From ca6da8498b724885d552f6b0740ca1b931711e7d Mon Sep 17 00:00:00 2001
From: "baptiste.coudray" <baptiste.coudray@etu.hesge.ch>
Date: Fri, 13 Aug 2021 13:09:56 +0200
Subject: [PATCH] Updated doc

---
 .DS_Store                                | Bin 10244 -> 0 bytes
 .gitignore                               |   2 +
 src/Makefile                             |   1 +
 src/figs/communication_1d.png            | Bin 5543 -> 8968 bytes
 src/figs/communication_2d.png            | Bin 11201 -> 14710 bytes
 src/figs/communication_3d.png            | Bin 13356 -> 14398 bytes
 src/figs/dispatch_3d.png                 | Bin 0 -> 38191 bytes
 src/figs/elem_result_and_speedup_cpu.png | Bin 83243 -> 85950 bytes
 src/figs/elem_result_and_speedup_gpu.png | Bin 75717 -> 72737 bytes
 src/figs/envelope_1d.png                 | Bin 20622 -> 14786 bytes
 src/figs/gol_result_and_speedup_cpu.png  | Bin 82568 -> 84730 bytes
 src/figs/gol_result_and_speedup_gpu.png  | Bin 69906 -> 72856 bytes
 src/figs/lbm_result_and_speedup_cpu.png  | Bin 76433 -> 83762 bytes
 src/figs/lbm_result_and_speedup_gpu.png  | Bin 69858 -> 73754 bytes
 src/figs/ring.png                        | Bin 35346 -> 28376 bytes
 src/templates/default.latex              |  10 +++
 src/text/01-references.md                |   2 +
 src/text/02-introduction.md              |   6 +-
 src/text/03-programmation-parallele.md   |  24 ------
 src/text/04-mpi.md                       |  16 ++--
 src/text/05-futhark.md                   |  55 +++-----------
 src/text/06-mpi-x-futhark.md             |  91 ++++++++++++++---------
 src/text/07-automate-elementaire.md      |  50 ++++++++-----
 src/text/08-jeu-de-la-vie.md             |  68 +++++++++++------
 src/text/09-lattice-boltzmann.md         |  20 ++---
 25 files changed, 176 insertions(+), 169 deletions(-)
 delete mode 100644 .DS_Store
 create mode 100644 .gitignore
 create mode 100644 src/figs/dispatch_3d.png

diff --git a/.DS_Store b/.DS_Store
deleted file mode 100644
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diff --git a/.gitignore b/.gitignore
new file mode 100644
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diff --git a/src/Makefile b/src/Makefile
index 6e77a67..15286ae 100644
--- a/src/Makefile
+++ b/src/Makefile
@@ -19,6 +19,7 @@ rapport.pdf: rapport.tex
 	xelatex --shell-escape $^
 	xelatex --shell-escape $^
 	xelatex --shell-escape $^
+	xelatex --shell-escape $^
 
 rapport.tex: config.yaml $(MD)
 	pandoc -s $(OPTIONS) $(PDFOPTIONS) $^ -o $@
diff --git a/src/figs/communication_1d.png b/src/figs/communication_1d.png
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diff --git a/src/figs/elem_result_and_speedup_cpu.png b/src/figs/elem_result_and_speedup_cpu.png
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diff --git a/src/figs/ring.png b/src/figs/ring.png
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diff --git a/src/templates/default.latex b/src/templates/default.latex
index e27f675..cb39994 100644
--- a/src/templates/default.latex
+++ b/src/templates/default.latex
@@ -300,6 +300,16 @@ $endif$
     \caption*{\textit{#4}}
   \end{figure}
 }
+\newcommand{\cimgl}[5]{% \img{path}{settings}{caption}{source}{label}
+  \begin{figure}
+    \centering
+    \includegraphics[#2]{#1}
+    \caption{#3}
+    \caption*{\textit{#4}}
+    \label{#5}
+  \end{figure}
+}
+
 % END OF CUSTOM ROUTINES
 
 \begin{document}
diff --git a/src/text/01-references.md b/src/text/01-references.md
index c2f3614..9483bd7 100644
--- a/src/text/01-references.md
+++ b/src/text/01-references.md
@@ -4,6 +4,8 @@
 
 \listoffigures
 
+\pagebreak
+
 \listoftables
 
 #### Reference of the URLs {-}
diff --git a/src/text/02-introduction.md b/src/text/02-introduction.md
index 1900e9f..bdae309 100644
--- a/src/text/02-introduction.md
+++ b/src/text/02-introduction.md
@@ -4,12 +4,12 @@
 
 Today, most computers are equipped with (+^GPU). They provide more and more computing cores and have become fundamental embedded high-performance computing tools. In this context, the number of applications taking advantage of these tools seems low at first glance. The problem is that the development tools are heterogeneous, complex, and strongly dependent on the (+GPU) running the code. Futhark is an experimental, functional, and architecture agnostic language; that is why it seems relevant to study it.  It allows generating code allowing a standard sequential execution (on a single-core processor), on (+GPU) (with (+CUDA) and (+OpenCL) backends), on several cores of the same processor (shared memory). To make it a tool that could be used on all high-performance platforms, it lacks support for distributed computing. This work aims to develop a library that can port any Futhark code to an (+MPI) library with as little effort as possible.
 
-To achieve that, we introduce the meaning of distributed high-performance computing, then what is (+MPI) and Futhark. We decide to implement a library that can parallelize cellular automaton in, one, two or three dimensions. By adding Futhark on top of (+MPI), the programmer will have the possibilities to compile his code in :
+To achieve that, we introduce the meaning of distributed high-performance computing, then what is (+MPI) and Futhark. We decide to implement a library that can parallelize cellular automaton in, one, two or three dimensions. By adding Futhark on top of (+MPI), the programmer will have the possibilities to compile his code in:
 
 * parallelized-sequential mode,
 * parallelized-multicore mode,
-* parallelized-OpenCL mode,
-* parallelized-CUDA mode.
+* parallelized-(+OpenCL) mode,
+* parallelized-(+CUDA) mode.
 
 Finally, we used this library by implementing a cellular automata in each dimension:
 
diff --git a/src/text/03-programmation-parallele.md b/src/text/03-programmation-parallele.md
index 7638906..70e1a67 100644
--- a/src/text/03-programmation-parallele.md
+++ b/src/text/03-programmation-parallele.md
@@ -8,28 +8,4 @@ Oracle's multithreading programming guide [@oracle_chapter_2010], define *concur
 
 Finally, parallel computing exploits the computing power of a graphics card thanks to the thousands of cores it has. This allows a gain in performance compared to the previously described calculation methods because the operations are performed simultaneously on the different cores.
 
