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\title { Robust parameter estimation in fluid flow models from aliased velocity measurements}
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%\author[Jeremías Garay Labra]
%{Jeremías Garay Labra}
\institute [University of Groningen]
{
Bernoulli Institute\\
Faculty of Sciences and Engineering\\
University of Groningen\\ [0.5cm]
%\includegraphics[height=1.5cm]{Imagenes/escudoU2014.pdf}
% \includegraphics[height=1cm]{Imagenes/fcfm.png} \\[0.5cm]
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Jeremías Garay Labra \emph { join with} Cristobal Bertoglio.}
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\date { \today }
\begin { document}
\frame { \titlepage }
% \onslide<1->
\begin { frame}
\frametitle { Index}
\tableofcontents
\end { frame}
\section [4D flow MRI] { 4D flow MRI}
\begin { frame}
\frametitle { 4D flow MRI}
\begin { columns} [c]
\column { .5\textwidth } % Left column and width
\footnotesize
\begin { itemize}
\item <2-> Full 3D coverage of the region of interest
\item <3-> Rich post-proccesing: derived parameters
\end { itemize}
\onslide <4-> Disadvantages:
\begin { itemize}
\item <5-> Long scan time
\end { itemize}
\column { .54\textwidth } % Right column and width
\onslide <1->
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.9\textwidth] { images/4dflow.png}
\caption { \footnotesize 4D flow MRI of a human thorax}
\end { center}
\end { figure}
\end { columns}
\end { frame}
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\section { The mathematical model}
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\begin { frame}
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\frametitle { The mathematical model}
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\begin { center}
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The mathematical model
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\end { center}
\end { frame}
\begin { frame}
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\frametitle { The mathematical model}
\begin { columns} [c]
\column { .5\textwidth } % Left column and width
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\footnotesize
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\column { .54\textwidth } % Right column and width
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=1.1\textwidth] { images/full_ aorta.png}
\end { center}
\end { figure}
\end { columns}
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\end { frame}
\begin { frame}
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\frametitle { The mathematical model}
\begin { columns} [c]
\column { .5\textwidth } % Left column and width
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\footnotesize
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\begin { itemize}
\item <2-> Incompressible Navier-Stokes equations:
\begin { equation}
\begin { cases}
\displaystyle \rho \frac { \partial \vec { u} } { \partial t} + \rho \big ( \vec { u} \cdot \nabla \big ) \vec { u} - \mu \Delta \vec { u} + \nabla p = 0 \\ [0.2cm]
\nabla \cdot \vec { u} = 0 \quad \text { in} \quad \Omega \\ [0.2cm]
\vec { u} = \vec { u} _ { inlet} \quad \text { on} \quad \Gamma _ { in} \\ [0.2cm]
\vec { u} = 0 \quad \text { on} \quad \Gamma _ { walls}
\end { cases}
\end { equation}
\item <3-> \emph { Three-element} Windkessel coupling at every outlet:
\begin { equation}
\begin { cases}
\displaystyle C_ { d,l} \frac { d \pi _ l} { dt} + \frac { \pi _ l} { R_ { d,l} } = Q_ l \\ [0.2cm]
P_ l = R_ { p,l} \ Q_ l + \pi _ l
\end { cases}
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\end { equation}
\end { itemize}
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\column { .54\textwidth } % Right column and width
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\onslide <1->
\begin { figure} [!hbtp]
\begin { center}
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\includegraphics [height=0.9\textwidth] { images/windk_ model.png}
\caption { \footnotesize Schematic of the model}
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\end { center}
\end { figure}
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\end { columns}
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\end { frame}
\begin { frame}
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\frametitle { The mathematical model}
\begin { columns} [c]
\column { .5\textwidth } % Left column and width
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\footnotesize
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\begin { itemize}
\item Incompressible Navier-Stokes equations:
\begin { equation}
\begin { cases}
\displaystyle \rho \frac { \partial \vec { u} } { \partial t} + \rho \big ( \vec { u} \cdot \nabla \big ) \vec { u} - \mu \Delta \vec { u} + \nabla p = 0 \\ [0.2cm]
\nabla \cdot \vec { u} = 0 \quad \text { in} \quad \Omega \\ [0.2cm]
\vec { u} = \vec { u} _ { inlet} \quad \text { on} \quad \Gamma _ { in} \\ [0.2cm]
\vec { u} = 0 \quad \text { on} \quad \Gamma _ { walls}
\end { cases}
\end { equation}
\item \emph { Three-element} Windkessel coupling at every outlet:
\begin { equation}
\begin { cases}
\displaystyle C_ { d,l} \frac { d \pi _ l} { dt} + \frac { \pi _ l} { R_ { d,l} } = Q_ l \\ [0.2cm]
P_ l = R_ { p,l} \ Q_ l + \pi _ l
\end { cases}
\end { equation}
\end { itemize}
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\column { .54\textwidth } % Right column and width
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\begin { figure} [!hbtp]
\begin { center}
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\includegraphics [height=0.9\textwidth] { images/ref.png}
\caption { \footnotesize Schematic of the model}
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\end { center}
\end { figure}
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\end { columns}
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\end { frame}
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\begin { frame}
\frametitle { The inverse problem}
\begin { itemize}
\item <1-> Upon this solution $ \Longrightarrow $ build a set of measurements
