} Velocity }

\centerline{
\textcolor{seafoamgreen}{{\setlength\unitlength{7pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} Pressure}

\newpage

%color diagram from slide 28 of Xian talk
\begin{center}
Degrees of freedom for 2-D RT0/BDDF1 elements on \\
an ``{\large\textcolor{orangered}{upscaled}}'' rectangular grid:
\end{center}

\begin{center}
{\setlength\unitlength{7pt}
\begin{picture}(72,48)(0,0)
\multiput(0,0)(0,16){3}{\multiput(0,0)(24,0){3}{
\textcolor{orangered}{%\linethickness{0.75pt}
\put(0, 4){\line(1,0){24}}\multiput(3, 4)(6,0){4}{\makebox(0,0){{$\times$}}}
\put(0, 8){\line(1,0){24}}\multiput(3, 8)(6,0){4}{\makebox(0,0){{$\times$}}}
\put(0,12){\line(1,0){24}}\multiput(3,12)(6,0){4}{\makebox(0,0){{$\times$}}}
\put( 6,0){\line(0,1){16}}\multiput( 6,2)(0,4){4}{\makebox(0,0){{$\times$}}}
\put(12,0){\line(0,1){16}}\multiput(12,2)(0,4){4}{\makebox(0,0){{$\times$}}}
\put(18,0){\line(0,1){16}}\multiput(18,2)(0,4){4}{\makebox(0,0){{$\times$}}}
}
\textcolor{seafoamgreen}{\multiput(2.5,0)(6,0){4}{\multiput(0,2)(0,4){4}{\makebox(0,0){\circle*{1}}}}}
\put(-1,0){
\put( 0,3){\makebox(0,0){{$\otimes$}}}\put( 0,13){\makebox(0,0){{$\otimes$}}}
\put(24,3){\makebox(0,0){{$\otimes$}}}\put(24,13){\makebox(0,0){{$\otimes$}}}
\put(5, 0){\makebox(0,0){{$\otimes$}}}\put(19.5, 0){\makebox(0,0){{$\otimes$}}}
\put(5,16){\makebox(0,0){{$\otimes$}}}\put(19.5,16){\makebox(0,0){{$\otimes$}}}
{\linethickness{1.5pt}\framebox(24,16){}}}
}}
\end{picture}}
\end{center}

\bigskip

\centerline{
$\otimes$ Coarse velocity (linear)\qquad
\textcolor{orangered}{$\times$} Subgrid velocity}

\centerline{
\textcolor{seafoamgreen}{{\setlength\unitlength{7pt}\begin{picture}(2,2)(0,0)
  \put(1,1){\circle*{1}}\end{picture}}} Pressure}

%\vspace{0.25in}
%\centerline{\setlength\unitlength{1pt}
%\begin{picture}(300,160)(0,0)
%  \multiput(0,0)(0,80){2}{
%  \multiput(0,0)(100,0){3}{
%    \thicklines
%    \put(0,0){\framebox(100,80){}}
%    \thinlines
%    \multiput(0,0)(100,0){2}{
%      \put(0,26){\makebox(0,0){\tiny $\otimes$}}
%      \put(0,54){\makebox(0,0){\tiny $\otimes$}}
%    }
%    \multiput(0,0)(0,80){2}{
%      \put(33,0){\makebox(0,0){\tiny $\otimes$}}
%      \put(67,0){\makebox(0,0){\tiny $\otimes$}}
%    }
%    \thinlines
%    \put(20,0){\framebox(20,80){}}
%    \put(60,0){\framebox(20,80){}}
%    \put(0,20){\framebox(100,20){}}
%    \put(0,60){\framebox(100,20){}}
%    \multiput(20,0)(20,0){4}
%      {\multiput(0,10)(0,20){4}{\makebox(0,0){\tiny $\times$}}}
%    \multiput(0,20)(0,20){3}
%      {\multiput(10,0)(20,0){5}{\makebox(0,0){\tiny $\times$}}}
%    \multiput(10,0)(20,0){5}{\multiput(0,10)(0,20){4}{\circle*{2}}}
%  }}
%\end{picture}}
%\medskip
%\centerline{\parbox{6.5in}{\tiny Figure~1 from~\cite{Arbogast_2002}.
%A 2-D mesh of $15\times8$ fine grid elements decomposed
%into a $3\times2$ coarse grid with $5\times4$ subgrid elements per
%coarse element.  The dots represent the pressure values, one per fine
%element.  The crosses represent the edge velocity fluxes.  The circled
%crosses apply to the coarse grid edges, and represent linear flux
%variation.  The other crosses represent constant fluxes across the
%subgrid edges internal to the coarse elements.}}

%\null\vfill
%    \begin{center}
%%  \includegraphics[scale=1.5,angle=0]{graphic}
%    \vspace{1.5in}  \fbox{image}\\
%    \bigskip\bigskip
%    \TextCcaption{%
%        Caption $x =\frac1{2a}\left(-b\pm\sqrt{b^2-4ac}\right)$.}
%    \label{fig1}
%    \end{center}
%\vfill\null
}


\newpage

\textcolor{greyblue}{\LARGE Advantages:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Subgrid problems are independent. \\
  Solve in parallel with no communication between coarse cells.
\item Many fewer degrees of freedom in subgrid problems than~the~whole \\
  (by about a factor of the number of coarse grid cells).
%\item In forward elimination, a subgrid problem has the same LHS for many
%  different RHSs $\Rightarrow$ opportunity for efficiency.
\item Many fewer degrees of freedom in coarse problem than the whole \\
  (by about a factor of the number of fine cells per coarse cell).
\item When $H$ is a small multiple of $h$,
  nearly all work in solving the upscaled problem is in solving the coarse problem.
%\item There is only one coarse grid (not many as in MG).
%\item No closure assumption is made regarding the permeability. \\
%  No analysis is needed of ad-hoc simplifying assumptions.
%\item Accuracy is simply how well $\WHh\times\VHh$ approximates
%  $W\times\mathbf{V}$.
\item The ``closure'' assumption is that coarse scale functions \\
  approximate all flux across coarse element faces.
\item Examples in this section use a higher order coarse basis \\
  (BDDF1 vs. RT0) to compensate for this grainy restriction.
\end{list}
}

