update

presentations/press_8ecm/press.tex

@@ -237,22 +237,11 @@ P_l = R_{p,l} \ Q_l + \pi_l
 \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}
-\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}
+\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}
 
 
@@ -260,7 +249,7 @@ P_l = R_{p,l} \ Q_l + \pi_l
 \begin{figure}[!hbtp]
  \begin{center}
   \includegraphics[height=0.9\textwidth]{images/ref.png}
-  \caption{\footnotesize Schematic of the model}
+  \caption{\footnotesize Reference solution at peak systole}
  \end{center}
  \end{figure}
 \end{columns} 
@@ -290,13 +279,14 @@ P_l = R_{p,l} \ Q_l + \pi_l
 \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: $u_z$
+\item<8-> Time interpolation $dt = 1 \ ms  \Longrightarrow dt = 30 \ ms$ 
 \end{itemize}
 \end{columns} 
 \end{frame}
 
 
 
-
+\section{The inverse problem}
 
 \begin{frame}
  \frametitle{The inverse problem}
@@ -381,20 +371,110 @@ J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0  ||^2_{P_0^{-1}}  + \
 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
+\item<1->  Amplitude of the inlet velocity: $U$ 
+\item<2-> Since $R_p << R_d$, we only consider an optimization dependent on $\big ( R_{d,l}, C_l \big )$ for $l=1,...,n_l$
 \end{itemize}
 
-\onslide<3-> $$\theta =  (U,\vec{R_d},\vec{C})$$ \\  with $\vec{R_d} = R_{d,l}$,  $\vec{C} = C_l$ for $l=1,..., n_l$
+\onslide<3-> $$\theta =  (U,\vec{R_d},\vec{C})$$ \\  with $\vec{R_d} = R_{d,l}$,  $\vec{C} = C_l$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm]
+\onslide<4-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed.
 
 \end{frame}
 
 
 
-\section{Numerical Experiments}
+\begin{frame}
+ \frametitle{Easy example}
+ \footnotesize
+\begin{itemize}
+\item<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
+\end{itemize}
+ \begin{columns}
+  \footnotesize
+ \column{.4\textwidth}
+ \begin{figure}
+ \onslide<2-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (8760,8760,8760)$
+  \onslide<3->
+        \includegraphics[width=1.2\textwidth]{images/U_Pb.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
+\end{figure}
+\column{.4\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}
+What happend when $venc < u_{max}$ ?
+\end{center}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Easy example: with $venc = 70 \% u_{max}$}
+ \footnotesize
+\begin{itemize}
+\item $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
+\end{itemize}
+ \begin{columns}
+  \footnotesize
+ \column{.4\textwidth}
+ \begin{figure}
+  \onslide<1-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (8760,8760,8760)$
+  \onslide<2->
+        \includegraphics[width=1.2\textwidth]{images/U_Pb.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
+\end{figure}
+\column{.4\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.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pc.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}
+
+
+
+
+
+
+
+
+
+
+\section{Numerical Experiments}
+
+
 \begin{frame}
  \frametitle{Numerical Experiments}
 \begin{center}