124 lines
5.2 KiB
TeX
124 lines
5.2 KiB
TeX
\documentclass[a4paper]{article}
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% To compile PDF run: latexmk -pdf {filename}.tex
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\usepackage{graphicx} % Used to insert images into the paper
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\graphicspath{ {} }
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\usepackage{float}
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\usepackage[justification=centering]{caption} % Used for captions
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\captionsetup[figure]{font=small} % Makes captions small
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\newcommand\tab[1][0.5cm]{\hspace*{#1}} % Defines a new command to use 'tab' in text
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% Math package
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\usepackage{amsmath}
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%enable \cref{...} and \Cref{...} instead of \ref: Type of reference included in the link
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\usepackage[capitalise,nameinlink]{cleveref}
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% Enable that parameters of \cref{}, \ref{}, \cite{}, ... are linked so that a reader can click on the number an jump to the target in the document
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\usepackage{hyperref}
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\usepackage[T1]{fontenc}
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\usepackage[utf8]{inputenc} %support umlauts in the input
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% Easier compilation
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\usepackage{bookmark}
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\usepackage{natbib}
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% \usepackage{graphicx}
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\begin{document}
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\title{Week 10 - Comparing Algorithms}
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\author{Kelvin Davis \and Jip J. Dekker\and Anthony Silvestere}
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\maketitle
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\section{Simulating Time}\label{simulating-time}
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In this section we use Cellular Automata to show how different updating
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methods can drastically affect the execution of a program.
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The cellular automata model takes a vector
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\(\mathbf{x}_t = (x_{t,1} \dots x_{t,n})\) where \(x_{t,i} \in \{0,1\}\)
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as the current state and returns another vector \(\mathbf{x}_{t+1}\).
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The next state of a cell \(x_{t+1, i}\) is dependent on its current
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state \(x_{t, i}\) and the state of the cells adjacent to it
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\(x_{t, i-1}\) and \(x_{t, i+1}\) based on a set of rules for the form
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\(f : \{0,1\}^3 \to \{0,1\}\).
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We are using the rule set
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\begin{eqnarray*}
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f(1,1,1) & = & 1\\
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f(1,0,0) & = & 1\\
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\text{otherwise }f(\_,\_,\_) & = & 0
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\end{eqnarray*}
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\subsection{Why do different patterns appear with different update
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rules?}\label{why-do-different-patterns-appear-with-different-update-rules}
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There are 6 updating schemes:
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\begin{enumerate}
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\def\labelenumi{\arabic{enumi}.}
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\tightlist
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\item
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Synchronous - updates cells all at once for each time step
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\item
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Random Independent - picks a cell at random to update
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\item
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Random Order - creates a random order to update cells for each time
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step
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\item
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Clocked - the cells update at a fixed interval but the intervals vary
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from cell to cell
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\item
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Cyclic - updates the cells one at a time from left to right
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\item
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Self-synchronising - initially updated at random, but by interacting
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with their neighbours, they gradually become synchronised
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\end{enumerate}
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The synchronous method updates all the cells at once for each time-step.
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As shown this results in a plot that is consistent. The Random
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Independent method allows any cell to be updated at any time
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\subsection{A common mistake in writing programs to run simulation
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models is to scan through an array updating each cell in turn, based on
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the current values of its neighbours. Which of the update schemes
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demonstrated corresponds to
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this?}\label{a-common-mistake-in-writing-programs-to-run-simulation-models-is-to-scan-through-an-array-updating-each-cell-in-turn-based-on-the-current-values-of-its-neighbours.-which-of-the-update-schemes-demonstrated-corresponds-to-this}
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The cycle option corresponds to updating each cell based on the current
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state of its neighbours. This can be verified by looking at the pattern
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produced be the bottom-middle plot in the figure. When the updating is
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done synchronously, the plot results in consistent parallel, diagonal
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lines, however when the cycle method is used, it can be seen that the
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updating is not dependent on \(x_{t, i-1}\) but rather \(x_{t+1, i-1}\).
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That and all of the other options incorporate an aspect of randomness
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which then results in sparse plots due to only two out of the eight
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possible rules being active i.e.
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\[\dfrac{\sum_{x\in\{0,1\}^3}{f(x)}}{|\{0,1\}^3|} = \dfrac{2}{8}\]
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\subsection{Suggest cases where the clock scheme or random asynchronous
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updating might bean appropriate way to model a system in the real
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world?}\label{suggest-cases-where-the-clock-scheme-or-random-asynchronous-updating-might-bean-appropriate-way-to-model-a-system-in-the-real-world}
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In cases where we are modelling systems over continuous time, then the
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clock scheme or random asynchronous updating would be appropriate to
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use. These might be systems like
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\section{Sensitivity analysis - critical mass in a
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nuclear}\label{sensitivity-analysis---critical-mass-in-a-nuclear}
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In this experiment, we model a chain reaction by nuclear fission and aim
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to see how the density of atoms affects the systems ability to create a
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runaway chain reaction. Specifically we are interested in find the
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critical mass as measured in density that leads to chain reactions. We
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simulate the system at varying densities between 0\% and 20\% and use
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the graphs showing the energy released from the system over time to
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gauge how where the runaway reaction occurs.
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We take measurements of the energy released at densities of 0\%, 5\%,
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8\%, 10\%, 11\%, 12\%, 13\%, 15\%, 17\% and 20\%, sampling at shorter
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intervals of density closer to the density at which the maximum reading
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of the energy released exceeds 10 .
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This breakout first happens at 12, and so we deem this to be the
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critical density of the system.
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\\
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\end{document}
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