\newpage

% ----------------------------------------------------------------------
% Pages 2-?: Some results for the upscaling technique
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Some Practical Results:}

\begin{center}
{\large Quarter Five-spot Oil Reservoir Water Flood }

Logarithm of the Permeability Field
\end{center}

\vspace{-0.5in}
\begin{center}
\includegraphics[height=4in,width=4in]{eigstruct/brent/log10permx}

{\small Fine $40\times40$}
\end{center}

\newpage

%\vspace*{0in}
\addtolength{\voffset}{-0.5truein}
\begin{center}
\begin{tabular}{c c c}

\multicolumn{3}{c}{\large Water saturation contours at 100 days} \\

\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{arbogast/s1.051.ps} &
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{arbogast/s0.048.ps} &
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{arbogast/s2.048.ps} \\

{\small Fine $40\times40$} &
{\small Fully upscaled to $5\times5$} &
{\small Coarse $5\times5$} \\

{\tiny ~} & {\tiny ~} & {\tiny ~} \\

\multicolumn{3}{c}{\large Water saturation contours at 500 days} \\

\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{arbogast/s1.124.ps} &
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{arbogast/s0.092.ps} &
\includegraphics[clip=true,keepaspectratio=true,angle=-90,width=2.75in]{arbogast/s2.092.ps} \\

{\small Fine $40\times40$} &
{\small Fully upscaled to $5\times5$} &
{\small Coarse $5\times5$} \\

\end{tabular}
\end{center}
\addtolength{\voffset}{0.5truein}

\newpage

% ----------------------------------------------------------------------
% Interlude: Outline
% ----------------------------------------------------------------------

\centerline{ \textcolor{obscureweakblue}{\huge Outline:} }

\begin{firstheadlineitemize}
\item \textcolor{paleblue}{The Problem}
\item \textcolor{paleblue}{Subgrid Upscaling}
\item \textcolor{greyblue}{Two-level Multigrid-like Scheme}
  \begin{secondheadlineitemize}
  \item \textcolor{black}{Algorithm Description}
  \item \textcolor{black}{Examples}
  \item \textcolor{black}{Further Study}
  \end{secondheadlineitemize}
\item \textcolor{paleblue}{Change the Matrix, not the RHS}
\item \textcolor{paleblue}{Applications}
\end{firstheadlineitemize}

\newpage

% ----------------------------------------------------------------------
% Pages 2-?: Description of the two-level scheme
% ----------------------------------------------------------------------

\vspace*{0.25in}
\textcolor{greyblue}{\LARGE Idea:}

{%\small
Use a two-level \textcolor{bo}{multigrid}-like scheme \\
to solve a fine-scale problem.

%Use a multiscale approximation (like subgrid upscaling) \\
%on the second level instead of a coarse approximation.
Multigrid (MG) uses a fine approximation as one level and a coarse approximation
as another.

\textcolor{bo}{Replace} the \textcolor{bo}{coarse} approximation
\textcolor{bo}{with} the \\ subgrid \textcolor{bo}{upscaling} approximation.

Upscaling retains more of the fine-scale information. \\
It's a better approximation for about the same computational cost.

An overall faster scheme results.
}


\newpage

%\vspace*{0.5in}
\textcolor{greyblue}{\LARGE Notation for Two-Level Scheme:}

{\scriptsize
\begin{list}{}{\setlength{\itemsep}{0pt} \setlength{\leftmargin}{1.0in}}
\item[\textcolor{bo}{$A$}:] matrix from pressure CCFD
\item[\textcolor{bo}{$\mathbf{x}$}:] pressure unknowns
\item[\textcolor{bo}{$\mathbf{b}$}:] information from sources, boundary terms,
and gravity
\end{list}
}

{\small
Then the problem is to solve the linear system
\textcolor{bo}{$A \mathbf{x} = \mathbf{b}$}.
}

{\scriptsize
\begin{list}{}{\setlength{\itemsep}{0pt} \setlength{\leftmargin}{1.0in}}
\item[\textcolor{bo}{$\mathbf{x}_i$}:] $i$th guess at pressure unknowns
\item[\textcolor{bo}{$\mathbf{r}_i$}:] $i$th-stage residual $\mathbf{b} - A \mathbf{x}_i$
\item[\textcolor{bo}{$\mathbf{c}_i$}:] $i$th-stage smoothing corrector
\item[\textcolor{bo}{$B_{\mathrm{J}}$}:] $\mathrm{diag}(A)$ for Jacobi smoothing step
\item[\textcolor{bo}{$\tilde{A}$}:] matrix for corresponding upscaling MFEM problem
\item[\textcolor{bo}{$\mathbf{\tilde{c}}_i$}:] $i$th-stage upscaling corrector
(pressure and velocity unknowns)
% \item[\textcolor{bo}{$\mathbf{\tilde{b}}$}:] upscaling problem data
\item[\textcolor{bo}{$R$}:] restriction operator
$R: \Wh \rightarrow \WHh \times \mathbf{0}$
\item[\textcolor{bo}{$P$}:] prolongation operator
$P: \WHh \times \VHh \rightarrow \Wh$
\end{list}
}

\newpage

\textcolor{greyblue}{\LARGE Two-Level Schematic:}

\begin{center}
{\setlength\unitlength{12pt}
\begin{picture}(60,30)(0,0)