-## Amdahl's law vs Gustafson–Barsis's law
-
-In parallel computing, two important laws give the theoretical speedup that can be expected of a system whose resources are improved, Amdahl's law and Gustafson-Barsis law.
-
-\pagebreak
-
-\cimg{figs/amdahls-law.png}{scale=0.6}{Amdahl's law}{Source: Taken from https://commons.wikimedia.org/, ref. URL01}
-
-Amdahl's law states that the program's overall speed is limited by the code that cannot be parallelized. Indeed, there will almost always be a sequential part in a code that cannot be parallelized. There is, therefore, a relationship between the ratio of parallelizable code and the overall execution speed of the program [@noauthor_amdahls_2021].
-
-In the graph above, we notice that if:
-
-* 50% of the code is parallelized, we obtain a maximum theoretical acceleration of x2 from 16 processors,
-* 75% of the code is parallelized, we obtain a maximum theoretical acceleration of x4 from 128 processors,
-* 90% of the code is parallelized, we obtain a maximum theoretical acceleration of x10 from 512 processors,
-* 95% of the code is parallelized, we obtain a maximum theoretical acceleration of x20 from 4096 processors.
-
-\pagebreak
-
-\cimg{figs/gustafson-law.png}{scale=0.75}{Gustafson–Barsis's law}{Source: Taken from https://commons.wikimedia.org/, ref. URL02}
-
-Gustafson's law says that the more significant the amount of data to be processed, the more advantageous it is to use many processors. Thus, the acceleration is linear, as can be seen on the graph [@noauthor_gustafsons_2021].
-On the graph, we notice, for example, that with a code that is 90% parallelized, we have a speedup of at least x100 with 120 processors, where Amdahl's law estimated a maximum speedup of x10 with 512 processors. Gustafson's law is therefore much more optimistic in terms of performance gain.
-
 \pagebreak
diff --git a/src/text/04-mpi.md b/src/text/04-mpi.md
index 1fb0aaf..8732d36 100644
--- a/src/text/04-mpi.md
+++ b/src/text/04-mpi.md
@@ -1,21 +1,21 @@
 # Message Passing Interface
 
-In order to realize parallel programming, the standard (+MPI) was created in 1993-1994 to standardize the passage of messages between several computers or in a computer with several processors/cores [@noauthor_message_2021]. (+MPI) is, therefore, a communication protocol and not a programming language. Currently, the latest version of (+MPI) is 4.0 which approved in 2021. There are several implementations of the standard:
+In order to realize distributed programming, the standard (+MPI) was created in 1993-1994 to standardize the passage of messages between several computers or in a computer with several processors/cores [@noauthor_message_2021]. (+MPI) is, therefore, a communication protocol and not a programming language. Currently, the latest version of (+MPI) is 4.0 which approved in 2021. There are several implementations of the standard:
 
 * MPICH, which support for the moment, MPI 3.1,
 * Open MPI, which support, for the moment, MPI 3.1
 
-We use Open MPI throughout this project on the cluster of the (+HES-GE).
+We use Open MPI throughout this project to distribute a cellular automaton across multiple compute nodes and exchange missing neighbors of a cell.
 
 \pagebreak
 
-## Example
+## Example by imitating a token ring network
 
 To understand the basis of (+MPI), let us look at an example mimicking a *token ring* network [@kendall_mpi_2018]. This type of network forces a process to send a message to the message in the console, for example, only if it has the token in its possession. Moreover, once it has emitted its message, the process must transmit the token to its neighbor.
 
-\cimg{figs/ring.png}{scale=0.4}{Imitation of a network in \textit{token ring}}{Source: Created by Baptiste Coudray}
+\cimgl{figs/ring.png}{scale=0.4}{Imitation of a network in \textit{token ring}}{Source: Created by Baptiste Coudray}{fig:ex-mpi}
 
-In this example, the node with the identifier zero has first the token that it will pass to node one, then it will give it to node two, and so on. The program ends when the token is back in possession of the process zero: node four sends the token to node zero.
+In this example (\ref{fig:ex-mpi}), the node with the rank zero has first the token that it will pass to node one, then it will give it to node two, and so on. The program ends when the token is back in possession of the process zero: node four sends the token to node zero.
 
 ```c
 int main(int argc, char** argv) {
@@ -54,13 +54,15 @@ int main(int argc, char** argv) {
 }
 ```
 
-Any parallel program using (+^MPI) must call the `MPI_Init` function to initialize the environment, otherwise, an error message is displayed. Then, `MPI_Comm_rank` allows us to retrieve our ID (the node number we have), and then with the `MPI_Comm_size` function, we get the number of nodes on which our program is running.
+Any parallel program using (+MPI) must call the `MPI_Init` function to initialize the environment, otherwise, an error message is displayed when using another MPI function. Then, `MPI_Comm_rank` allows us to retrieve our ID (the node number we have), and then with the `MPI_Comm_size` function, we get the number of nodes on which our program is running, in this case five.
 Thanks to the node number, the node with the identifier zero, sends the token to its neighbor via the `MPI_Send` function.
 
 So, once sent, it waits for node four to send the token via the function `MPI_Recv`. Then, the other nodes are waiting to receive the token from their neighbor to pass the token in turn.
 The nodes communicate through the communicator `MPI_COMM_WORLD`, a macro-constant designating all nodes associated with the current program.
 
-Finally, every program must terminate with the `MPI_Finalize()` function; otherwise, the execution ends with an error message.
+The function `MPI_Send` and `MPI_Recv` can be blocking or non-blocking, depending on the size of the message that we send. Indeed, there is an internal message buffer; while it is not full, the functions will be non-blocking; otherwise, they will be blocking.
+
+Finally, every program must terminate with the `MPI_Finalize` function; otherwise, the execution ends with an error message.
 
 ```bash
 mpicc ring.c -o ring
diff --git a/src/text/05-futhark.md b/src/text/05-futhark.md
index a16b1db..7530224 100644
--- a/src/text/05-futhark.md
+++ b/src/text/05-futhark.md
@@ -11,19 +11,11 @@ The main goal of Futhark is to write generic code that can compile into either:
 * sequential C,
 * sequential Python.
 
-Although a Futhark code can compile into an executable, this feature reserves for testing purposes because there is no (+IO). Thus, the main interest is to write particular functions that you would like to speed up thanks to parallel programming and compile in library mode to use in a C program.
+Although a Futhark code can compile into an executable, this feature reserves for testing purposes because there is no (+IO). Thus, the main interest is to write particular functions that you would like to speed up thanks to parallel programming and compile in library mode. In this mode, the compiler convert our Futhark code in sequential C, multi-threaded (+POSIX) C, (+OpenCL), or (+CUDA) code depending on which backend we want.
 
 \pagebreak
 
-To see the performance of Futhark, Here is an example from the Futhark site that compares the resolution time of the (+MSS) problem. The (+MSS) problem is the task of finding a contiguous subarray with the largest sum, within a given one-dimensional array A[1...n] of numbers [@noauthor_maximum_2021].
-
-\cimg{figs/mss_bench.png}{scale=0.35}{MSS runtime (lower is better)}{Source: Taken from https://futhark-lang.org/performance.html, ref. URL04}
-
-This graph shows performance of a maximum segment sum implementation in Futhark and Thrust (a C++ library developed by NVIDIA for (+GPU) programming). The sequential runtime is for Futhark code compiled to sequential (+CPU) code and the Futhark runtime is for code compiled to (+CUDA) [@henriksen_gotta_2021]. As we can see, the Futhark version is much faster than the sequential and Thrust versions, which justify using this language in this project.
-
-\pagebreak
-
-## Example 1
+## Example in pure Futhark
 
 To better understand Futhark, here is a simple example: calculating the factorial of a number [@henriksen_basic_2021].
 
@@ -32,7 +24,7 @@ let fact (n: i32): i32 = reduce (*) 1 (1...n)
 let main (n: i32): i32 = fact n
 ```
 
