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%                   LATEX TEMPLATE FOR PhD THESIS
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% based on Harish Bhanderi's PhD/MPhil template, then Uni Cambridge
% http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/ThesisStyle/
% corrected and extended in 2007 by Jakob Suckale, then MPI-CBG PhD programme
% and made available through OpenWetWare.org - the free biology wiki


%: Style file for Latex
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%: ----- font
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%: ----- title of chapters
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\makeglossaries
% use makeglossaries thesis in terminal

%: ----- theorem environment
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\newtheorem{proposition}[theorem]{Proposition}
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%: Macro file for Latex
% Macros help you summarise frequently repeated Latex commands.
% Here, they are placed in an external file /Latex/Macros/MacroFile1.tex
% An macro that you may use frequently is the figuremacro (see introduction.tex)
\include{Latex/Macros/MacroFile1}

%: ----------------------------------------------------------------------
%:                  TITLE PAGE: name, degree,..
% ----------------------------------------------------------------------
% below is to generate the title page with crest and author name

%if output to PDF then put the following in PDF header
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    \pdfinfo { /Title  (Assimilation of velocity data into fluid dynamics simulations, an application to computational hemodynamics)
               /Creator (TeX)
               /Producer (pdfTeX)
               /Author (Marta D'Elia mdelia2@emory.edu)
               /CreationDate (D:20110726)  %format D:YYYYMMDDhhmmss
               /ModDate (D:20110726)
               /Subject (applied mathematics)
               /Keywords (computational fluid dynamics, inverse problems, optimal control, data assimilation, hemodynamics) }
    \pdfcatalog { /PageMode (/UseOutlines)
                  /OpenAction (fitbh)  }
\fi


\title{Assimilation of velocity data into fluid dynamics simulations, an application to computational hemodynamics}



% ----------------------------------------------------------------------
% The section below defines www links/email for author and institutions
% They will appear on the title page of the PDF and can be clicked
\ifpdf
  \author{\href{mailto:mdelia2@emory.edu}{Marta D'Elia}}
%  \cityofbirth{born in XYZ} % uncomment this if your university requires this
%  % If city of birth is required, also uncomment 2 sections in PhDthesisPSnPDF
%  % Just search for the "city" and you'll find them.
  \collegeordept{\href{http://www.mathcs.emory.edu}{Mathematics and Computer Science department}}
  \university{\href{http://www.emory.edu}{Emory University}}

  % The crest is a graphics file of the logo of your research institution.
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% If you are not creating a PDF then use the following. The default is PDF.
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  \author{Marta D'Elia}
%  \cityofbirth{born in XYZ}
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  \crest{\includegraphics[width=4cm]{logo}}
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%\renewcommand{\submittedtext}{change the default text here if needed}
\degree{Philosophi\ae Doctor (PhD) in Applied Mathematics}
\degreedate{2011 November}


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%: --------------------------------------------------------------
%:                  FRONT MATTER: dedications, abstract,..
% --------------------------------------------------------------



\begin{document}


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%: ----------------------- distribution agreement ------------------------
\pagestyle{empty}
\noindent {\bf Distribution Agreement}\\

\noindent In presenting this thesis or dissertation as a partial fulfillment of the requirements for an
advanced degree from Emory University, I hereby grant to Emory University and its agents
the non-exclusive license to archive, make accessible, and display my thesis or dissertation
in whole or in part in all forms of media, now or hereafter known, including display on
the world wide web. I understand that I may select some access restrictions as part of
the online submission of this thesis or dissertation. I retain all ownership rights to the
copyright of the thesis or dissertation. I also retain the right to use in future works (such
as articles or books) all or part of this thesis or dissertation.

\vspace{3cm}

\noindent Signature:

\vspace{.5cm}

\noindent \line(1,0){150} \hspace{1.5cm} \line(1,0){70} \\
\noindent Marta D'Elia \hspace{5.4cm} Date


\cleardoublepage
\newpage

%: ----------------------- approval sheet ------------------------
\begin{center}

\large{Assimilation of velocity data into fluid dynamic simulations, 

an application to computational hemodynamics}

\vspace{.5cm}

\large{by}

\vspace{.5cm}

\large{Marta D'Elia}

\large{Doctor of Philosophy}

\vspace{.5cm}

\large{Mathematics and Computer Science}\\

\vspace{1.cm}

\line(1,0){250}

\large{Alessandro Veneziani, Ph.D.}

\large{Advisor}

\vspace{1.cm}

\line(1,0){250}

\large{Michele Benzi, Ph.D.}

\large{Committee Member}

\vspace{1.cm}

\line(1,0){250}

\large{James Nagy, Ph.D.}

\large{Committee Member}

\vspace{1.cm}

Accepted:

\line(1,0){250}

\large{Lisa A. Tedesco, Ph.D.}

\large{Dean of the James T. Laney School of Graduate Studies}

\vspace{1.5cm}

\line(1,0){120}

\large{Date}

\end{center}

%: ----------------------- abstract ------------------------
\cleardoublepage
\newpage

\begin{singlespace}
\begin{center}
Assimilation of velocity data into fluid dynamic simulations, \\
an application to computational hemodynamics

\vspace{2cm}

By

\vspace{0.7cm}

Marta D'Elia

M.S. in Mathematical Engineering, Politecnico di Milano, 2007

B.S. in Mathematical Engineering, Politecnico di Milano, 2005

\vspace{2cm}

Advisor: Alessandro Veneziani, Ph.D.