\put(0,10){\framebox(12,10){\shortstack[l]{
\textcolor{bo}{\small START \strut} \\
\textcolor{greyblue}{\footnotesize $i=0$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{x}_0 = \mathbf{0}$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{r}_0 = \mathbf{b} - A \mathbf{x}_0$ \strut}
}}}

\put(16,0){\framebox(12,12){\shortstack[l]{
\textcolor{bo}{\small UPSCALE \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{\tilde{c}}_i = \tilde{A}^{-1} R \mathbf{r}_i$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{x}_{i+1} = \mathbf{x}_i + P \mathbf{\tilde{c}}_i$ \strut} \\
\textcolor{greyblue}{\footnotesize $i \gets i+1$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{r}_i = \mathbf{b} - A \mathbf{x}_i$ \strut}
}}}

\put(16,18){\framebox(12,12){\shortstack[l]{
\textcolor{bo}{\small SMOOTH \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{c}_i = \alpha B_{\mathrm{J}}^{-1} \mathbf{r}_i$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{x}_{i+1} = \mathbf{x}_i + \mathbf{c}_i$ \strut} \\
\textcolor{greyblue}{\footnotesize $i \gets i+1$ \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{r}_i = \mathbf{b} - A \mathbf{x}_i$ \strut}
}}}

\put(32,12){\framebox(12,6){\shortstack[l]{
\textcolor{bo}{\small DONE? \strut} \\
\textcolor{greyblue}{\footnotesize Is $\| \mathbf{r}_i \|$ small? \strut}
}}}

\put(48,12){\framebox(12,6){\shortstack[l]{
\textcolor{bo}{\small FINISH \strut} \\
\textcolor{greyblue}{\footnotesize $\mathbf{x} \approx \mathbf{x}_i$ \strut}
}}}

%{\linethickness{2pt}
%\put(12,15){\vector(4,8){8.9442719}}
%\put(28,23){\vector(4,-6){7.2111026}}
%\put(32,13){\vector(-4,-8){8.9442719}}
%\put(22,12){\vector(0,6){6}}
%\put(44,15){\vector(4,0){4}}
\put(12,15){\vector(1,2){4}}
%\put(28,7){\vector(2,3){4}}
\put(32,13){\vector(-2,-3){4}}
%\put(32,17){\vector(-1,2){4}}
\put(28,25){\vector(1,-2){4}}
\put(22,12){\vector(0,1){6}}
\put(44,15){\vector(1,0){4}}
%}

\put(30,9){ \textcolor{seafoamgreen}{\footnotesize NO} }
\put(44,16){ \textcolor{seafoamgreen}{\footnotesize YES} }

\end{picture}}
\end{center}
%start with a guess for the solution $\mathbf{x}_0$
%
%compute a corrector using smoothing
%$\mathbf{r}_0 = \mathbf{b} - A \mathbf{x}_0$
%$\mathbf{c}_0 = \alpha B^{-1}_{\mathrm{J}} \mathbf{r}_0$
%
%apply it
%$\mathbf{x}_{1} = \mathbf{x}_0 + \mathbf{c}_0$
%
%proceed to question with $i=1$
%
%does the residual $\mathbf{r}_i = \mathbf{b} - A \mathbf{x}_i$ still have too
%large a norm?
%
%yes:
%
%compute a corrector using upscaling
%$\mathbf{\tilde{c}}_i = \tilde{A}^{-1} (R \mathbf{r}_i)$
%$\mathbf{c}_i = P \mathbf{\tilde{c}}_i$
%
%apply it
%$\mathbf{x}^*_i = \mathbf{x}_i + \mathbf{c}_i$
%
%compute a corrector using smoothing
%$\mathbf{r}^*_i = \mathbf{b} - A \mathbf{x}^*_i$
%$\mathbf{c}^*_i = \alpha B^{-1}_{\mathrm{J}} \mathbf{r}^*_i$
%
%apply it
%$\mathbf{x}_{i+1} = \mathbf{x}^*_i + \mathbf{c}^*_i$
%
%back to question with $i \leftarrow i+1$
%
%no:
%
%stop.  $\mathbf{x}_i$ is the approximate solution

\newpage

\centerline{ \textcolor{greyblue}{\LARGE A Simple Test Case:} }

\begin{center}
\includegraphics[keepaspectratio=true,width=4.5in]{eigstruct/twolevel/simple-perm-diagram}

{\small A Permeability Field}
\end{center}

\newpage

% FIX ME: \raiseboxes don't help?
\addtolength{\voffset}{-0.5truein}
\begin{center}

\begin{tabular}{c c c c}
\multicolumn{4}{c}{ \textcolor{greyblue}{\LARGE A ``Typical'' Result:} } \\
\multicolumn{4}{c}{\small Eigenstructure of Smoother and Upscaler} \\
\rotatebox{90}{\hspace{0.5in}\textcolor{bo}{\small small}\hspace{0.5in}Eigenvalue\hspace{0.5in}\textcolor{bo}{\small large}} &
\multicolumn{2}{c}{
\includegraphics[keepaspectratio=true,width=6in]{eigstruct/twolevel/rt0/simplek-24-08-1e-1-hack}} &
\raisebox{3.55in}[0pt]{ $\frac{\mathrm{amplify}}{\mathrm{attenuate}}$} \\
& \multicolumn{2}{c}{\textcolor{bo}{\small low}\hspace{0.75in}Eigen-Frequency\hspace{0.75in}\textcolor{bo}{\small high}} & \\
& \raisebox{0ex}[0ex]{\textcolor{cyan}{$\bullet$} {\tiny Coarse Smoothing Preconditioner}} &
\raisebox{0ex}[0ex]{\textcolor{magenta}{$\bullet$} {\tiny Coarse Exact Preconditioner}} & \\
& \raisebox{0ex}[0ex]{\textcolor{blue}{$\bullet$} {\tiny Fine Smoothing Preconditioner}} &
\raisebox{0ex}[0ex]{\textcolor{red}{$\bullet$} {\tiny Subgrid Upscaling Preconditioner}} & \\
\end{tabular}