-Futhark does not handle recursion, so the factorial of a number is defined as the successive multiplication of numbers from one to `n`. In Futhark, this operation defines the reduction of an array with the multiplication of each value as the operation. The program's entry point, `main`, takes as parameter a number and calls the function `fact`.
+As we can see, the function `fact` defines the factorial of a number as the successive multiplication of numbers from one to `n` through the function `reduce`. The program's entry point, `main`, takes as parameter a number `n` and calls the function `fact` with this number.
 
 ```bash
 futhark opencl fact.fut
@@ -55,21 +47,21 @@ Here we compile in (+OpenCL) to run the program on the graphics card, and we run
 
 The program calculates the factorial of 12 and therefore returns 479 001 600.
 
-\pagebreak
-## Example 2
+## Example using Futhark C API
 
-In this other example, we use Futhark in a C program to perform a very specific operation, in this case to calculate the factorial of a number.
+In this other example, we use Futhark in a C program to perform a very specific operation, in this case to calculate the factorial of a number. We use the library mode of the Futhark compiler, and we use the (+OpenCL) backend to convert the Futhark code.
 
 ```futhark
 entry fact (n: i32): i32 = reduce (*) 1 (1...n)
 ```
-Functions that can be used in C code must be defined with the `entry` keyword. Therefore, the `fact` function respects this rule.
+
+In a Futhark file (`fact.fut`), we define the function `fact` like the previous example. In library mode, the compiler convert all functions preceded with the keyword `entry`. Thus, the `fact` function is respecting this rule in order to use it in a C code.
 
 ```bash
 futhark opencl --lib fact.fut
 ```
 
-Then you have to compile the Futhark code in library mode and specify the backend. Here, the factorial program is compiled in (+OpenCL). Finally, it generates a `fact.h` and `fact.c` file, which can be included in a C program.
+Then you have to convert the Futhark code in library mode and specify the backend. Here, the factorial program is converted in (+OpenCL). Finally, it generates a `fact.h` and `fact.c` file, which can be included in a C program.
 
 ```c
 #include <stdio.h>
@@ -95,8 +87,7 @@ int main(int argc, char **argv) {
 }
 ```
 
-The program initializes a Futhark configuration and a Futhark context; then, the program calls the `fact` function, which has been compiled in C. It is called via the function `futhark_entry_fact`, which takes as arguments the Futhark context, an integer pointer to store the result, and the number whose factorial desire.
-
+The program initializes a Futhark configuration and a Futhark context (`futhark_context_config_new`, `futhark_context_new`); then, the program calls the `fact` function, which has been converted in (+OpenCL). It is called via the function `futhark_entry_fact`, which takes as arguments the Futhark context, an integer pointer to store the result, and the number whose factorial desire.
 