\vspace{4cm}

An abstract of

A dissertation submitted to the Faculty of the

James T. Laney School of Graduate Studies of Emory University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in Mathematics and Computer Science

2011

\end{center}
\end{singlespace}


\cleardoublepage
\newpage
%: ----------------------- abstract text ------------------------
\begin{singlespace}
\begin{center}

{\bf Abstract}\\

\vspace{1cm}

Assimilation of velocity data into fluid dynamic simulations, \\
an application to computational hemodynamics

By Marta D'Elia
\end{center}

\vspace{1cm}


Cardiovascular applications recently fostered the development of numerical methods for fluid dynamics. Furthermore, thanks to new precise measurement devices and efficient image processing techniques, medicine is experiencing a tremendous increment of available data, inevitably affected by noise. Beyond validation, these data can be combined with numerical simulations in order to develop mathematical tools, known as data assimilation (DA) methods, of clinical impact. In the context of hemodynamics, accuracy and reliability of assimilated solutions are particularly crucial in view of possible applications in the clinical practice. Hence, it is fundamental to quantify the uncertainty of numerical results.

In this thesis, we propose a robust DA technique for the inclusion of noisy velocity measures, collected from magnetic resonance imaging, into the simulation of hemodynamics equations, namely the incompressible Navier-Stokes equations (NSE). The technique is formulated as a control problem where a weighted misfit between velocity and data is minimized under the constraint of the NSE; the optimization problem is solved with a discretize then optimize approach relying on the finite element method. The control variable is the normal stress on the inflow section of the vessel, which is usually unknown in real applications. We design deterministic and statistical estimators (the latter based on a Bayesian approach to inverse problems) for the estimation of the blood velocity and its statistical properties and of related variables of medical relevance, such as the wall shear stress. We also derive conditions on data location that guarantee the existence of an optimal solution. 

Numerical simulations on 2-dimensional and axisymmetric 3-dimensional geometries show the consistency and accuracy of the method with synthetic noise-free and noisy data. Simulations on 2-dimensional geometries approximating blood vessels demonstrate the applicability of the approach for hemodynamics applications.
\end{singlespace}

\cleardoublepage
\newpage

%: ----------------------- dissertation ------------------------

\begin{singlespace}
\begin{center}
Assimilation of velocity data into fluid dynamic simulations, \\
an application to computational hemodynamics

\vspace{2cm}

By

\vspace{0.7cm}

Marta D'Elia

M.S. in Mathematical Engineering, Politecnico di Milano, 2007

B.S. in Mathematical Engineering, Politecnico di Milano, 2005

\vspace{2cm}

Advisor: Alessandro Veneziani, Ph.D.

\vspace{4cm}

A dissertation submitted to the Faculty of the

James T. Laney School of Graduate Studies of Emory University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in Mathematics and Computer Science

2011

\end{center}
\end{singlespace}



\begin{doublespace}

\cleardoublepage
\newpage

%\frontmatter
\include{0_frontmatter/acknowledgement}

\cleardoublepage
\newpage

\include{0_frontmatter/dedication}


%: ----------------------- contents ------------------------

\pagestyle{empty}
%\setcounter{secnumdepth}{3} % organisational level that receives a numbers
%\setcounter{tocdepth}{3}    % print table of contents for level 3
\tableofcontents            % print the table of contents
% levels are: 0 - chapter, 1 - section, 2 - subsection, 3 - subsection


%: ----------------------- list of figures/tables ------------------------
\cleardoublepage
\listoffigures	% print list of figures

\listoftables  % print list of tables




%: --------------------------------------------------------------
%:                  MAIN DOCUMENT SECTION
% --------------------------------------------------------------

% the main text starts here with the introduction, 1st chapter,...

\mainmatter
\setboolean{@twoside}{true} 
\pagestyle{fancy}
\renewcommand{\chaptername}{} % uncomment to print only "1" not "Chapter 1"

%: ----------------------- glossary ------------------------

% Tie in external source file for definitions: /0_frontmatter/glossary.tex
%% Glossary entries can also be defined in the main text. See glossary.tex
\include{0_frontmatter/glossary} 
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\printglossaries
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%: ----------------------- subdocuments ------------------------

% Parts of the thesis are included below. Rename the files as required.
% But take care that the paths match. You can also change the order of appearance by moving the include commands.

\include{1_introduction/introduction}	% background information
\include{2/preliminary}						% preliminary analysis
\include{3/da_nse}			
\include{4/da_bayesian}	
\include{5/da_unsteady}
\include{6/conclusion}

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%:                  BACK MATTER: appendices, refs,..
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% the back matter: appendix and references close the thesis
%\include{9_backmatter/appendix}

%: ----------------------- bibliography ------------------------

% The section below defines how references are listed and formatted
% The default below is 2 columns, small font, complete author names.
% Entries are also linked back to the page number in the text and to external URL if provided in the BibTex file.

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\renewcommand{\bibname}{References} % changes the header; default: Bibliography

\bibliography{9_backmatter/references} % adjust this to fit your BibTex file

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% Various bibliography styles exit. Replace above style as desired.

% in-text refs: (1) (1; 2)
% ref list: alphabetical; author(s) in small caps; initials last name; page(s)
%\bibliographystyle{Latex/Classes/PhDbiblio-case} % title forced lower case
%\bibliographystyle{Latex/Classes/PhDbiblio-bold} % title as in bibtex but bold
%\bibliographystyle{Latex/Classes/PhDbiblio-url} % bold + www link if provided

% % calls style file jmb.bst
% \bibliographystyle{Latex/Classes/jmb} in-text refs: author (year) without brackets
% ref list: alphabetical; author(s) in normal font; last name, initials; page(s)

%\bibliographystyle{plainnat} % calls style file plainnat.bst
% in-text refs: author (year) without brackets
% (this works with package natbib)
% --------------------------------------------------------------

\end{document}