\end{center}
\addtolength{\voffset}{0.5truein}

\newpage

%\vspace*{0in}
\addtolength{\voffset}{-0.5truein}
\begin{center}
\begin{tabular}{c c}

\multicolumn{2}{c}{\large Vary $h$ ; Fix $H/h=3$ ; Fix $k_2=0.1$} \\

\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-06-02-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-12-04-1e-1} \\

{\small $6\times6$} &
{\small $12\times12$} \\

{\tiny ~} & {\tiny ~} \\

\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-24-08-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-48-16-1e-1} \\

{\small $24\times24$} &
{\small $48\times48$} \\

\end{tabular}
\end{center}
\addtolength{\voffset}{0.5truein}

\newpage

%\vspace*{0in}
\addtolength{\voffset}{-0.5truein}
\begin{center}
\begin{tabular}{c c}

\multicolumn{2}{c}{\large Fix $h=1/24$ ; Vary $H/h$ ; Fix $k_2=0.1$} \\

\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-24-03-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-24-04-1e-1} \\

{\small $H/h=8$} & {\small $H/h=6$} \\
%{\tiny ~} & {\tiny ~} \\

\end{tabular}
\begin{tabular}{c c c}

\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-24-06-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-24-08-1e-1} &
\includegraphics[keepaspectratio=true,width=3.25in]{eigstruct/twolevel/rt0/simplek-24-12-1e-1} \\

{\small $H/h=4$} & {\small $H/h=3$} & {\small $H/h=2$} \\

\end{tabular}
\end{center}
\addtolength{\voffset}{0.5truein}
%
%\newpage
%
\textcolor{greyblue}{\LARGE Two Other Permeabililty Fields:}

\begin{center}
\begin{tabular}{c c}
{\large ``Medium''} & {\large ``Hard''} \\
\includegraphics[height=4in,width=4in]{eigstruct/arcoperm/log10perm} &
\includegraphics[height=4in,width=4in]{eigstruct/brent/log10permx} \\
{\small log10min = -0.2916} & {\small log10min = -0.6909} \\
{\small log10max = 0.1477} & {\small log10max = 3.5230} \\
\end{tabular}
\end{center}

\newpage

\textcolor{greyblue}{\LARGE Sample Convergence Histories:}

\begin{center}
\begin{tabular}{c c c}
{\small Simple} & {\small Medium} & {\small Hard} \\
\includegraphics[keepaspectratio=true,width=3in]{eigstruct/twolevel/rt0/simplek-24-06-1e-1} &
\includegraphics[keepaspectratio=true,width=3in]{eigstruct/arcoperm/rt0/arcoperm-40-08} &
\includegraphics[keepaspectratio=true,width=3in]{eigstruct/brent/rt0/brent-40-08} \\
\includegraphics[keepaspectratio=true,width=3in]{eigstruct/twolevel/simple-errors} &
\includegraphics[keepaspectratio=true,width=3in]{eigstruct/arcoperm/arcoperm-errors} &
\includegraphics[keepaspectratio=true,width=3in]{eigstruct/brent/brent-errors} \\
{\small $\nu=1$} & {\small $\nu=1$} & {\small $\nu=40$} \\
\end{tabular}
\end{center}

\newpage

\textcolor{greyblue}{\LARGE Over- and Under-Estimating \\ Number of Smoothings:}

\begin{center}
\begin{tabular}{c c}
{\small Over-estimating $\nu$ for Medium} & {\small Under-estimating $\nu$ for Hard} \\
\includegraphics[keepaspectratio=true,width=4in]{eigstruct/arcoperm/arcoperm-better-errors} &
\includegraphics[keepaspectratio=true,width=4in]{eigstruct/brent/brent-mistake-errors} \\
{\small $\nu=2$} & {\small $\nu=5$} \\
\end{tabular}
\end{center}

\newpage

\textcolor{greyblue}{\LARGE Some Conclusions:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}

\item Although we cannot prove it yet, these test cases suggest:
\begin{list}{\textcolor{darkorange}{\small $\circ$}}{}
\item performance depends only on $H/h$ and heterogeneity in $\mathbf{\kappa}$ \\
  (and on geometric factors),
\item convergence factor is independent of $h$ \\
  (like MG but unlike Jacobi PCG),
\item and the number of smoothing steps $\nu$ is adjustable \\
  to obtain a contraction for any $H/h$ and any $\mathbf{\kappa}$.
\end{list}

\item Because there is only one coarse mesh, not several, \\
  this scheme is a bit easier to apply than MG.

\end{list}
}

\newpage

% ----------------------------------------------------------------------
% Page ?: further study
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Work to be Done:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Use MG techniques to prove convergence for some $\nu$ \\
  (with a convergence factor independent of $h$).
%\item $\nu$ depends on $\mathbf{k}$.
\item For a given $\nu$ and $H/h$, will this scheme converge more quickly
  than a corresponding two-level MG scheme?
\item Add a strategy to improve performance on middle frequencies \\
  (staggered grids, semi-coarsened grids, \\
  Krylov methods, or an ordinary coarse smoother).
\item Use super coarsening for large problems \\ (for a true \emph{multi}-level method).
%\item upscaling as coarse-level smoother
\end{list}}

\textcolor{greyblue}{\LARGE And Beyond:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Apply to DG and expanded mixed methods.
\item Apply to other physical problems.
\end{list}}

\newpage

% ----------------------------------------------------------------------
% Interlude: Outline
% ----------------------------------------------------------------------