 ```bash
 futhark opencl --library fact.fut
@@ -109,32 +100,6 @@ gcc main.c -o fact fact.o -lOpenCL -lm
 479001600
 ```
 
-The program's execution with the factorial of 12 returns the correct value, i.e. 479 001 600.
-
-## Functional programming
-
-Functional programming is a programming paradigm that considers computation as an evaluation of
-mathematical functions. The origin of functional programming comes from lambda-calculus, a formal system
-invented by Alonzo Church where everything is a function [@noauthor_programmation_2021].
-
-This paradigm avoids side effects by prohibiting the change of the value of a variable that is not defined in the current scope. Thus, it facilitates concurrent programming because shared variables cannot be modified, which reduces the bugs resulting from concurrent programmings, such as data race conditions.
-
-A functional language is a language that encourages functional programming by providing the features
-necessary to respect this programming paradigm.
-
-### Higher-order functions
-
-In functional programming, a higher-order function is a function with at least one of these characteristics:
-
-* it takes one or more functions as parameters,
-* it returns a function as a result.
-
-Thus, the best known higher-order functions are:
-
-* `map([A], A -> B) -> [B]`, takes as a parameter an array of data of type A and a function that applies to each element of the array to create a new array of data of type B,
-* `filter([A], A -> Boolean) -> [A]`, takes as a parameter an array of type A and a predicate (a function returning a boolean) which applies on each element of the array. Thus, the function returns only those elements that answer true to the predicate.
-* `reduce([A], (A, A) -> A) -> A`, takes as a parameter an array of type A and an accumulator function. This function allows to reduce an array to a single value. For example, to add the elements of an array.
-
-Using these functions becomes faster to write code and understand the code created by a person.
+After compiling in library mode our Futhark code, we compile the generated code with our main code via (+GCC). The program's execution with the factorial of 12 returns the correct value, i.e. 479 001 600.
 
 \pagebreak
diff --git a/src/text/06-mpi-x-futhark.md b/src/text/06-mpi-x-futhark.md
index aad7324..bce1044 100644
--- a/src/text/06-mpi-x-futhark.md
+++ b/src/text/06-mpi-x-futhark.md
@@ -2,7 +2,7 @@
 
 A cellular automaton consists of a regular grid of cells, each in one of a finite number of states. The grid can be in any finite number of dimensions. For each cell, a set of cells called its neighborhood is defined relative to the specified cell. An initial state (time $t = 0$) is selected by assigning a state for each cell. A new generation is created (advancing t by 1), according to some fixed rule (generally, a mathematical function) that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood. Typically, the rule for updating the state of cells is the same for each cell and does not change over time [@noauthor_automate_2021].
 
-The neighborhood of a cell is defined either by the Moore neighborhood or by the Von Neumann neighborhood. The first one defines that a cell has in a two-dimensional cellular automaton eight neighbors while the second one, four.
+The neighborhood of a cell is defined either by the Moore neighborhood or by the Von Neumann neighborhood. The first one defines that a cell has in a two-dimensional cellular automaton four neighbors while the second one, eight.
 
 \cimg{figs/neighbours.png}{scale=0.5}{Comparison between Von Neumann (left) and Moore (right) neighborhoods}{Source: Created by Baptiste Coudray}
 
@@ -22,8 +22,8 @@ These values are valid for a cellular automaton of dimension two and a Chebyshev
 Our library allows parallelizing cellular automata automatically so that the programmer only has to write the Futhark function to update his cellular automaton. Our library supports cellular automata of one, two, and three dimensions and with any types of data. The use of the Futhark language allows to quickly update the state of the cellular automaton thanks to the different backend available. Therefore, several modes are available:
 
 * parallelized-sequential, the Futhark code executes sequentially,
-* parallelized-multicore, the Futhark code executes concurrently to POSIX threads,
-* parallelized-OpenCL/CUDA, the Futhark code executes on the graphics card.
+* parallelized-multicore, the Futhark code executes concurrently to (+POSIX) threads,
+* parallelized-OpenCL/(+CUDA), the Futhark code executes on the graphics card.
 
 ### Communication
 
@@ -34,74 +34,97 @@ Communication between the different MPI tasks is necessary to recover the missin
 
 #### One dimension
 
-\cimg{figs/communication_1d.png}{scale=0.60}{Example of Cartesian virtual topology in one dimension}{Source: Created by Baptiste Coudray}
+\cimgl{figs/communication_1d.png}{scale=0.60}{Example of Cartesian virtual topology in one dimension}{Source: Created by Baptiste Coudray}{fig:comm-1d}
 
-In a one-dimensional Cartesian topology, we notice that the rows can communicate directly with their left and right neighbors even if they are at the ends of the network. Indeed, the MPI communicator is defined as cyclic, which avoids having to traverse the $N - 2$ neighbors that separate them. 
+In a one-dimensional Cartesian topology like \ref{fig:comm-1d}, we notice that the rows can communicate directly with their left and right neighbors even if they are at the ends of the network. Indeed, the MPI communicator is defined as cyclic, which avoids having to traverse the $N - 2$ neighbors that separate them. 
 
 #### Two dimensions
 
-\cimg{figs/communication_2d.png}{scale=0.60}{Example of Cartesian virtual topology in two dimensions}{Source: Created by Baptiste Coudray}
+\cimgl{figs/communication_2d.png}{scale=0.60}{Example of Cartesian virtual topology in two dimensions}{Source: Created by Baptiste Coudray}{fig:comm-2d}
 
-In a two-dimensional Cartesian topology, we notice that rows can communicate directly with their left, right, top, and bottom neighbors. When a row needs to communicate with its diagonal neighbor, we use the default communicator (`MPI_COMM_WORLD`) to communicate directly with each other without going through a neighbor.
+In a two-dimensional Cartesian topology like \ref{fig:comm-2d}, we notice that each rank can communicate directly with their left, right, top, and bottom neighbors. For each row there is an MPI communicator containing the ranks available in it:
 
-\pagebreak
+* first communicator has `Rank 0` and `Rank 1`,
+* second communicator has `Rank 2` and `Rank 3`.
+
+For each column, there is an MPI communicator containing the ranks available in it:
+
+* first communicator has `Rank 0` and `Rank 2`,
+* second communicator has `Rank 1` and `Rank 3`.
+
+Each communicator created is cyclic, so the first rank in each the communicator created can communicate with the last rank available in it. 
+
+When a rank needs to communicate with its diagonal neighbor, we use the default communicator (`MPI_COMM_WORLD`) to communicate directly with each other without going through a neighbor.
 
 ### Three dimensions
 
-\cimg{figs/communication_3d.png}{scale=0.60}{Example of Cartesian virtual topology in three dimensions}{Source: Created by Baptiste Coudray}
+\cimgl{figs/communication_3d.png}{scale=0.60}{Example of Cartesian virtual topology in three dimensions}{Source: Created by Baptiste Coudray}{fig:comm-3d}
 
-In a three-dimensional Cartesian topology, we notice that the rows have the same communication capabilities as a two-dimensional topology, but, in addition, they can communicate with their front and back neighbors. As for the other dimensions, the communicator of this dimension is cyclic.
