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\title{Robust parameter estimation in fluid flow models from aliased velocity measurements}
%\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]
Jeremías Garay Labra \emph{join with} Cristobal Bertoglio.}
\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-> Velocities encoded into the magnetization phase
\item<3-> Rich post-proccesing: derived parameters
\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}
\section{The mathematical model}
\begin{frame}
\frametitle{The mathematical model}
\begin{center}
The mathematical model
\end{center}
\end{frame}
\begin{frame}
\frametitle{The mathematical model}
\begin{columns}[c]
\column{.5\textwidth} % Left column and width
\footnotesize
\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}
\end{frame}
\begin{frame}
\frametitle{The mathematical model}
\begin{columns}[c]
\column{.5\textwidth} % Left column and width
\footnotesize
\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}
\end{equation}
\end{itemize}
\column{.54\textwidth} % Right column and width
\onslide<1->
\begin{figure}[!hbtp]
\begin{center}
\includegraphics[height=0.9\textwidth]{images/windk_model.png}
\caption{\footnotesize Schematic of the model}
\end{center}
\end{figure}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{The mathematical model}
\begin{columns}[c]
\column{.5\textwidth} % Left column and width
\footnotesize
\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}
\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}
\column{.54\textwidth} % Right column and width
\begin{figure}[!hbtp]
\begin{center}
\includegraphics[height=0.9\textwidth]{images/ref.png}
\caption{\footnotesize Reference solution at peak systole}
\end{center}
\end{figure}
\end{columns}
\end{frame}
%\begin{frame}
% \frametitle{The mathematical model}
%
% \begin{columns}[c]
%\column{.5\textwidth} % Left column and width
%\footnotesize
%\begin{itemize}
%\item<1-> $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ the weaveform.
%\item<2-> Fractional step scheme.
%\item<3-> Semi-implicit Windkessel model.
%\item<4-> Stabilized $\mathbb{P}1/\mathbb{P}1$ finite elements.
%\item<4-> Implemented in FEniCS.
%\end{itemize}
%
%
%\column{.54\textwidth} % Right column and width
%\begin{figure}[!hbtp]
% \begin{center}
% \includegraphics[height=0.9\textwidth]{images/ref.png}
% \caption{\footnotesize Reference solution at peak systole}
% \end{center}
% \end{figure}
%\end{columns}
%
%\end{frame}
\begin{frame}
\frametitle{The inverse problem}
\begin{itemize}
\item<1-> Upon this solution $\Longrightarrow$ build a set of measurements
\item<2-> $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
\item<3-> reconstructed velocity: $u \in \big ( -venc, + venc \big )$
\item<4-> $VNR \sim 1/venc$
\end{itemize}
\centering
\onslide<5-> \textbf{The measurements:}
\begin{itemize}
\item<6-> Gaussian noise into the magnetization
\item<7-> Spatial and temporal interpolation
\item<8-> Only using the dominant component of the velocity: $u_z$
\item<9-> Different levels of aliasing varying the $venc$ parameter
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{The inverse problem}
\begin{itemize}
\item Upon this solution $\Longrightarrow$ build a set of measurements
\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
\item reconstructed velocity: $u \in \big ( -venc, + venc \big )$
\item $VNR \sim 1/venc$
\end{itemize}
\begin{figure}
\includegraphics[width=0.7\textwidth]{images/supra_venc.png}
\caption*{Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
\hfill
\end{figure}
\end{frame}
\begin{frame}
\frametitle{The inverse problem}
\begin{itemize}
\item Upon this solution $\Longrightarrow$ build a set of measurements
\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
\item reconstructed velocity: $u \in \big ( -venc, + venc \big )$
\item $VNR \sim 1/venc$
\end{itemize}
\begin{figure}
\includegraphics[width=0.7\textwidth]{images/coartation.png}
\caption*{ Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
\hfill
\end{figure}
\end{frame}
\section{The inverse problem}
\begin{frame}
\frametitle{The inverse problem}
\begin{center}
Parameter optimization
\end{center}
\end{frame}
\begin{frame}
\frametitle{The Kalman Filter}
\begin{itemize}
\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$:
\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<4-> 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}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{The Kalman Filter}
The parameter vector:
\begin{itemize}
\item<1-> Amplitude of the inlet velocity: $U$
\item<2-> Only the higher resistence: $R_d$
\end{itemize}
\onslide<4-> $$\theta = (U,\vec{R_d})$$ \\ with $\vec{R_d} = R_{d,l}$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm]
\onslide<5-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed.
