\documentclass[xcolor=dvipsnames,notheorem,mathserifs]{beamer}
\usepackage{amsmath}
%\documentclass{beamer}
\usepackage[english]{babel}
%\usepackage[latin1]{inputenc}
\usepackage{multicol} % indice en 2 columnas
\usepackage[utf8]{inputenc}

\usepackage{helvet}
\usefonttheme{serif} 
%\usepackage{ccfonts}     % Font family: Concrete Math
\usepackage[T1]{fontenc}

%\usepackage{graphicx}
%\usepackage{movie15}
%\usepackage{media9}[2013/11/04]
\usepackage{xcolor}


\usepackage{graphicx}
\usepackage{multimedia}
\usepackage{media9}
\usepackage{listings,xcolor,caption, mathtools, wrapfig}
\usepackage{amsfonts}
\usepackage{amssymb,graphicx,enumerate}
\usepackage{subcaption}
\usepackage{hyperref}

\usepackage[normalem]{ulem} % for strike out command \sout





%\usetheme{default}
%\usetheme{AnnArbor}
%\usetheme{Antibes}
%\usetheme{Bergen}
%\usetheme{Berkeley}
%\usetheme{Berlin}
%\usetheme{Boadilla}
%\usetheme{CambridgeUS}
%\usetheme{Copenhagen}
%\usetheme{Darmstadt}
%\usetheme{Dresden}
%\usetheme{Frankfurt}
%\usetheme{Goettingen}
%\usetheme{Hannover}
%\usetheme{Ilmenau}
%\usetheme{JuanLesPins}
%\usetheme{Luebeck}
%\usetheme{Madrid}
%\usetheme{Malmoe}
%\usetheme{Marburg}
%\usetheme{Montpellier}
%\usetheme{PaloAlto}
%\usetheme{Pittsburgh}
%\usetheme{Rochester}
%\usetheme{Singapore}
%\usetheme{Szeged}
\usetheme{Warsaw}

%\usecolortheme{albatross}
%\usecolortheme{beaver}
%\usecolortheme{beetle}
\usecolortheme{crane}
%\usecolortheme{dolphin}
%\usecolortheme{dove}
%\usecolortheme{fly}
%\usecolortheme{lily}
%\usecolortheme{orchid}
%\usecolortheme{rose}
%\usecolortheme{seagull}
%\usecolortheme{seahorse}
%\usecolortheme{whale}
%\usecolortheme{wolverine}

%\useoutertheme{infolines}
%\useoutertheme{miniframes}
%\useoutertheme{sidebar}
\useoutertheme{smoothbars}
%\useoutertheme{shadow}
%\useoutertheme{smoothtree}
%\useoutertheme{split}
%\useoutertheme{tree}


\usepackage{amssymb,mathrsfs,amsmath,latexsym,amsthm,amsfonts}
\useinnertheme{rectangles}


\setbeamertemplate{navigation symbols}{} % quitar simbolitos


\setbeamerfont{page number in head/foot}{size=\large}
%\setbeamertemplate{footline}[frame number] number in footer
\setbeamertemplate{footline}{}



\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}