+In a three-dimensional Cartesian topology like \ref{fig:comm-3d}, we notice that the rows have the same communication capabilities as a two-dimensional topology, but, in addition, they can communicate with their front and back neighbors. Thus, two depth communicators are created:
+
+* one containing `Rank 0` and `Rank 1`,
+* one containing `Rank 2` and `Rank 3`.
 
 ### Data dispatching
 
-The cellular automaton is shared as equally possible among the available tasks to perform more or less the same amount of work. Thus, each task has a chunk that is a part of the cellular automaton. This chunk can be of dimensions one, two, or three.
+The cellular automaton is shared as equally possible among the available tasks to perform more or less the same amount of work. Thus, each task has a chunk that is a part of the cellular automaton. This chunk match the dimension of the cellular automaton, so it can be of dimensions one, two, or three.
+
+\pagebreak
 
 #### One dimension
 
-\cimg{figs/dispatch_1d.png}{scale=0.60}{Example of sharing a cellular automaton in one dimension}{Source: Created by Baptiste Coudray}
+\cimgl{figs/dispatch_1d.png}{scale=0.50}{Example of sharing a cellular automaton in one dimension}{Source: Created by Baptiste Coudray}{fig:dd-1d}
 
-In this example, a cell automaton of dimension one, size 8, is split between three processes. As the division of the cellular automaton is not an integer, rank two have only two cells, unlike the others, which have three.
+In this example (\ref{fig:dd-1d}), a cellular automaton of dimension one, size eight, is split between three processes. As the division of the cellular automaton is not an integer, rank two have only two cells, unlike the others, which have three.
 
 #### Two dimensions
-\cimg{figs/dispatch_2d.png}{scale=0.60}{Example of sharing a cellular automaton in two dimensions}{Source: Created by Baptiste Coudray}
+\cimgl{figs/dispatch_2d.png}{scale=0.50}{Example of sharing a cellular automaton in two dimensions}{Source: Created by Baptiste Coudray}{fig:dd-2d}
 
-In this example, the cellular automaton is in two dimensions and of size $9 \times 9$. With four tasks available, it can be separated into four sub-matrices of $3 \times 3$.
+In this example (\ref{fig:dd-2d}), the cellular automaton is in two dimensions and of size $6 \times 6$. With four tasks available, it can be separated into four sub-matrices of $3 \times 3$.
+
+\pagebreak
 
 #### Three dimensions
 
-In three dimensions, the cellular automaton partitioning representation is challenging to make understandable. Thus, based on the two-dimensional partitioning, each task divides the third dimension.
-For example, a cellular automaton of size $4 \times{} 4 \times{} 4$, each process has a piece of size $2 \times{} 2 \times{} 2$.
+\cimgl{figs/dispatch_3d.png}{scale=0.40}{Example of sharing a cellular automaton in three dimension}{Source: Created by Baptiste Coudray}{fig:dd-3d}
 
-### Envelope
+In this example (\ref{fig:dd-3d}), the cellular automaton is in three dimensions and of size $4 \times{} 4 \times{} 2$. With four tasks available, it can be separated into four sub-cubes of $2 \times{} 2 \times{} 2$.
 
-The envelope of a chunk represents the missing neighbours of the cells at the extremities of the chunk. These missing cells are needed to compute the next iteration of the chunk of the cellular automaton that the process has.
+### Envelope
 
-\pagebreak
+The envelope of a chunk represents the missing neighbours (at a Chebyshev distance chose by the programmer) of the cells at the extremities of the chunk. These missing cells are needed to compute the next iteration of the chunk of the cellular automaton that the process has.
 
 #### One dimension
-\cimg{figs/envelope_1d.png}{scale=0.60}{Example of the envelope of a chunk in one dimension}{Source: Created by Baptiste Coudray}
 
-In one dimension, the Moore neighborhood of a cell includes the west-neighbor and the east-neighbor. We notice that the envelope of $R_{n}$ includes the last cell of $R_{(n-1)\:\%\:N}$ and the first cell of $R_{(n+1)\:\%\:N}$. For example, the envelope for R1 is the west-neighbor (three) and the east neighbor (seven). Thus, the ranks exchange data via MPI using the Cartesian virtual topology.
+\cimgl{figs/envelope_1d.png}{scale=0.50}{Example of the envelope of a chunk in one dimension}{Source: Created by Baptiste Coudray}{fig:env-1d}
 
-\pagebreak
+Using this one dimension cellular automaton (\ref{fig:dd-1d}), the Moore neighborhood of a cell includes the west-neighbor and the east-neighbor. We notice that the envelope of $R_{n}$ includes the last cell of $R_{(n-1)\:\%\:N}$ and the first cell of $R_{(n+1)\:\%\:N}$. 
+
+For example, the envelope for R0 is the west-neighbor of one, so eight and the east neighbor of three, so four. The envelope for R1 is the west-neighbor of four, so three and the east neighbor of six, so seven. Finally, the envelope for R2 is the west-neighbor of seven, so six and the east neighbor of eight, so one. As we can see, the Chebyshev distance of all these envelopes is one. The ranks exchange data via MPI using the Cartesian virtual topology.
 
 #### Two dimensions
 
-\cimg{figs/envelope_2d.png}{scale=0.60}{Example of the envelope of a chunk in two dimensions}{Source: Created by Baptiste Coudray}
+\cimgl{figs/envelope_2d.png}{scale=0.50}{Example of the envelope of a chunk in two dimensions}{Source: Created by Baptiste Coudray}{fig:env-2d}
 
-Using the two-dimensional cellular automaton described above, the chunk envelope of R0 requires eight communications. This example uses a Cartesian topology of size $2\times 2$ (m\times n), the neighbors are recovered as follows:
+Using the two-dimensional cellular automaton described above (\ref{fig:dd-2d}), the chunk envelope of R0 requires eight communications. This example uses a Cartesian topology of size $2\times 2$ (m\times n), the neighbors are recovered as follows:
 
-1. `North West Neighbors` are sent by $R_{ ((y - 1)\:\%:m,\:(x - 1)\%:n) }$,
-2. `North Neighbors`, are sent by $R_{ ((y - 1)\%:m,\:x) }$,
-3. `North East Neighbors`, are sent by $R_{ ((y - 1)\:\%:m,\:(x + 1)\:\%:n) }$,
-4. `East Neighbors`, are sent by $R_{ (y,\:(x + 1)\%:n) }$,
-5. `South East Neighbors`, are sent by $R_{ ((y + 1)\:\%:m,\:(x + 1)\%:n) }$,
-6. `South Neighbors`, are sent by $R_{ ((y + 1)\%:m,\:x) }$,
-7. `South West Neighbors`, are sent by $R_{ ((y + 1)\%:m,\:(x - 1)\%:n) }$,
-8. `West Neighbors`, are sent by $R_{ (y,\:(x - 1)\%:n) }$.
+1. `North West Neighbors` are sent by $R_{ ((y - 1)\:\%\:m,\:(x - 1)\:\%\:n) }$,
+2. `North Neighbors`, are sent by $R_{ ((y - 1)\:\%\:m,\:x) }$,
+3. `North East Neighbors`, are sent by $R_{ ((y - 1)\:\%\:m,\:(x + 1)\:\%\:n) }$,
+4. `East Neighbors`, are sent by $R_{ (y,\:(x + 1)\:\%\:n) }$,
+5. `South East Neighbors`, are sent by $R_{ ((y + 1)\:\%\:m,\:(x + 1)\:\%\:n) }$,
+6. `South Neighbors`, are sent by $R_{ ((y + 1)\:\%\:m,\:x) }$,
+7. `South West Neighbors`, are sent by $R_{ ((y + 1)\:\%\:m,\:(x - 1)\:\%\:n) }$,
+8. `West Neighbors`, are sent by $R_{ (y,\:(x - 1)\:\%\:n) }$.
+
+As we can see, the Chebyshev distance of this envelope is one.
 
 #### Three dimensions
 
-With a three-dimensional cellular automaton, the envelope of a chunk requires 26 MPI communications.
+With a three-dimensional cellular automaton like (\ref{fig:dd-3d}, the envelope of a chunk requires 26 MPI communications because we send:
+
+* the eight vertices,
+* the twelve edges,
+* and the six faces around the chunk. 
 
 \pagebreak
diff --git a/src/text/07-automate-elementaire.md b/src/text/07-automate-elementaire.md
index cccbdca..1a7eaf8 100644
--- a/src/text/07-automate-elementaire.md
+++ b/src/text/07-automate-elementaire.md
@@ -33,14 +33,25 @@ Iteration 0 is the initial state and only cell two is alive. To perform the next
 With the created library, we implement this (+SCA) previously described. To do this, we create a Futhark `elementary.fut` file, which is used to calculate the next state of a part of the cellular automaton.
 
