Commit 7c320fec authored by conmccoid's avatar conmccoid
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Temp: CV and Research Statement updates/init commit

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%%%%%%%%%%%%%%%%% NAME OF APPLICANT %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% NAME OF APPLICANT %%%%%%%%%%%%%%%%%%%
\noindent \LARGE{\textbf{Conor McCoid}} \\ \noindent \LARGE{\textbf{Conor McCoid}} \\
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conmccoid@gmail.com & \hspace{1in} \\ conmccoid@gmail.com & \hspace{1in} \\
+41 76 465 08 92 & \hspace{1in} \\ +41 76 465 08 92 & \hspace{1in} \\
\url{www.unige.ch/~mccoid} & \hspace{1in} \\ \url{www.unige.ch/~mccoid} & \hspace{1in} \\
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% The main body is contained in a tabular environment. To move sections onto the next page, simply end the tabular environment and begin a new tabular environment. % The main body is contained in a tabular environment. To move sections onto the next page, simply end the tabular environment and begin a new tabular environment.
\noindent \begin{tabular}{@{} l l} \noindent \begin{tabular}{@{} l l}
\Large{Education} & \textbf{Doctorat en mathématiques (en cours)} \\ \Large{\'Education} & \textbf{Doctorat en mathématiques (en cours)} \\
& Université de Genève, 2018 - maintenant \\ & Université de Genève, 2018 - 2022 (anticipé) \\
& Sujet: Analyse numérique \\ & Sujet: Analyse numérique \\
& Directeur: Martin J. Gander \\
& \\ & \\
& \textbf{Masters en mathématiques appliquées} \\ & \textbf{Master en mathématiques appliquées} \\
& Simon Fraser University, 2016 - 2018 \\ & Simon Fraser University, 2016 - 2018 \\
& Sujet: Analyse numérique \\ & Sujet: Analyse numérique \\
& Directeur: Manfred Trummer \\
& \\ & \\
\Large{Recherche} & \textbf{Préconditionnement nonlinéaire} \\ \Large{Recherche} & \textbf{Préconditionnement nonlinéaire} \\
& Décomposition de domaines, Newton-Raphson, triangulations \\ & Décomposition de domaines, Newton-Raphson, triangulations \\
& Université de Genève, 2018 - maintenant \\ & Université de Genève, 2018 - maintenant \\
& Directeur: Martin Gander \\
& \\ & \\
& \textbf{Solution numérique des problèmes aux limites} \\ & \textbf{Solution numérique des problèmes aux limites} \\
& Méthodes spectrales, préconditionnement, perturbations singulières \\ & Méthodes spectrales, préconditionnement, perturbations singulières \\
& Simon Fraser University, 2015 - 2018 \\ & Simon Fraser University, 2015 - 2018 \\
& Directeur: Manfred Trummer \\
& \\ & \\
\Large{Prix et } & \textbf{Canada Graduate Scholarship - Master's Program} \\ \Large{Prix et } & \textbf{Canada Graduate Scholarship - Master's Program} \\
\Large{Bourses} & National Sciences and Engineering Research Council of Canada \\ \Large{Bourses} & National Sciences and Engineering Research Council of Canada \\
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\noindent \begin{tabular}{@{} l l} \noindent \begin{tabular}{@{} l l}
\Large{Affiches} & \textbf{Robust algorithm for intersetcions of triangles} \\ \Large{Posters} & \textbf{Robust algorithm for intersetcions of triangles} \\
& CUSO \'ecole d'hiver, 2020 \\ & CUSO \'ecole d'hiver, 2020 \\
& Villars-sur-Ollon, Suisse \\ & Villars-sur-Ollon, Suisse \\
& \\ & \\
...@@ -136,11 +138,11 @@ ...@@ -136,11 +138,11 @@
& Analyse Num\'erique, 2019-2021 \\ & Analyse Num\'erique, 2019-2021 \\
& \indent Travaux pratiques, en fran\c{c}ais et anglais \\ & \indent Travaux pratiques, en fran\c{c}ais et anglais \\
& Statistiques et M\'ethodologie Pharmaceutique, 2020 \\ & Statistiques et M\'ethodologie Pharmaceutique, 2020 \\
& \indent Exercises, en fran\c{c}ais \\ & \indent Exercices, en fran\c{c}ais \\
