added figures for ndarray decomposition

\subsection{Distributed multidimensional array}

In the \emph{distributed} configuration, a global multidimensional array is decomposed into several subarrays and each MPI rank typically stores the data associated to one subarray.

Since a multidimensional array stores structured data, we will consider only \emph{structured conforming} decompositions, where the array is split by hyperplanes perpendicular to one of the axes.

\Cref{fig:ndarray decomposition}\todo{draw the figure} shows a typical two-dimensional decomposition of a two-dimensional array into 9 subarrays.

\Cref{fig:ndarray decomposition} shows a typical two-dimensional decomposition of a two-dimensional array into 9 subarrays.

\begin{figure}[bt]

    \centering

    \includegraphics[scale=1.0]{./figures/ndarray_decomposition/2D_without_overlaps.pdf}

    \caption{Typical two-dimensional decomposition of a two-dimensional array into 9 subarrays assigned to ranks 0-8.}

    \label{fig:ndarray decomposition}

\end{figure}

In order to represent a distributed multidimensional array, each MPI rank needs to have the following variables:

\begin{itemize}

\inline{Introduce the concept of overlaps}

\inline{Describe the synchronizer}

\begin{figure}[bt]

    \centering

    \includegraphics[scale=1.0]{./figures/ndarray_decomposition/2D_with_overlaps.pdf}

    \caption{Typical two-dimensional decomposition of a two-dimensional array into 9 subarrays with overlaps between neighbors.}

    \label{fig:ndarray decomposition overlaps}

\end{figure}

\section{Sparse matrices}

\label{sec:sparse matrices}

\inline{Supporting paper: "Segments: Data abstraction for parallel computations on sparse data"}

pen PEN = fontsize(10) + linewidth(1.5);

unitsize(1cm);

real x0 = 0;

real x1 = x0 + 3;

real x2 = x1 + 3;

real x3 = x2 + 3;

real y0 = 0;

real y1 = y0 + 2;

real y2 = y1 + 2;

real y3 = y2 + 2;

pair p00 = (x0, y0);

pair p10 = (x1, y0);

pair p20 = (x2, y0);

pair p30 = (x3, y0);

pair p01 = (x0, y1);

pair p11 = (x1, y1);

pair p21 = (x2, y1);

pair p31 = (x3, y1);

pair p02 = (x0, y2);

pair p12 = (x1, y2);

pair p22 = (x2, y2);

pair p32 = (x3, y2);

pair p03 = (x0, y3);

pair p13 = (x1, y3);

pair p23 = (x2, y3);

pair p33 = (x3, y3);

draw(p00--p10--p11--p01--cycle, PEN);

draw(p10--p20--p21--p11, PEN);

draw(p20--p30--p31--p21, PEN);

draw(p02--p12--p13--p03--cycle, PEN);

draw(p12--p22--p23--p13, PEN);

draw(p22--p32--p33--p23, PEN);

draw(p01--p02, PEN);

draw(p11--p12, PEN);

draw(p21--p22, PEN);

draw(p31--p32, PEN);

label("Rank 0", (p00 + p11) / 2);

label("Rank 1", (p10 + p21) / 2);

label("Rank 2", (p20 + p31) / 2);

label("Rank 3", (p01 + p12) / 2);

label("Rank 4", (p11 + p22) / 2);

label("Rank 5", (p21 + p32) / 2);

label("Rank 6", (p02 + p13) / 2);

label("Rank 7", (p12 + p23) / 2);

label("Rank 8", (p22 + p33) / 2);

pen PEN2 = fontsize(10) + linewidth(1.0) + dotted;

real o = 0.4;

draw(p00-(o,o)--p30-(-o,o), PEN2);

draw(p00+(-o,o)--p30+(o,o), PEN2);

draw(p01-(o,o)--p31-(-o,o), PEN2);

draw(p01+(-o,o)--p31+(o,o), PEN2);

draw(p02-(o,o)--p32-(-o,o), PEN2);

draw(p02+(-o,o)--p32+(o,o), PEN2);

draw(p03-(o,o)--p33-(-o,o), PEN2);

draw(p03+(-o,o)--p33+(o,o), PEN2);

draw(p00-(o,o)--p03-(o,-o), PEN2);

draw(p00+(o,-o)--p03+(o,o), PEN2);

draw(p10-(o,o)--p13-(o,-o), PEN2);

draw(p10+(o,-o)--p13+(o,o), PEN2);

draw(p20-(o,o)--p23-(o,-o), PEN2);

draw(p20+(o,-o)--p23+(o,o), PEN2);

draw(p30-(o,o)--p33-(o,-o), PEN2);

draw(p30+(o,-o)--p33+(o,o), PEN2);

draw(p00-(o,0)--p00, PEN2);

draw(p01-(o,0)--p01, PEN2);

draw(p02-(o,0)--p02, PEN2);

draw(p03-(o,0)--p03, PEN2);

draw(p30+(o,0)--p30, PEN2);

draw(p31+(o,0)--p31, PEN2);

draw(p32+(o,0)--p32, PEN2);

draw(p33+(o,0)--p33, PEN2);

draw(p00-(0,o)--p00, PEN2);

draw(p10-(0,o)--p10, PEN2);

draw(p20-(0,o)--p20, PEN2);

draw(p30-(0,o)--p30, PEN2);

draw(p03+(0,o)--p03, PEN2);

draw(p13+(0,o)--p13, PEN2);

draw(p23+(0,o)--p23, PEN2);

draw(p33+(0,o)--p33, PEN2);

pen PEN = fontsize(10) + linewidth(1.5);

unitsize(1cm);

real x0 = 0;

real x1 = x0 + 3;

real x2 = x1 + 3;

real x3 = x2 + 3;

real y0 = 0;

real y1 = y0 + 2;

real y2 = y1 + 2;

real y3 = y2 + 2;

pair p00 = (x0, y0);

pair p10 = (x1, y0);

pair p20 = (x2, y0);

pair p30 = (x3, y0);

pair p01 = (x0, y1);

pair p11 = (x1, y1);

pair p21 = (x2, y1);

pair p31 = (x3, y1);

pair p02 = (x0, y2);

pair p12 = (x1, y2);

pair p22 = (x2, y2);

pair p32 = (x3, y2);

pair p03 = (x0, y3);

pair p13 = (x1, y3);

pair p23 = (x2, y3);

pair p33 = (x3, y3);

draw(p00--p10--p11--p01--cycle, PEN);

draw(p10--p20--p21--p11, PEN);

draw(p20--p30--p31--p21, PEN);

draw(p02--p12--p13--p03--cycle, PEN);

draw(p12--p22--p23--p13, PEN);

draw(p22--p32--p33--p23, PEN);

draw(p01--p02, PEN);

draw(p11--p12, PEN);

draw(p21--p22, PEN);

draw(p31--p32, PEN);

label("Rank 0", (p00 + p11) / 2);

label("Rank 1", (p10 + p21) / 2);

label("Rank 2", (p20 + p31) / 2);

label("Rank 3", (p01 + p12) / 2);

label("Rank 4", (p11 + p22) / 2);

label("Rank 5", (p21 + p32) / 2);

label("Rank 6", (p02 + p13) / 2);

label("Rank 7", (p12 + p23) / 2);

label("Rank 8", (p22 + p33) / 2);