 ```
-let compute_next_elems [n] (chunk_elems :[n]i8) :[]i8 = ...
+let get_neighbours [n] (elem: [n]i8) : [n](i8,i8) = ...
 
 entry next_chunk_elems [n] (chunk_elems :[n]i8) :[]i8 =
-    let next_elems = compute_next_elems chunk_elems
+    let neighbours = get_neighbours chunk_elems
+    let next_elems = ...
     in next_elems[1:n-1]
 ```
 
-Therefore, the `elementary.fut` file contains only a function that applies the rules on the cellular automaton. Note that the function returns the cellular automaton without the envelope.
+Therefore, the `elementary.fut` file contains only a function that applies the rules on the cellular automaton.
+Note that the function returns the cellular automaton without the envelope.
+
+```c
+void init_chunk_elems(chunk_info_t *ci) {
+    int8_t *data8 = ci->data;
+    for (int i = 0; i < ci->dimensions[1]; ++i) {
+        data8[i] = rand() % 2;
+    }
+}
+```
 
 ```c
 void compute_next_chunk_board(struct dispatch_context *dc, 
@@ -71,13 +82,16 @@ int main(int argc, char *argv[]) {
 
     for (int j = 0; j < N_ITERATIONS; ++j) {
         compute_next_chunk_elems(disp_context, fut_context, &ci);
+        int8_t *sca = get_data(disp_context);
     }
 