\centerline{ \textcolor{obscureweakblue}{\huge Outline:} }

\begin{firstheadlineitemize}
\item \textcolor{paleblue}{The Problem}
\item \textcolor{paleblue}{Subgrid Upscaling}
\item \textcolor{paleblue}{Two-level Multigrid-like Scheme}
\item \textcolor{greyblue}{Change the Matrix, not the RHS}
  \begin{secondheadlineitemize}
  \item \textcolor{black}{Algorithm Description}
  \item \textcolor{black}{Example}
  \item \textcolor{black}{Further Study}
  \end{secondheadlineitemize}
\item \textcolor{paleblue}{Applications}
\end{firstheadlineitemize}

\newpage

% ----------------------------------------------------------------------
% Page ?: non-linear stuff
% ----------------------------------------------------------------------

\vspace*{0.75in}
\textcolor{greyblue}{\LARGE Goal:}

{\Large
Obtain fine-scale solution $(\ph,\uh)$ as a \\
direct result of an upscaled computation.
}

\newpage

\textcolor{greyblue}{\LARGE Problem of Spaces:}

\begin{center}
\begin{minipage}{7in}
{\large
Fine-scale solution is
\begin{align*}
(\ph,\uh) &\in \Wh\times\Vh. \\
\intertext{Upscaled solution is}
(\pHh,\uHh) &\in \WHh\times\VHh.
\end{align*}
But almost always
$$\ph\neq\pHh \qquad\text{and}\qquad \uh\neq\uHh.$$
Even though
$$\ph\in\WHh=\Wh \qquad\text{and}\qquad \VHh\subset\Vh,$$
what's worse is \ldots
}
\end{minipage}
\end{center}

\newpage

\textcolor{greyblue}{\LARGE Difficulty:}

\begin{center}
\textcolor{bo}{\LARGE $\uh\notin\VHh$}

\begin{tabular}{c c}
{\setlength\unitlength{7pt}
\begin{picture}(36,12)(4,0)
\multiput(0,0)(18,0){2}{
\textcolor{orangered}{%\linethickness{0.75pt}
\put(0, 4){\line(1,0){18}}\multiput(3, 4)(6,0){3}{\makebox(0,0){$\times$}}
\put(0, 8){\line(1,0){18}}\multiput(3, 8)(6,0){3}{\makebox(0,0){$\times$}}
\put( 6,0){\line(0,1){12}}\multiput( 6,2)(0,4){3}{\makebox(0,0){$\times$}}
\put(12,0){\line(0,1){12}}\multiput(12,2)(0,4){3}{\makebox(0,0){$\times$}}
}
\textcolor{seafoamgreen}{\multiput(1.75,0)(6,0){3}{\multiput(0,2)(0,4){3}{\makebox(0,0){$\bullet$}}}}
\put(-1.25,0){\linethickness{1.5pt}\framebox(18,12){}}}
%\thicklines
%\put(0,0){\line(1,0){18}}\put(0,12){\line(1,0){18}}
%\put(0,0){\line(0,1){12}}\put(18,0){\line(0,1){12}}}
\multiput(19.5,2)(0,4){3}{\makebox(0,0){$\times$}}
\end{picture}} &
{\setlength\unitlength{7pt}
\begin{picture}(36,12)(4,0)
\multiput(0,0)(18,0){2}{
\textcolor{orangered}{%\linethickness{0.75pt}
\put(0, 4){\line(1,0){18}}\multiput(3, 4)(6,0){3}{\makebox(0,0){$\times$}}
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\textcolor{seafoamgreen}{\multiput(1.75,0)(6,0){3}{\multiput(0,2)(0,4){3}{\makebox(0,0){$\bullet$}}}}
\put(-1.25,0){\linethickness{1.5pt}\framebox(18,12){}}}
\put(19.5,6){\makebox(0,0){$\otimes$}}
\end{picture}} \\
\end{tabular}

\parbox{6in}{
Coarse elements are too restrictive: \\
all flow across a coarse edge is constant
along that edge.
}
\end{center}

\newpage

\textcolor{greyblue}{\LARGE Work-Around:}

Allow coarse elements to vary along coarse edges: \\
\centerline{this is a \textcolor{bo}{macro-element}!}

For an upscaled computation, \\
\textcolor{bo}{fix} the \textcolor{bo}{shape} along the coarse edge, \\
and use the \textcolor{bo}{one DOF} to control its \textcolor{bo}{scale}.

Use parameters \textcolor{bo}{$\bvec{\beta}$}, a vector of weights, \\
to control the \textcolor{bo}{shape}.

\newpage

\textcolor{greyblue}{\LARGE Coarse Basis Elements:}

\begin{center}
\begin{tabular}{c c}
Original (Unmodified) & Modified \\
Elements $\VH$ & Elements $\VtwidH$ \\
\includegraphics[keepaspectratio=true,width=2.5in]{nonlinear/ramp-schematic} &
\includegraphics[keepaspectratio=true,width=2.5in]{nonlinear/multiramp-schematic} \\
\includegraphics[keepaspectratio=true,width=3in]{nonlinear/ramp} &
\includegraphics[keepaspectratio=true,width=3in]{nonlinear/multiramp} \\
``$\bvec{\beta}=\bvec{1}$'' & $\bvec{\beta}\neq\bvec{1}$ \\
\end{tabular}
\end{center}

\newpage

$\VtwidH(\bvec{\beta})$ is an RT0-like parameterized family of \\
approximation subspaces of $\Vc$.
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item $\mathrm{dim}(\VH)=\mathrm{dim}(\VtwidH)$,
\item spatial support of elements stays the same,
\item $\dlVh$ and $\WHh$ stay the same,
\item and usually $\VH\neq\VtwidH(\bvec{\beta})$ but $\VH=\VtwidH(\bvec{1})$.
\end{list}

\begin{center}
\begin{minipage}{6in}
Like
\begin{align*}
\VHh &= \VH\oplus\dlVh, \\
\intertext{let}
\VtwidHh(\bvec{\beta}) &= \VtwidH(\bvec{\beta})\oplus\dlVh.
\end{align*}
\end{minipage}
\end{center}