\end{frame}
\section{Numerical Experiments}
\begin{frame}
\frametitle{Numerical Experiments}
\begin{center}
Numerical Experiments
\end{center}
\end{frame}
\begin{frame}
\frametitle{Numerical Experiments}
\footnotesize
\onslide<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
\begin{columns}
\footnotesize
\column{.45\textwidth}
\begin{figure}
\onslide<2-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$
\onslide<3-> \includegraphics[width=1.2\textwidth]{images/U_Pb.png}
\includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
\end{figure}
\column{.45\textwidth}
\begin{figure}
\onslide<2-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
\onslide<4->
\includegraphics[width=1.2\textwidth]{images/U_Pc.png}
\includegraphics[width=1.2\textwidth]{images/Rd_Pc.png}
\end{figure}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Aliased data}
\begin{center}
\onslide<1-> What happend when $venc < u_{max}$ ?
\begin{figure}
\onslide<2-> \includegraphics[width=0.45\textwidth]{images/v120.png}
\caption{Measurement set with $venc = 120 \% u_{max}$}
\end{figure}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Aliased data}
\begin{center}
What happend when $venc < u_{max}$ ?
\begin{figure}
\includegraphics[width=0.45\textwidth]{images/v70.png}
\caption{Measurement set with $venc = 70 \% u_{max}$}
\end{figure}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Easy example: with $venc = 70 \% u_{max}$}
\footnotesize
$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
\begin{columns}
\footnotesize
\column{.45\textwidth}
\begin{figure}
\onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$
\onslide<2->
\includegraphics[width=1.2\textwidth]{images/U_Pb_V70.png}
\includegraphics[width=1.2\textwidth]{images/Rd_Pb_V70.png}
\end{figure}
\column{.45\textwidth}
\begin{figure}
\onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
\onslide<2->
\includegraphics[width=1.2\textwidth]{images/U_Pc_V70.png}
\includegraphics[width=1.2\textwidth]{images/Rd_Pc_V70.png}
\end{figure}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{The Kalman Filter: A frequency defined functional}
\begin{itemize}
\item<1-> We proposed a change in the cost functional:
\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}} + \color{red} \sum_{k=1}^N 1- cos \big ( \frac{\pi}{venc} \cdot ( Z_k - \mathbb{H} X_k ) \big )
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Easy example: with $venc = 70 \% u_{max}$}
\footnotesize
$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
\begin{columns}
\footnotesize
\column{.45\textwidth}
\begin{figure}
\onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$
\onslide<2->
\includegraphics[width=1.2\textwidth]{images/HU_Pb_V70.png}
\includegraphics[width=1.2\textwidth]{images/HRd_Pb_V70.png}
\end{figure}
\column{.45\textwidth}
\begin{figure}
\onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
\onslide<3->
\includegraphics[width=1.2\textwidth]{images/HU_Pc_V70.png}
\includegraphics[width=1.2\textwidth]{images/HRd_Pc_V70.png}
\end{figure}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Aliased data}
\begin{center}
Or even higher aliasing...
\begin{figure}
\includegraphics[width=0.45\textwidth]{images/v30.png}
\caption{Measurement set with $venc = 30 \% u_{max}$}
\end{figure}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Easy example: with $venc = 30 \% u_{max}$}
\footnotesize
$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
\begin{columns}
\footnotesize
\column{.45\textwidth}
\begin{figure}
\onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$
\onslide<2->
\includegraphics[width=1.2\textwidth]{images/HU_Pb_V30.png}
\includegraphics[width=1.2\textwidth]{images/HRd_Pb_V30.png}
\end{figure}
\column{.45\textwidth}
\begin{figure}
\onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
\onslide<3->
\includegraphics[width=1.2\textwidth]{images/HU_Pc_V30.png}
\includegraphics[width=1.2\textwidth]{images/HRd_Pc_V30.png}
\end{figure}
\end{columns}
\end{frame}
\section{Conclusions}
\begin{frame}
\begin{center}
Conclusions
\end{center}
\end{frame}
\begin{frame}
\frametitle{Conclusions}
\footnotesize
\begin{itemize}
\item<1-> 4D Flow measurements are promising for extracting data via inverse problems
\item<2-> Noise and aliasing are the typical artifacts involved.
\item<3-> Using a suitable Kalman filter, have shown to bypass aliasing defining the funcional in terms of the frequencies.
\end{itemize}
\onslide<4-> Future Work
\begin{itemize}
\item<5-> To include the capacitancies in the inverse problem (adding some pressure meas.)
\item<6-> Real data!
\end{itemize}
\end{frame}
\begin{frame}
\begin{center}
Thank you for your time!
\end{center}
\end{frame}
\end{document}
%\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}