     /* ... Free resources ... */
 }
 ```
 
-Finally, a C file `main.c` is needed to create the program's entry point. We initialize the MPI and Futhark environment. Then, our library, via the function `dispatch_context_new`, by specifying the size of the cellular automaton, its data type, and the number of dimensions (one in this case). Finally, via the `compute_next_chunk_board` function, we retrieve the chunk of the cellular automaton with the envelope and call the previously created Futhark function to obtain the chunk of the cellular automaton at time $t+1$. 
+Finally, a C file `main.c` is needed to create the program's entry point. We initialize the MPI and Futhark environment. Then, our library, via the function `dispatch_context_new`, by specifying the size of the cellular automaton (600), its data type by using a predefined type in MPI (`MPI_INT8_T`), and the number of dimensions (one in this case). The function `get_chunk_info` is called to get the chunk of the cellular automaton attributed to the current rank. Thus, we initiate it with the values that we want with the function `init_chunk_elems` We call our Futhark function `futhark_entry_next_chunk_elems` to compute the new values. Given that the function is asynchronous, the function `futhark_context_sync` waits for the action to finish. Thus, we retrieve the new values with `futhark_values_i8_1d`.
+
+The entire cellular automaton is retrieved on the root node by calling the function `get_data`. The variable `sca` is not `NULL` if it is the root node.
 
 ## CPU Benchmark
 
@@ -101,6 +115,10 @@ The sequential and multicore benchmarks are performed as follows:
 | 128 | 5.316 [s] | ± 0.191 [s] | x123.7 | 15 |
 Table: Results for the parallelized-sequential version of (+SCA)
 
+This table contains the results obtained by using the backend `sequential C` of Futhark.
+
+\pagebreak
+
 | Number of tasks | Average [s] | Standard Derivation [s] | Speedup | Number of measures |
 |:---:|:---:|:---:|:---:|:---:|
 | 1 | 708.689 [s] | ± 16.036 [s] | x1.0 | 15 |
@@ -113,25 +131,23 @@ Table: Results for the parallelized-sequential version of (+SCA)
 | 128 | 5.816 [s] | ± 0.045 [s] | x121.8 | 15 |
 Table: Results for the parallelized-multicore version of (+SCA)
 
-\pagebreak
+This table contains the results obtained by using the backend `multicore` of Futhark.
 
-\cimg{figs/elem_result_and_speedup_cpu.png}{width=\linewidth}{Benchmarks of the SCA in parallelized-sequential/multicore}{Source: Realized by Baptiste Coudray}
+\cimgl{figs/elem_result_and_speedup_cpu.png}{width=\linewidth}{Benchmarks of the SCA in parallelized-sequential/multicore}{Source: Realized by Baptiste Coudray}{fig:bench-cpu-sca}
 
 We compare the average computation time for each task and each version (sequential and multicore) on the left graph. On the right graph, we compare the ideal speedup with the parallelized-sequential and multicore version speedup.
 
-The more we increase the number of tasks, the more the execution time is reduced. Thus, the parallelized-sequential or multicore version speedup follows the curve of the ideal speedup. We can see that concurrent computing does not provide a significant performance gain over sequential computing.
+The more we increase the number of tasks, the more the execution time is reduced. Thus, the parallelized-sequential or multicore version speedup follows the curve of the ideal speedup. We can see that concurrent computing does not provide a significant performance gain over sequential computing because of the overhead of creating threads.
 
 ## GPU Benchmark
 
 The (+OpenCL) and (+CUDA) benchmarks are performed as follows:
 
 * the cellular automaton has $300'000'000$ cells,
-* the number of tasks varies between $2^0$ and $2^6$.
+* the number of tasks varies between $2^0$ and $2^3$.
 * 15 measurements are performed, one measurement corresponds to one iteration,
-* the iteration is computed $50'000$ times.
-* From $2^0$ to $2^3$ tasks, an NVIDIA GeForce RTX 3090 is allocated for each task; beyond that, the eight graphics cards are shared equally among the ranks.
-
-\pagebreak
+* the iteration is computed $50'000$ times,
+* an NVIDIA GeForce RTX 3090 is allocated for each task.
 
 | Number of tasks | Number of GPUs | Average [s] | Standard Derivation [s] | Speedup | Number of measures |
 |:---:|:---:|:---:|:---:|:---:|:---:|
@@ -139,9 +155,6 @@ The (+OpenCL) and (+CUDA) benchmarks are performed as follows:
 | 2 | 2 | 83.339 [s] | ± 0.099 [s] | x2.0 | 15 |
 | 4 | 4 | 42.122 [s] | ± 0.078 [s] | x3.9 | 15 |
 | 8 | 8 | 21.447 [s] | ± 0.031 [s] | x7.7 | 15 |
-| 16 | 8 | 31.675 [s] | ± 0.056 [s] | x5.2 | 15 |
-| 32 | 8 | 43.65 [s] | ± 0.102 [s] | x3.8 | 15 |
-| 64 | 8 | 67.096 [s] | ± 0.118 [s] | x2.5 | 15 |
 Table: Results for the parallelized-OpenCL version of (+SCA)
 
 | Number of tasks | Number of GPUs | Average [s] | Standard Derivation [s] | Speedup | Number of measures |
@@ -150,15 +163,12 @@ Table: Results for the parallelized-OpenCL version of (+SCA)
 | 2 | 2 | 80.434 [s] | ± 0.094 [s] | x2.0 | 15 |
 | 4 | 4 | 40.64 [s] | ± 0.073 [s] | x3.9 | 15 |
 | 8 | 8 | 20.657 [s] | ± 0.046 [s] | x7.8 | 15 |
-| 16 | 8 | 30.749 [s] | ± 0.069 [s] | x5.2 | 15 |
-| 32 | 8 | 42.352 [s] | ± 0.117 [s] | x3.8 | 15 |
-| 64 | 8 | 65.228 [s] | ± 0.042 [s] | x2.5 | 15 |
 Table: Results for the parallelized-CUDA version of (+SCA)
 
 \pagebreak
 
-\cimg{figs/elem_result_and_speedup_gpu.png}{width=\linewidth}{Benchmarks of the SCA in parallelized-OpenCL/CUDA}{Source: Realized by Baptiste Coudray}
+\cimgl{figs/elem_result_and_speedup_gpu.png}{width=\linewidth}{Benchmarks of the SCA in parallelized-OpenCL/CUDA}{Source: Realized by Baptiste Coudray}{fig:bench-gpu-sca}
 
-With this performance test, we notice that the computation time is essentially the same in (+OpenCL) as in (+CUDA). Moreover, the parallelization follows the ideal speedup curve when the number of processes equals the number of graphics cards. However, when the eight graphics cards are shared, the speedup in (+OpenCL)/(+CUDA) crashes, and the computation time increases. Moreover, we notice that parallel computation is up to four times faster than sequential/concurrent computation when executing with a single task/graphical card.
+With this performance test (\ref{fig:bench-gpu-sca}), we notice that the computation time is essentially the same in (+OpenCL) as in (+CUDA). Moreover, the parallelization follows the ideal speedup curve. Finally, we notice that parallel computation is up to four times faster than sequential/concurrent computation when executing with a single task/graphical card.
 
 \pagebreak
diff --git a/src/text/08-jeu-de-la-vie.md b/src/text/08-jeu-de-la-vie.md
index a90e597..14e0654 100644
--- a/src/text/08-jeu-de-la-vie.md
+++ b/src/text/08-jeu-de-la-vie.md
@@ -7,12 +7,10 @@ The Game of Life is a zero-player game designed by John Horton Conway in 1970. I
 3. a dead cell with exactly three living neighbors becomes alive,
 4. a living cell with two or three living neighbors stays alive; otherwise, it dies [@noauthor_jeu_2021].
 
-## Example
+## Example with the blinker
 
 \cimg{figs/gol_blinker1.png}{scale=0.40}{First state of blinker}{Source: Taken from \url{https://commons.wikimedia.org/}, ref. URL05. Re-created by Baptiste Coudray}
 
-\pagebreak
-
 A basic example is a blinker:
 
 * the cell (one, one) and (one, three) die because they have seven dead neighbors and one living neighbor (rule n°4),
@@ -28,9 +26,45 @@ Thus, after the application of the rules, the horizontal line becomes a vertical
 We create the game of life with our library to test it with a two-dimensional cellular automaton.
 The code is relatively the same as the previous example; therefore, it is not explained, but you can find it in the Git repository.
 
+```futhark
+let count_neighbours [n][m] (board: [n][m]i8) :[n][m]i8 = ...
+let compute_next_board [n][m] (chunk_board :[n][m]i8) 
+                              (neighbours: [n][m]i8) :[][]i8 = ...
+
+entry next_chunk_board [n][m] (chunk_board :[n][m]i8) :[][]i8 =
+    let neighbours = count_neighbours chunk_board
+    let next_board = compute_next_board chunk_board neighbours
+    in next_board[1:n-1, 1:m-1]
+```
+This is a sneak peek of the `gol.fut` file. Like the (+SCA), we only update our cellular automaton.
+
+```c
+void compute_next_chunk_board(struct dispatch_context *dc, 
+        struct futhark_context *fc, chunk_info_t *ci) { 
+    struct futhark_i8_2d *fut_chunk_with_envelope = 
+            get_chunk_with_envelope(dc, fc, 1, futhark_new_i8_2d);
+    struct futhark_i8_2d *fut_next_chunk_board;
+    futhark_entry_next_chunk_board(fc, &fut_next_chunk_board,
+                                   fut_chunk_with_envelope);
+    /* ... */
+}
+
+int main(int argc, char *argv[]) {
+    /* ... */
+    int board_dimensions[2] = {800, 800};
+    struct dispatch_context *disp_context = 
+            dispatch_context_new(board_dimensions, MPI_INT8_T, 2);
+    /* ... */ 
+    for (int i = 0; i < N_MEASURES; ++i) { 