\newpage

\textcolor{greyblue}{\LARGE Original Fine-Scale Problem:}

With $\ph\in\Wh$ and $\uh\in\Vh$, solve

{\Huge
$$\begin{bmatrix}
\mathit{0} & B^T \\
-B & D \\
\end{bmatrix}
\begin{bmatrix}
\ph \\ \uh \\
\end{bmatrix}
=
\begin{bmatrix}
b \\ c \\
\end{bmatrix}$$
}

\newpage

\textcolor{greyblue}{\LARGE Decompose:}
{\large
\begin{alignat*}{2}
\pHh &= \ph\in\Wh &&= \WHh \\
\utwidHh+\ucheckHh &\equiv \uh\in\Vh &&\equiv \VtwidHh\oplus\VcheckHh
\end{alignat*}}

\textcolor{greyblue}{\LARGE And Partition:}
{\Huge
$$\begin{bmatrix}
\mathit{0} & \tilde{B}^T & \check{B}^T \\
-\tilde{B} & \Tilde{\Tilde{D}} & \Tilde{\Check{D}} \\
-\check{B} & \Check{\Tilde{D}} & \Check{\Check{D}} \\
\end{bmatrix}
\begin{bmatrix}
\ph \\ \utwidHh \\ \ucheckHh \\
\end{bmatrix}
=
\begin{bmatrix}
b \\ \tilde{c} \\ \check{c} \\
\end{bmatrix}$$
}

\newpage

\textcolor{paleblue}{
\textcolor{greyblue}{\LARGE Decompose:}
{\large
\begin{alignat*}{2}
\pHh &= \ph\in\Wh &&= \WHh \\
\textcolor{black}{\utwidHh+\ucheckHh} &\equiv \uh\in\Vh &&\equiv \textcolor{black}{\VtwidHh\oplus\VcheckHh}
\end{alignat*}}
}

\textcolor{paleblue}{
\textcolor{greyblue}{\LARGE And Partition:}
{\Huge
$$\begin{bmatrix}
\mathit{0} & \textcolor{black}{\tilde{B}^T} & \textcolor{black}{\check{B}^T} \\
\textcolor{black}{-\tilde{B}} & \textcolor{black}{\Tilde{\Tilde{D}}} & \textcolor{black}{\Tilde{\Check{D}}} \\
\textcolor{black}{-\check{B}} & \textcolor{black}{\Check{\Tilde{D}}} & \textcolor{black}{\Check{\Check{D}}} \\
\end{bmatrix}
\begin{bmatrix}
\ph \\ \textcolor{black}{\utwidHh} \\ \textcolor{black}{\ucheckHh} \\
\end{bmatrix}
=
\begin{bmatrix}
b \\ \textcolor{black}{\tilde{c}} \\ \textcolor{black}{\check{c}} \\
\end{bmatrix}$$
}
}

\newpage

\textcolor{paleblue}{
\textcolor{greyblue}{\LARGE Decompose:}
{\large
\begin{alignat*}{2}
\textcolor{black}{\pHh} &= \ph\in\Wh &&= \textcolor{black}{\WHh} \\
\textcolor{black}{\utwidHh}+\ucheckHh &\equiv \uh\in\Vh &&\equiv \textcolor{black}{\VtwidHh}\oplus\VcheckHh
\end{alignat*}}
}

\textcolor{paleblue}{
\textcolor{greyblue}{\LARGE And Partition:}
{\Huge
$$\begin{bmatrix}
\textcolor{black}{\mathit{0}} & \textcolor{black}{\tilde{B}^T} & \check{B}^T \\
\textcolor{black}{-\tilde{B}} & \textcolor{black}{\Tilde{\Tilde{D}}} & \Tilde{\Check{D}} \\
-\check{B} & \Check{\Tilde{D}} & \Check{\Check{D}} \\
\end{bmatrix}
\begin{bmatrix}
\textcolor{black}{\ph} \\ \textcolor{black}{\utwidHh} \\ \ucheckHh \\
\end{bmatrix}
=
\begin{bmatrix}
\textcolor{black}{b} \\ \textcolor{black}{\tilde{c}} \\ \check{c} \\
\end{bmatrix}$$
}
}

\newpage

\textcolor{greyblue}{\LARGE Uzawa-like Method:}

\textcolor{bo}{Goal}:
 Find $\bvec{\beta}^*$ so that $\mathbf{\check{u}}(\bvec{\beta}^*)=\mathbf{0}$.

\textcolor{darkpeagreen}{Initialize}:
 $m=0$ and $\bvec{\beta}^0=\bvec{1}$.

\textcolor{darkpeagreen}{Solve}: $$\begin{bmatrix}
\mathit{0} & \left(\tilde{B}^m\right)^T \\
-\tilde{B}^m & \Tilde{\Tilde{D}}^m \\
\end{bmatrix}
\begin{bmatrix}
p^m \\ \mathbf{\tilde{u}}^m \\
\end{bmatrix}
=
\begin{bmatrix}
b \\ \tilde{c}^m \\
\end{bmatrix}$$

\textcolor{darkpeagreen}{Compute}:
 $\mathbf{\check{r}}^m=\check{c}^m+\check{B}^m p^m-\Check{\Tilde{D}}^m \mathbf{\tilde{u}}^m$.

\textcolor{darkpeagreen}{Test}:
 If $\norm{\mathbf{\check{r}}^m}$ is small, go to \textcolor{darkpeagreen}{Stop}.

\textcolor{darkpeagreen}{Update}:
 $\bvec{\beta}^{m+1}=\bvec{\beta}^m+R\omega\Check{\Check{D}}^{-1}\mathbf{\check{r}}^m$
 and $m\leftarrow m+1$. \\ \hspace*{1.5in} Go to \textcolor{darkpeagreen}{Solve}.