& M\'ethodes It\`eratives, 2020 \\ & M\'ethodes It\`eratives, 2020 \\
& \indent Exercises, en fran\c{c}ais \\ & \indent Exercices, en fran\c{c}ais \\
& Analyse I, 2019 \\ & Analyse I, 2019 \\
& \indent Exercises, en fran\c{c}ais \\ & \indent Exercices, en fran\c{c}ais \\
& Math\'ematiques G\'en\'erales, 2018 \\ & Math\'ematiques G\'en\'erales, 2018 \\
& \indent Travaux pratiques, en fran\c{c}ais et anglais \\ & \indent Travaux pratiques, en fran\c{c}ais et anglais \\
& \\ & \\
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& MACM (Math and Computing) 316, 2016-2018 \\ & MACM (Math and Computing) 316, 2016-2018 \\
& \indent Coordinateur d'atelier, en anglais \\ & \indent Coordinateur d'atelier, en anglais \\
& Math 155, 2017 \\ & Math 155, 2017 \\
& \indent Exercises, en anglais \\ & \indent Exercices, en anglais \\
& Applied Calculus Workshop, 2017 \\ & Applied Calculus Workshop, 2017 \\
& \indent Travaux pratiques, en anglais \\ & \indent Travaux pratiques, en anglais \\
& \\ & \\
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\rhead{Conor McCoid}
\begin{document}
\begin{center}
\textbf{\LARGE Research Statement}
\end{center}
%Motivation for applying to Oxford
%Relevant experience and education
%Specific areas interested in
%Introduction - What am I interested in? Why am I applying for a PhD?
To date my research has focused on numerical analysis, in particular preconditioning of methods for solving nonlinear problems.
This has included pseudospectral methods and domain decomposition methods.
In studying these topics I have delved into boundary layer problems, Newton's method, extrapolation methods, Krylov subspace methods and intersection algorithms for simplices.
%Past experiences - work with Chebyshev differentiation
%Include its possible usefulness
I worked with Professor Manfred Trummer of Simon Fraser University from 2015 to 2018.
My focus during this time was on Chebyshev pseudospectral methods.
These methods involve solving differential equations on a restricted set of collocation points by forming differentiation matrices, thereby turning nonlinear problems into linear ones.
The matrices for many of these problems are inherently ill-conditioned, making it difficult to find accurate solutions.
During the course of my masters program I developed a preconditioner for these problems, in some cases reducing the problem to a matrix-vector multiplication (nb: cite).
This was a generalization of earlier work by Wang et al. (nb: cite?) to accomodate a broader range of boundary conditions and differential equations.
In essence, this preconditioner acts as an integration matrix in much the same way one constructs differentiation matrices for the same class of methods.
Also during this time I looked into combining two spectral methods:
a resampling procedure suggested by Driscoll and Hale (nb: cite?) and;
an iterated sine-transformation designed for boundary layer problems by Tang and Trummer (nb: cite?).
It became clear that the latter of these introduced artificial boundary conditions when combined with the former.
We added a regularization which removed these conditions and improved the overall accuracy (nb: cite).
%Current experiences
Since 2018 I've been working with Professor Martin J. Gander at the University of Geneva.
I have continued to study preconditioning, though now primarily in the context of domain decomposition methods.
In a domain decomposition method the problem, usually untenably large, is subdivided into several smaller problems.
These smaller problems are solved and their solutions passed back and forth between them, informing each other until the solution to the larger problem is found.