+        compute_next_chunk_board(disp_context, fut_context, &ci);
+    }
+    /* ... */
+}
+```
+
 ## CPU Benchmarks
 
-We perform benchmarks to validate the scalability of our two-dimensional parallelization when compiling in sequential, multicore, (+^OpenCL), or (+^CUDA) mode. The benchmarks are performed on the (+^HES-GE) cluster (Baobab/Yggdrasil).
+We perform benchmarks to validate the scalability of our two-dimensional parallelization when compiling in sequential, multicore, (+OpenCL), or (+CUDA) mode. The benchmarks are performed on the (+HES-GE) cluster (Baobab/Yggdrasil).
 
 The sequential and multicore benchmarks are performed as follows:
 
@@ -63,23 +97,19 @@ Table: Results for the parallelized-sequential version of Game of Life
 | 128 | 14.008 [s] | ± 0.335 [s] | x153.8 | 15 |
 Table: Results for the parallelized-multicore version of Game of Life
 
-\pagebreak
-
-\cimg{figs/gol_result_and_speedup_cpu.png}{width=\linewidth}{Benchmarks of the game of life in parallelized-sequential/multicore}{Source: Realized by Baptiste Coudray}
+\cimgl{figs/gol_result_and_speedup_cpu.png}{width=\linewidth}{Benchmarks of the game of life in parallelized-sequential/multicore}{Source: Realized by Baptiste Coudray}{fig:bench-cpu-gol}
 
 We notice an apparent difference between the parallelized-sequential and multicore version when there is only one task. The multicore version is $1.6$ times faster than the sequential version. Nevertheless, both versions have a perfect speedup. The multicore version even gets a maximum speedup of x154 with 128 tasks. This performance can be explained by the caching of data in the processor and the use of threads.
 
 ## GPU Benchmarks
 
-The (+^OpenCL) and (+^CUDA) benchmarks are performed as follows:
+The (+OpenCL) and (+CUDA) benchmarks are performed as follows:
 
 * the cellular automaton has $900'000'000$ cells,
-* the number of tasks varies between $2^0$ and $2^6$.
+* the number of tasks varies between $2^0$ and $2^3$.
 * 15 measurements are performed, one measurement corresponds to one iteration,
-* the iteration is computed $8'000$ times.
-* From $2^0$ to $2^3$ tasks, an NVIDIA GeForce RTX 3090 is allocated for each task; beyond that, the eight graphics cards are shared equally among the ranks.
-
-\pagebreak
+* the iteration is computed $8'000$ times,
+* an NVIDIA GeForce RTX 3090 is allocated for each task.
 
 | Number of tasks | Number of GPUs | Average [s] | Standard Derivation [s] | Speedup | Number of measures |
 |:---:|:---:|:---:|:---:|:---:|:---:|
@@ -87,9 +117,6 @@ The (+^OpenCL) and (+^CUDA) benchmarks are performed as follows:
 | 2 | 2 | 115.4 [s] | ± 0.07 [s] | x2.0 | 15 |
 | 4 | 4 | 58.019 [s] | ± 0.104 [s] | x4.0 | 15 |
 | 8 | 8 | 29.157 [s] | ± 0.061 [s] | x7.9 | 15 |
-| 16 | 8 | 30.579 [s] | ± 0.085 [s] | x7.5 | 15 |
-| 32 | 8 | 32.323 [s] | ± 0.045 [s] | x7.1 | 15 |
-| 64 | 8 | 35.551 [s] | ± 0.133 [s] | x6.5 | 15 |
 Table: Results for the parallelized-OpenCL version of Game of Life
 
 | Number of tasks | Number of GPUs | Average [s] | Standard Derivation [s] | Speedup | Number of measures |
@@ -98,15 +125,10 @@ Table: Results for the parallelized-OpenCL version of Game of Life
 | 2 | 2 | 109.598 [s] | ± 0.109 [s] | x2.0 | 15 |
 | 4 | 4 | 55.039 [s] | ± 0.1 [s] | x4.0 | 15 |
 | 8 | 8 | 27.737 [s] | ± 0.05 [s] | x7.9 | 15 |
-| 16 | 8 | 29.174 [s] | ± 0.079 [s] | x7.5 | 15 |
-| 32 | 8 | 30.844 [s] | ± 0.051 [s] | x7.1 | 15 |
-| 64 | 8 | 34.192 [s] | ± 0.12 [s] | x6.4 | 15 |
 Table: Results for the parallelized-CUDA version of Game of Life
 
-\pagebreak
-
-\cimg{figs/gol_result_and_speedup_gpu.png}{width=\linewidth}{Benchmarks of the game of life in parallelized-OpenCL/CUDA}{Source: Realized by Baptiste Coudray}
+\cimgl{figs/gol_result_and_speedup_gpu.png}{width=\linewidth}{Benchmarks of the game of life in parallelized-OpenCL/CUDA}{Source: Realized by Baptiste Coudray}{fig:bench-gpu-gol}
 
-With this performance test, we notice that the computation time is essentially the same in OpenCL as in CUDA. Moreover, the parallelization follows the ideal speedup curve when the number of processes equals the number of graphics cards. However, when the eight graphics cards are shared, the speedup in OpenCL/CUDA stabilize, and the computation time increases ($+7\:[s]$ between eight tasks and 64 tasks).
+With this performance test (\ref{fig:bench-gpu-gol}), we notice that the computation time is essentially the same in OpenCL as in CUDA. Moreover, the parallelization follows the ideal speedup curve. Furthermore, we notice that parallel computation is up to 15 times faster than sequential/concurrent computation when executing with a single task/graphical card.
 
 \pagebreak
diff --git a/src/text/09-lattice-boltzmann.md b/src/text/09-lattice-boltzmann.md
index 0c33cbc..832f751 100644
--- a/src/text/09-lattice-boltzmann.md
+++ b/src/text/09-lattice-boltzmann.md
@@ -10,7 +10,7 @@ fundamental research, as it keeps the cycle between the elaboration of a theory
 
 ## Parallelized version
 
-We implement the Lattice-Boltzmann Method with our library to test it with a three-dimensional cellular automaton.
+We implement the Lattice-Boltzmann Method with our library to test it with a three-dimensional cellular automaton. Each cell is containing an array of 27 floats.
 
 ## CPU Benchmark
 
@@ -48,7 +48,7 @@ Table: Results for the parallelized-multicore version of (+LBM)
 
 \pagebreak
 
-\cimg{figs/lbm_result_and_speedup_cpu.png}{width=\linewidth}{Benchmarks of the LBM in parallelized-sequential/multicore}{Source: Realized by Baptiste Coudray}
+\cimgl{figs/lbm_result_and_speedup_cpu.png}{width=\linewidth}{Benchmarks of the LBM in parallelized-sequential/multicore}{Source: Realized by Baptiste Coudray}{fig:bench-cpu-lbm}
 
 Contrary to the previous benchmarks, the speedups do not follow the ideal speedup curve. Indeed, whether in sequential or multicore, we obtain a maximum speedup with 128 tasks of x41 when we were hoping to have a speedup of x128.
 
@@ -57,10 +57,10 @@ Contrary to the previous benchmarks, the speedups do not follow the ideal speedu
 The (+OpenCL) and (+CUDA) benchmarks are performed as follows:
 
 * the cellular automaton has $27'000'000$ cells,
-* the number of tasks varies between $2^0$ and $2^6$.
+* the number of tasks varies between $2^0$ and $2^3$.
 * 15 measurements are performed, one measurement corresponds to one iteration,
-* the iteration is computed $3'000$ times.
-* From $2^0$ to $2^3$ tasks, an NVIDIA GeForce RTX 3090 is allocated for each task; beyond that, the eight graphics cards are shared equally among the ranks.
+* the iteration is computed $3'000$ times,
+* an NVIDIA GeForce RTX 3090 is allocated for each task.
 
 | Number of tasks | Number of GPUs | Average [s] | Standard Derivation [s] | Speedup | Number of measures |
 |:---:|:---:|:---:|:---:|:---:|:---:|
@@ -68,9 +68,6 @@ The (+OpenCL) and (+CUDA) benchmarks are performed as follows:
 | 2 | 2 | 99.677 [s] | ± 0.038 [s] | x2.1 | 15 |
 | 4 | 4 | 40.71 [s] | ± 0.076 [s] | x5.2 | 15 |
 | 8 | 8 | 20.8 [s] | ± 0.031 [s] | x10.1 | 15 |
-| 16 | 8 | 22.88 [s] | ± 0.064 [s] | x9.2 | 15 |
-| 32 | 8 | 22.47 [s] | ± 0.036 [s] | x9.4 | 15 |
-| 64 | 8 | 23.848 [s] | ± 0.035 [s] | x8.8 | 15 |
 Table: Results for the parallelized-OpenCL version of (+LBM)
 
 | Number of tasks | Number of GPUs | Average [s] | Standard Derivation [s] | Speedup | Number of measures |
@@ -79,13 +76,10 @@ Table: Results for the parallelized-OpenCL version of (+LBM)
 | 2 | 2 | 99.177 [s] | ± 0.056 [s] | x2.1 | 15 |
 | 4 | 4 | 40.24 [s] | ± 0.074 [s] | x5.2 | 15 |
 | 8 | 8 | 20.459 [s] | ± 0.037 [s] | x10.2 | 15 |
-| 16 | 8 | 22.837 [s] | ± 0.037 [s] | x9.1 | 15 |
-| 32 | 8 | 22.361 [s] | ± 0.024 [s] | x9.3 | 15 |
-| 64 | 8 | 23.688 [s] | ± 0.051 [s] | x8.8 | 15 |
 Table: Results for the parallelized-CUDA version of (+LBM)
 
-\cimg{figs/lbm_result_and_speedup_gpu.png}{width=\linewidth}{Benchmarks of the LBM in parallelized-OpenCL/CUDA}{Source: Realized by Baptiste Coudray}
+\cimgl{figs/lbm_result_and_speedup_gpu.png}{width=\linewidth}{Benchmarks of the LBM in parallelized-OpenCL/CUDA}{Source: Realized by Baptiste Coudray}{fig:bench-gpu-gol}
 
-Like the other benchmarks, there is very little difference between the OpenCL and CUDA versions (computation time and speedup). We get a more than ideal speedup with 2, 4, and 8 tasks/GPUs (x2.1, x5.2, and x10.2, respectively). When several tasks use the same graphics card, the computation time stabilizes at 22 seconds, and the speedup stops increasing.
+Like the other benchmarks (\ref{fig:bench-gpu-elem}, \ref{fig:bench-gpu-gol}), there is very little difference between the OpenCL and CUDA versions (computation time and speedup). We get a more than ideal speedup with 2, 4, and 8 tasks/GPUs (x2.1, x5.2, and x10.2, respectively). Finally, we notice that parallel computation is up to 3 times faster than sequential/concurrent computation when executing with a single task/graphical card.
 
 \pagebreak
-- 
GitLab