\textcolor{darkpeagreen}{Stop}:
 $\bvec{\beta}^*\approx\bvec{\beta}^m$

\newpage

\textcolor{greyblue}{\LARGE Convergence History of Method:}

\begin{center}
Example Domain and DOFs \rule[-1ex]{0em}{2ex} \\
{\setlength\unitlength{7pt}
\begin{picture}(36,12)(4,0)
\multiput(0,0)(18,0){2}{
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\put(0, 8){\line(1,0){18}}\multiput(3, 8)(6,0){3}{\makebox(0,0){$\times$}}
\put( 6,0){\line(0,1){12}}\multiput( 6,2)(0,4){3}{\makebox(0,0){$\times$}}
\put(12,0){\line(0,1){12}}\multiput(12,2)(0,4){3}{\makebox(0,0){$\times$}}
}
\textcolor{seafoamgreen}{\multiput(1.75,0)(6,0){3}{\multiput(0,2)(0,4){3}{\makebox(0,0){$\bullet$}}}}
\put(-1.25,0){\linethickness{1.5pt}\framebox(18,12){}}}
%\thicklines
%\put(0,0){\line(1,0){18}}\put(0,12){\line(1,0){18}}
%\put(0,0){\line(0,1){12}}\put(18,0){\line(0,1){12}}}
\multiput(19.5,2)(0,4){3}{\makebox(0,0){$\times$}}
\end{picture}}

\rule{0em}{0ex}

\begin{tabular}{c c}
\raisebox{0.75in}[0pt]{
\begin{tabular}{c c}
& {\tiny iteration number \rule{0in}{1.0in}} \\
\rotatebox{90}{\tiny log$_{10}$ residual \& error} &
\includegraphics[keepaspectratio=true,width=3in]{nonlinear/jacobi-like-conv-hist} \\
\end{tabular}
} &
\begin{tabular}{c c}
\raisebox{1.5in}[0pt]{\small $\beta_1$} &
\includegraphics[keepaspectratio=true,width=3in]{nonlinear/jacobi-like-conv-graph} \\
& {\small $\beta_2$} \\
\end{tabular} \\
\end{tabular}
\end{center}

\newpage

\textcolor{greyblue}{\LARGE Another way to look at the problem:}

{\Large
Find $\bvec{\beta^*}$ so $\mathbf{\check{r}}(\bvec{\beta^*})=\bvec{0}$.

$\mathbf{\check{r}}(\bvec{\beta})$ is a rational polynomial function of $\bvec{\beta}$.

We want its root\strut.
}

\newpage

\textcolor{greyblue}{\LARGE Another way to look at the problem:}

{\Large
Find $\bvec{\beta^*}$ so $\mathbf{\check{r}}(\bvec{\beta^*})=\bvec{0}$.

$\mathbf{\check{r}}(\bvec{\beta})$ is a rational polynomial function of $\bvec{\beta}$.

We want its root\strut.
}

\begin{center}
\textcolor{bo}{\LARGE Use Newton's Method!} \\
{\small (or Broyden's etc.)}
\end{center}

\newpage

% ----------------------------------------------------------------------
% Page ?: newton's method in uzawa framework
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Uzawa-like Method:}

\textcolor{bo}{Goal}:
 Find $\bvec{\beta}^*$ so that $\mathbf{\check{u}}(\bvec{\beta}^*)=\mathbf{0}$.

\textcolor{darkpeagreen}{Initialize}:
 $m=0$ and $\bvec{\beta}^0=\bvec{1}$.

\textcolor{darkpeagreen}{Solve}: $$\begin{bmatrix}
\mathit{0} & \left(\tilde{B}^m\right)^T \\
-\tilde{B}^m & \Tilde{\Tilde{D}}^m \\
\end{bmatrix}
\begin{bmatrix}
p^m \\ \mathbf{\tilde{u}}^m \\
\end{bmatrix}
=
\begin{bmatrix}
b \\ \tilde{c}^m \\
\end{bmatrix}$$

\textcolor{darkpeagreen}{Compute}:
 $\mathbf{\check{r}}^m=\check{c}^m+\check{B}^m p^m-\Check{\Tilde{D}}^m \mathbf{\tilde{u}}^m$.

\textcolor{darkpeagreen}{Test}:
 If $\norm{\mathbf{\check{r}}^m}$ is small, go to \textcolor{darkpeagreen}{Stop}.

\textcolor{darkpeagreen}{Update}:
 $\bvec{\beta}^{m+1}=\bvec{\beta}^m+R\omega\Check{\Check{D}}^{-1}\mathbf{\check{r}}^m$
 and $m\leftarrow m+1$. \\ \hspace*{1.5in} Go to \textcolor{darkpeagreen}{Solve}.

\textcolor{darkpeagreen}{Stop}:
 $\bvec{\beta}^*\approx\bvec{\beta}^m$

\newpage

\textcolor{greyblue}{\LARGE Nonlinear Corrector Method:}

\textcolor{bo}{Goal}:
 Find $\bvec{\beta}^*$ so that $\mathbf{\check{r}}(\bvec{\beta}^*)=\mathbf{0}$.

\textcolor{darkpeagreen}{Initialize}:
 $m=0$ and $\bvec{\beta}^0=\bvec{1}$.

\textcolor{darkpeagreen}{Solve}: $$\begin{bmatrix}
\mathit{0} & \left(\tilde{B}^m\right)^T \\
-\tilde{B}^m & \Tilde{\Tilde{D}}^m \\
\end{bmatrix}
\begin{bmatrix}
p^m \\ \mathbf{\tilde{u}}^m \\
\end{bmatrix}
=
\begin{bmatrix}
b \\ \tilde{c}^m \\
\end{bmatrix}$$

\textcolor{darkpeagreen}{Compute}:
 $\mathbf{\check{r}}^m=\check{c}^m+\check{B}^m p^m-\Check{\Tilde{D}}^m \mathbf{\tilde{u}}^m$.