In this context I am examining Newton's method as a way to accelerate these methods towards the final result.
By focusing on the information passed from one of the smaller problems to another and how it changes with each iteration, one can view these domain decomposition methods as highly nonlinear functions.
Applying Newton's method to these functions accelerates these methods, but only in the basin of attraction surrounding the solution.
I have found counterexamples where one can induce cycling in these methods by applying Newton's method (nb: cite in press), and I am looking to expand this list of counterexamples with the hope of finding \textit{a priori} indicators of convergence.
As well, I am investigating the link between Newton's method, extrapolation methods and Krylov subspace methods.
All three can be connected through the multisecant equations, which are themselves the generalization of the secant method to higher dimensions.
Through this connection extrapolation methods and Krylov subspace methods can be included in a broad family of methods that are in some way or another extensions of the multisecant equations.
I am also developing an algorithm for the intersection of simplices in arbitrary dimension.
Some domain decomposition methods require passing information between finite elements and volumes, and robust intersection algorithms are crucial for their success.
Having written a provably robust algorithm for the intersection of triangles (nb: cite submitted revised), the goal is now to generalize to higher dimensions.
This development has brought up a number of interesting problems, most of them concerned with how to manage the computational complexity.
%Future experiences - where do you want to take your research?
%Where do you want to take your career?
Moving forward, I am interested in both broadening my research interests and expanding on the work I've already completed.
There are many fields of mathematics, even within applied mathematics, that I would be excited to learn more about.
In particular, I am eager to apply my knowledge to more concrete applications.
We live in a world facing a myriad of crises and we must each do our part to solve them.
Two avenues of research I would be interested in continuing are pseudospectral integration matrices and parsimonious algorithms.
The integration matrices can be further generalized to inversion matrices, matrices representing the inverse operator of a given problem.
I have started developing these matrices for linear operators with constant coefficients but the theory may be applicable more broadly.
Meanwhile, the philosophy of parsimony used in the intersection algorithms has far-reaching applications.
Many other methods can benefit from applying this principle.
% Rough first draft, expand on intro paragraph, remove jargon
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\textbf{\LARGE Projet de Recherche}
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%Motivation for applying to Oxford
%Relevant experience and education
%Specific areas interested in
%Introduction - What am I interested in? Why am I applying for a PhD?
Jusqu'à maintenant ma recherche concernait l'analyse numérique, en particulier le préconditionnement de méthodes pour résoudre les problèmes non-linéaires.
Cela inclut les méthodes pseudo-spectrales et les méthodes de décomposition de domaine.
Durant ces études j'ai examiné les problèmes de couche limite, la méthode de Newton, les méthodes d'extrapolation, les méthodes de Krylov, et les algorithmes de découpe des simplexes.
%Past experiences - work with Chebyshev differentiation
%Include its possible usefulness
J'ai travaillé avec le Professeur Manfred Trummer de l'Université Simon Fraser de 2015 à 2018 en premier cycle et master.
Pendant ces années j'ai travaillé sur les méthodes pseudo-spectrales de Chebyshev.
Ces méthodes résolvent les équations différentielles par la constructions de dérivées numériques sur un ensemble de collocation avec l'aide des matrices différentielles.
Souvent, les matrices sont intrinsèquement mal conditionnées, et il est difficile de résoudre le problème avec précision.
Au cours de mon master j'ai développé une matrice de préconditionnement pour ces problèmes, qui, dans certains cas, réduit le problème à une multiplication matrice-vecteur \cite{mccoid2018preconditioning}.
Ce travail est dans l'esprit de Wang et al. \cite{wang2014well} où on inclut plus de conditions limites et des équations différentielles.
Il s'agit d'une matrice d'intégration construite d'une manière similaire aux matrices différentielles.
Aussi, pendant cette période, je me suis penché sur la combinaison de deux méthodes spectrales:
une procédure de ré-échantillonnage de Driscoll et Hale \cite{Driscoll} et
une transformation sinusoide itérative de Tang et Trummer pour les couches limites.
