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\title{{\huge Parallel Programming}\\Presentation of the exercises}
\author[N. Richart, E. Lanti]{Nicolas Richart \\ Emmanuel Lanti}
\date{\nth{14} - \nth{18} of November 2022}
\begin{document}
\begin{frame}[plain]
\titlepage
\end{frame}
\begin{frame}[t,fragile]
\frametitle{PI}
\framesubtitle{}
\begin{itemize}
\item The goal of this exercise is to compute an approximate value of $\pi$
\item To compute $\pi$, we will use two identities:
\begin{equation}
\label{eq:1}
\arctan(1) = \frac{\pi}{4}
\end{equation}
\begin{equation}
\label{eq:2}
\int_{0}^{x} \frac{1}{1+y^{2}} {\rm d}y = arctan(x)
\end{equation}
Then, it follows that
\begin{equation}
\label{eq:4}
\pi = 4 \int_{0}^{1} \frac{1}{1+x^{2}} {\rm d}x
\end{equation}
\item Using a Riemann sum, we can approximate the integral in Eq.\,\eqref{eq:2}
\begin{equation}
\label{eq:3}
\pi \approx \frac{4}{n}\sum_{i=0}^{n}\frac{1}{1+(i/n)^{2}}
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}[t,fragile]
\frametitle{Poisson}
\framesubtitle{}
\begin{itemize}
\item Let us consider the following Poisson problem
\begin{equation}
\label{eq:5}
\left\{ \begin{array}{rcl}
\Delta u = f(x, y) & \text{in} & \Omega, \\
u(x, y) = 0 & \text{on} & \partial\Omega.
\end{array}\right.
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}[t,fragile]
\frametitle{Poisson}
\framesubtitle{}
\begin{itemize}
\item The 2D domain is uniformly discretized using $N+1$ points in each
direction:
\begin{equation}
\label{eq:6}
\left\{ \begin{array}{l}
(x_{i}, y_{j}), \\
\{x_{i}\}, i = 1, ..., N+1, \\
\{j_{i}\}, i = 1, ..., N+1.
\end{array}\right.
\end{equation}
The grid spacings are
$\Delta x = \Delta y = 1 / N$ (we assume domains of length one).
\item The Laplacian is approximated using second-order centered
finite-differences:
\begin{align*}
\Delta u(x, y) &= \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} \approx \frac{u_{i+1, j}- 2u_{i,j} + u_{i-1, j}}{\Delta x^{2}} + \frac{u_{i, j+1}- 2u_{i,j} + u_{i, j-1}}{\Delta y^{2}}\\
&= \frac{u_{i+1, j} + u_{i-1, j} + u_{i, j+1} + u_{i, j-1} - 4u_{i, j}}{\Delta x^{2}}
\end{align*}
\end{itemize}
\end{frame}
\begin{frame}[t, fragile]
\frametitle{Poisson}
\framesubtitle{}
\begin{itemize}
\item This defines a 4-point stencil
\end{itemize}
\vspace{4cm}
\begin{itemize}
\item The system of linear equations is solved using the Jacobi method
\item Iterations are done until the error is lower than a prescribed tolerance
\end{itemize}
\addimage[width=3cm]{figures/stencil}{6.5cm}{3.8cm}
\end{frame}
\begin{frame}[t,fragile]
\frametitle{Poisson}
\framesubtitle{}
\begin{itemize}
\item Let solve the Poisson problem with
\begin{equation}
u(x, y) = 0\label{eq:8}
\end{equation}
on the boundaries and
\begin{equation}
f = -200\pi^{2}\sin(10\pi x)\sin(10\pi y)\label{eq:7}
\end{equation}
\end{itemize}
\addimage[width=6.5cm]{figures/poisson}{4.5cm}{0.5cm}
\end{frame}
\end{document}
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