\textcolor{darkpeagreen}{Test}:
 If $\norm{\mathbf{\check{r}}^m}$ is small, go to \textcolor{darkpeagreen}{Stop}.

\textcolor{darkpeagreen}{Update}:
 $\bvec{\beta}^{m+1}=\bvec{\beta}^m+\mathcal{C}\left(R\omega\Check{\Check{D}}^{-1}\mathbf{\check{r}}^m\right)$
 and $m\leftarrow m+1$. \\ \hspace*{1.5in} Go to \textcolor{darkpeagreen}{Solve}.

\textcolor{darkpeagreen}{Stop}:
 $\bvec{\beta}^*\approx\bvec{\beta}^m$

\newpage

% ----------------------------------------------------------------------
% Page ?: sample result
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Convergence History of Method \\ \hspace*{0.5in}(With Newton Corrector):}

\begin{center}
%\rule{0in}{0.0in}
%
\begin{tabular}{c c}
& {\small \hspace*{0.5in}iteration number} \\
\raisebox{1in}[0in]{\rotatebox{90}{\small log$_{10}$ error}} &
\includegraphics[keepaspectratio=true,width=5in]{nonlinear/newton-conv-hist} \\
\end{tabular}

\rule{0in}{0.0in}

\textcolor{bo}{\large Quadratic convergence!}
\end{center}

\newpage

% ----------------------------------------------------------------------
% Page ?: work to be done
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Further Study:}

\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Theoretical framework for $\bvec{\beta}$, $\VcheckHh$, normalization, \\
  and application of corrector.
\item Conditions under which we can prove convergence.
\item Different correctors. \\ Time, memory, and parallelization.  Initialization.
\end{list}

\newpage

\textcolor{greyblue}{\LARGE Further Study:}

\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Theoretical framework for $\bvec{\beta}$, $\VcheckHh$, normalization, \\
  and application of corrector.
\item Conditions under which we can prove convergence.
\item Different correctors. \\ Time, memory, and parallelization.  Initialization.
\end{list}

\textcolor{greyblue}{\LARGE And Beyond:}

{\small
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Apply to other physical problems.
\item DG, expanded mixed, and $hp$ methods.
\end{list}}

\begin{center}
\begin{tabular}{c c}
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\begin{picture}(36,12)(4,0)
\multiput(0,0)(18,0){2}{
\textcolor{orangered}{%\linethickness{0.75pt}
\put(0, 4){\line(1,0){18}}\multiput(3, 4)(6,0){3}{\makebox(0,0){$\times$}}
\put(0, 8){\line(1,0){18}}\multiput(3, 8)(6,0){3}{\makebox(0,0){$\times$}}
\put( 6,0){\line(0,1){12}}\multiput( 6,2)(0,4){3}{\makebox(0,0){$\times$}}
\put(12,0){\line(0,1){12}}\multiput(12,2)(0,4){3}{\makebox(0,0){$\times$}}
}
\textcolor{seafoamgreen}{\multiput(1.75,0)(6,0){3}{\multiput(0,2)(0,4){3}{\makebox(0,0){$\bullet$}}}}
\put(-1.25,0){\linethickness{1.5pt}\framebox(18,12){}}}
%\thicklines
%\put(0,0){\line(1,0){18}}\put(0,12){\line(1,0){18}}
%\put(0,0){\line(0,1){12}}\put(18,0){\line(0,1){12}}}
\multiput(19.5,2)(0,4){3}{\makebox(0,0){$\times$}}
\end{picture}} &
{\setlength\unitlength{7pt}
\begin{picture}(36,12)(4,0)
\multiput(0,0)(18,0){2}{
\textcolor{orangered}{%\linethickness{0.75pt}
\put(0, 4){\line(1,0){18}}\multiput(3, 4)(6,0){3}{\makebox(0,0){$\times$}}
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\end{tabular}
\end{center}

\newpage

% ----------------------------------------------------------------------
% Interlude: Outline
% ----------------------------------------------------------------------

\centerline{ \textcolor{obscureweakblue}{\huge Outline:} }

\begin{firstheadlineitemize}
\item \textcolor{paleblue}{The Problem}
\item \textcolor{paleblue}{Subgrid Upscaling}
\item \textcolor{paleblue}{Two-level Multigrid-like Scheme}
\item \textcolor{paleblue}{Change the Matrix, not the RHS}
\item \textcolor{greyblue}{Applications}
\end{firstheadlineitemize}

\newpage

% ----------------------------------------------------------------------
% Page ?: describe some computationally infeasible problems
% ----------------------------------------------------------------------

\textcolor{greyblue}{\LARGE Goal:}

{\large Faster simulations. \\ (Bigger problems, Better sampling, Etc.)}

\newpage

\textcolor{greyblue}{\LARGE Goal:}

{\large Faster simulations. \\ (Bigger problems, Better sampling, Etc.)

\rule{0in}{0in}

Some current computationally infeasible problems:
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Lawyer Canyon
\item Vuggy Data
%\item Optimization Benchmark
\item Geostatistical correlation length Vs.\ $H$
\end{list}}

\newpage

\textcolor{greyblue}{\LARGE Goal:}

{\large Faster simulations. \\ (Bigger problems, Better sampling, Etc.)

\rule{0in}{0in}

More applications:
\begin{list}{\textcolor{darkorange}{\small $\bullet$}}{}
\item Other phyical problems
\item Other FEMs
\end{list}}

%{\tiny are the vugs connected? new realizations w/different
%spacing/degree-of-connectedness of vugs -> how does this affect flow?}

\end{document}
