asd

presentation/pres03.tex

@@ -184,7 +184,7 @@ University of Groningen\\[0.5cm]
 
 \onslide<3-> And a corrector field $\vec{w}$ which satisfies:
 \onslide<4-> \begin{align}
-\vec{u}  & \approx   \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector} \\
+\vec{u}  & =   \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector} \\
 \nabla \cdot \vec w  & =   0  \quad \text{in}  \quad \Omega \label{eq:correctorDiv} \\
 \vec w & =   \vec 0  \quad \text{on} \quad \partial \Omega \label{eq:correctorBC}
 \end{align}
@@ -198,7 +198,8 @@ University of Groningen\\[0.5cm]
  \frametitle{The corrector field: Continuum problem}
 \footnotesize
 
-\onslide<1-> Applying the decomposition $\vec{u}   \approx   \vec{u}_{meas} + \vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have the following: Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that 
+\onslide<1-> Applying the decomposition $\vec{u}   =  \vec{u}_{meas} + \vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have the following:\\ 
+Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that 
 \onslide<2-> \begin{equation*} 
 \int_{\Omega} \rho \frac{\partial \vec{w}}{\partial t} \cdot \vec{v} +  \rho \big ( ( \vec{u}_{meas} + \vec w)  \cdot \nabla \big) \vec{w} \cdot \vec{v} + \rho \big ( \vec{w}  \cdot \nabla \big) \vec{u}_{meas} \cdot \vec{v}  + \mu \nabla \vec{w} : \nabla \vec{v} - p \nabla \cdot \vec{v} + q \nabla \cdot \vec{w} \notag 
 \end{equation*}
@@ -256,12 +257,12 @@ $
 \footnotesize
 \onslide<1->
 \begin{theorem} 
-There exists a unique solution of Problem \ref{eq:Corrector_discrete} under  condition: $$\rho/\tau + C_\Omega^{-2} \mu/2 - \rho 3 \| \nabla\vec u_{meas}^k\|_\infty > 0$$ for all $k>0$.
+There exists a unique solution of Problem (\ref{eq:Corrector_discrete}) under the condition: $$\rho/\tau + C_\Omega^{-2} \mu/2 - \rho 3 \| \nabla\vec u_{meas}^k\|_\infty > 0$$ for all $k>0$.
 \end{theorem}
 \onslide<2->
 We can furthermore prove the following energy balance:
 \onslide<3->
-\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem \ref{eq:Corrector_discrete},  with $\ell_j(\vec v,q)=0$ it holds
+\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem (\ref{eq:Corrector_discrete}),  with $\ell_j(\vec v,q)=0$ it holds
 \begin{equation*}\label{eq:energy}
    \| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)} 
 \end{equation*}
@@ -359,7 +360,7 @@ Also perturbations were added into the measurements:
 \begin{figure}[!hbtp]
  \begin{center}
   \includegraphics[height=0.5\textwidth]{images/perturbation_pres.png}
-\caption{Different perturbation scenarios}
+\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
  \end{center}
  \end{figure}
 
@@ -557,6 +558,13 @@ Experiments using real 4D flow data
 
 \section{Conclusions}
 
+\begin{frame}
+ \frametitle{Experiments}
+\begin{center}
+Conclusions
+\end{center}
+\end{frame}
+
 
 \begin{frame}
  \frametitle{Conclusions and future work}