\item <2-> Induce typical arifacts via a Magnetization vector: $ M ( \vec { x } ,t ) = M _ 0 ( \vec { x } ) exp ( i \phi _ 0 + i \frac { \pi } { venc } u ( \vec { x } ,t ) ) $
\end { itemize}
\onslide <3->
\begin { columns}
\column { .3\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/ref_ int.png}
\caption * { (a) Interpolated reference solution}
\end { figure}
\column { .67\textwidth }
\centering
\onslide <4-> \textbf { The measurements:}
\begin { itemize}
\item <5-> Gaussian noise into the magnetization
\item <6-> Different levels of aliasing varying the $ venc $ parameter
\item <7-> Only using the dominant component of the velocity: $ u _ z $
\end { itemize}
\end { columns}
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\end { frame}
\begin { frame}
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\frametitle { The inverse problem}
\begin { itemize}
\item Upon this solution $ \Longrightarrow $ build a set of measurements
\item Induce typical arifacts via a Magnetization vector: $ M ( \vec { x } ,t ) = M _ 0 ( \vec { x } ) exp ( i \phi _ 0 + i \frac { \pi } { venc } u ( \vec { x } ,t ) ) $
\end { itemize}
\begin { columns}
\column { .3\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/ref_ int.png}
\caption * { (a) Interpolated reference solution}
\end { figure}
\column { .67\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/supra_ venc.png}
\caption * { (b) Aliased measurements with different $ vencs = 120 , 70 , 30 \% $ of $ u _ { max } $ }
\hfill
\end { figure}
\end { columns}
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\end { frame}
\begin { frame}
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\frametitle { The inverse problem}
\begin { itemize}
\item Upon this solution $ \Longrightarrow $ build a set of measurements
\item Induce typical arifacts via a Magnetization vector: $ M ( \vec { x } ,t ) = M _ 0 ( \vec { x } ) exp ( i \phi _ 0 + i \frac { \pi } { venc } u ( \vec { x } ,t ) ) $
\end { itemize}
\begin { columns}
\column { .3\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/ref_ int.png}
\caption * { (a) Interpolated reference solution}
\end { figure}
\column { .67\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/coartation.png}
\caption * { (b) Aliased measurements with different $ vencs = 120 , 70 , 30 \% $ of $ u _ { max } $ }
\hfill
\end { figure}
\end { columns}
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\end { frame}
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\begin { frame}
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\frametitle { The Kalman Filter}
\begin { itemize}
\item <1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $ \theta $ solving the next optimization problem:
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\onslide <2->
\begin { equation*}
\hat { \theta } = arg \min _ { \theta } J(\theta )
\end { equation*}
\begin { equation}
J(\theta ) = \displaystyle \frac { 1} { 2} || \theta - \theta _ 0 ||^ 2_ { P_ 0^ { -1} } + \sum _ { k=1} ^ N \frac { 1} { 2} || Z_ k - \mathbb { H} X_ k(\theta ) ||^ 2_ { W^ { -1} }
\end { equation}
\onslide <3-> Where:
\begin { itemize}
\item <4-> $ Z $ the measurements and $ X = ( \vec { u } , \pi ) $ the state variable
\item <5-> $ \mathbb { H } $ observation operator
\item <6-> $ \theta _ 0 $ is the initial guess for the parameters
\item <7-> $ P _ 0 $ is the associated covariance matrix
\item <8-> $ W $ is the associated covariance matrix to the meas. noise
\end { itemize}
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\end { itemize}
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\end { frame}
\begin { frame}
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\frametitle { The Kalman Filter}
The parameter vector:
\begin { itemize}
\item <1-> Plug flow at the inlet: $ u _ { inlet } = - U f ( t ) \hat { n } $ , with $ f ( t ) $ is the weaveform which simulate a cardiac cycle and $ \hat { n } $ is the outward normal vector.
\item <2-> Since $ R _ p << R _ d $ , we only consider an optimization dependent on $ R _ d, C $ for every 3D-0D coupled outlet
\end { itemize}
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\onslide <3-> $$ \theta = ( U, \vec { R _ d } , \vec { C } ) $$ \\ with $ \vec { R _ d } = R _ { d,l } $ , $ \vec { C } = C _ l $ for $ l = 1 ,..., n _ l $
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\end { frame}
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\section { Numerical Experiments}
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\begin { frame}
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\frametitle { Numerical Experiments}
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\begin { center}
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Numerical Experiments
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\end { center}
\end { frame}
\begin { frame}
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\frametitle { Numerical Experiments}
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\end { frame}
\section { Conclusions}
\begin { frame}
\begin { center}
Conclusions
\end { center}
\end { frame}
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\end { document}
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%\begin{frame}
% \frametitle{Results}
%\footnotesize
%
%\begin{figure}
%\begin{subfigure}{.31\textwidth}
% \centering
% % \includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/u_15.png}
% \caption*{(a) $\vec{u}_{meas}$}
%\end{subfigure}
%\begin{subfigure}{.01\textwidth}
% \hfill
%\end{subfigure}
%\begin{subfigure}{.31\textwidth}
% \centering
% %\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/w_15.png}
% \caption*{(b) $\vec{w}$}
%\end{subfigure}
%\begin{subfigure}{.01\textwidth}
% \hfill
%\end{subfigure}
%\begin{subfigure}{.31\textwidth}
% \centering
% %\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/uc_15.png}
% \caption*{(c) $\vec{u}_{meas}+\vec{w}$}
%\end{subfigure}
%\caption{Measurements, corrector fields and corrected velocities for all the cases.}
%\label{fig:phantom_resolution}
%\end{figure}
%
%\end{frame}