Il est devenu clair que la dernière introduit les conditions limites artificielles en combinaison avec la première.
Nous avons ajouté une régularisation qui a enlevé ces conditions et a amélioré la précision \cite{mccoid2019improved}.
%Current experiences
Depuis 2018, j'ai travaillé avec le Professeur Martin J. Gander de l'Université de Genève.
Je continue à m'intéresser au préconditionnement de problèmes, mais maintenant principalement les méthodes de décomposition de domaines.
Une méthode de décomposition de domaine subdivise un problème de très grande taille en problèmes petits et nombreux.
Puis, on résoud ces petits problèmes et on permute leurs solutions, jusqu'à ce que l'on trouve la solution du problème initial.
Dans ce contexte pour les problèmes non-linéaires, j'examine la méthode de Newton comme moyen d'accélérer ces méthodes.
En mettant l'accent sur l'information passée entre les problèmes petits et comment elle change itérativement, on peut considérer ces méthodes comme des fonctions non-linéaires.
On applique la méthode de Newton à ces fonctions pour accélérer les méthodes de décomposition de domaine, mais seulement dans le bassin d'attrait qui entoure la solution.
J'ai trouvé des contre-exemples où on peut créer des suites périodiques, et donc divergentes, par l'application de la méthode de Newton \cite{mccoid2021cycles} et je poursuis la recherche de ces contre-exemples dans le but de trouver des indicateurs de convergence.
De plus, je détermine le lien entre la méthode de Newton, les méthodes d'extrapolation et les méthodes de Krylov.
Les trois sont liées par les équations multisécantes, qui elles-mêmes sont la généralisation de la méthode de la sécante en dimension plus élevée.
Par cette connexion les méthodes d'extrapolation et les méthodes de Krylov appartiennent à une famille de méthodes qui sont des généralisations des équations multisécantes.
Je développe aussi un algorithme de découpe des simplexes en dimension arbitraire.
Quelques méthodes de décomposition de domaine échangent l'information entre des éléments finis, et des algorithmes robustes de découpe sont nécessaires pour que l'intégration numérique soit efficace.
J'ai écrit un algorithme de découpe des triangles dont la robustesse est prouvée \cite{mccoid2021provably} et le but est de le généraliser aux dimensions plus élevées.
Ce développement introduit plusieurs problèmes intéressants de la gestion de complexité des calculs informatiques.
%Future experiences - where do you want to take your research?
%Where do you want to take your career?
A l'avenir, j'aimerais élargir mes domaines de recherches et développer la recherche que j'ai déjà faite.
Il y a beaucoup de sujets de mathématiques, appliqués ou non, où je serais intéressé d'en apprendre plus.
En particulier, j'ai hâte de travailler sur des applications plus concrètes.
Il y a aussi deux pistes de recherche que j'aimerais à continuer:
les matrices d'intégrations pseudo-spectrales et les algorithmes robustes.
On peut généraliser les matrices d'intégrations aux matrices d'inversion qui représentent les opérateurs inverses.
J'ai commencé le développement de celles-ci par les opérateurs linéaires avec des coefficients constants, mais on peut les appliquer plus généralement.
D'ailleurs, l'idée d'économie utilisée dans la construction d'algorithmes de découpe a des applications vastes.
Beaucoup d'autres méthodes pourraient bénéficier de cette idée.
\bibliographystyle{siam}
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\usepackage{fancyhdr}
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\pagestyle{fancy}
\rhead{Conor McCoid}
\begin{document}
\begin{center}
\textbf{\LARGE Statement of Purpose}
\end{center}
%Motivation for applying to Oxford
%Relevant experience and education
%Specific areas interested in
\paragraph{}
%Introduction - What am I interested in? Why am I applying for a PhD?
I am applying to the PhD program in mathematics at ... to continue my studies and work towards a career as a professor.
My research and studies have focused on numerical analysis, especially spectral methods.
\paragraph{}
%Past experiences - work with Chebyshev differentiation
%Include its possible usefulness
For the past three years I have worked with Chebyshev pseudospectral methods under Dr. Manfred Trummer at Simon Fraser University.
I have generalized a method to precondition collocation matrices to work with a broader range of boundary conditions and differential equations.
\paragraph{}
%Work experience as an educator
While studying as a Masters student, I worked as both a teaching assistant and workshop coordinator.
I have enjoyed my time teaching, and look forward to improving as an educator.
\paragraph{}
%Future experiences - where do you want to take your research?
%Where do you want to take your career?
By generalizing the preconditioning, certain aspects of the pseudospectral methods became ambiguous.
Specifically, the process of applying the boundary conditions could be done in a number of ways.
It remains to find an optimal way to include boundary conditions in the method.
I would greatly appreciate the opportunity to study this further.
% Rough first draft, expand on intro paragraph, remove jargon
\end{document}
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@article{wang2014well,
Author = {Wang, Li-Lian and Samson, Michael Daniel and Zhao, Xiaodan},
Date-Added = {2016-01-22 01:00:30 +0000},
Date-Modified = {2016-01-22 01:00:30 +0000},
Journal = {SIAM Journal on Scientific Computing},
Number = {3},
Pages = {A907--A929},
Publisher = {SIAM},
Title = {A well-conditioned collocation method using a pseudospectral integration matrix},
Volume = {36},
Year = {2014}}
@article{Driscoll,
issn = "0272-4979",
abstract = "Boundary conditions in spectral collocation methods are typically imposed by removing some rows of the discretized differential operator and replacing them with others that enforce the required conditions at the boundary. A new approach based upon resampling differentiated polynomials into a lower-degree subspace makes differentiation matrices, and operators built from them, rectangular without any row deletions. Then, boundary and interface conditions can be adjoined to yield a square system. The resulting method is both flexible and robust, and avoids ambiguities that arise when applying the classical row deletion method outside of two-point scalar boundary-value problems. The new method is the basis for ordinary differential equation solutions in Chebfun software, and is demonstrated for a variety of boundary-value, eigenvalue and time-dependent problems.",
journal = "IMA Journal of Numerical Analysis",
pages = "108--132",
volume = "36",
publisher = "Oxford University Press",
number = "1",
year = "2016",
title = "Rectangular spectral collocation",
author = "Driscoll, Tobin A. and Hale, Nicholas",
keywords = "Spectral Method ; Collocation ; Boundary Conditions ; Chebyshev",
month = "January",
}
@article{mccoid2018preconditioning,
title={Preconditioning of spectral methods via {B}irkhoff interpolation},
author={McCoid, Conor and Trummer, Manfred R},
journal={Numerical Algorithms},
volume={79},
number={2},
pages={555--573},
year={2018},
publisher={Springer}
}
@article{mccoid2019improved,
title={Improved Resolution of Boundary Layers for Spectral Collocation},
author={McCoid, Conor and Trummer, Manfred R},
journal={SIAM Journal on Scientific Computing},
volume={41},
number={5},
pages={A2836--A2849},
year={2019},
publisher={SIAM}
}
@inproceedings{mccoid2021cycles,
title={Cycles in {N}ewton-{R}aphson preconditioned by {S}chwarz ({ASPIN} and its cousins)},
author={McCoid, Conor and Gander, Martin J.},
booktitle={Domain Decomposition Methods in Science and Engineering XXVI},
year={in press},
publisher={Springer}
}
@article{mccoid2021provably,
title={A provably robust algorithm for triangle-triangle intersections in floating-point arithmetic},
author={McCoid, Conor and Gander, Martin J.},
journal={Transactions on Mathematical Software},
year={submitted after revisions},
publisher={ACM}
}
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