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% Author, Title, etc.

\title[] 
{%
	Data assimilation on the Kalman filter
}

\author[Garay]
{
  Jeremias Garay %\inst{1}
  %\textcolor{green!50!black}{Till~Tantau}\inst{5}
}

%\institute[University of Groningen]
%{
%  \inst{1}%
%  University of Groningen, The Netherlands
%  \and
%  \vskip-2mm
%}

\date



% The main document

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}{Outline}
%  \tableofcontents
%\end{frame}




\section{Introduction}

\begin{frame}
\begin{center}
\large{Introduction}
\end{center}
\end{frame}




\begin{frame}{Stationary Case: Least square estimation}

\onslide<1->
\textit{Assume we want to find an estimator $\hat{X}$ of a unknown vector $X$, with a certain guess available $\hat{X}^-$, associated with a confidence matrix $(P^-)^{-1}$. Assume also that we have partial observation $Z$, satisfying $Z = HX + \zeta^Z$, associated with a confidence matrix $W^{-1}$.}  \\[0.5cm]

\onslide<2->
A quantity taking care of $\hat{X}^{-}$ and $Z$ can be obtained minimizing the cuadratic cost functional:

\onslide<3->
\begin{equation}
J(\hat{X}) = \frac{1}{2} (\hat{X} - \hat{X}^-) (P^-)^{-1} (\hat{X} - \hat{X}^-)   +    \frac{1}{2} (Z -H\hat{X}) W^{-1} (Z - H\hat{X})
\end{equation}

\end{frame}




\begin{frame}{Stationary Case: Least square estimation}

\onslide<1->
Find the optimal state imposing: $\frac{dJ}{d\hat{X}}(\hat{X}^+) = 0$:

\onslide<2->
\begin{eqnarray*}
-H^T W^{-1} Z + H^T W^{-1} H \hat{X} - (P^-)^{-1} \hat{X}^- + (P^-)^{-1} \hat{X} \equiv 0
\end{eqnarray*}

\onslide<3->
or reordering terms:

\onslide<4->
\begin{equation*}
\hat{X}^+ = \hat{X}^- + K (Z-H\hat{X}^-)
\end{equation*}
\vspace{0.4cm}

With $K = P^+ H^T W^{-1}$ the Kalman matrix and $P^+ = ((P^-)^{-1} + H^T W^{-1} H)^{-1}$.

\end{frame}



\begin{frame}{Time dependent problems}

\onslide<1->
The method could be easily expanded into time-dependent systems ($\dot{X} = AX + F$):


\begin{itemize}
\item[1.]<2-> Assume that $\hat{X}^{+}_{n-1}$ is known with a covariance $P^+_{n-1}$

    \begin{exampleblock}{Prediction}
	$$\hat{X}^{-}_n = A_n  \hat{X}^{+}_{n-1}  + F_n$$
	by linearity of $A_n$, the covariance of $\hat{X}^-_n$ is equal to $A_n P_{n-1}^+ A_n^T$
    \end{exampleblock}

\item[2. ]<3-> Afterwards

    \begin{exampleblock}{Correction}
	$$\hat{X}^{+}_n = \hat{X}^{-}_n + K_n (Z_n - H_n \hat{X}_n^-) $$

    \end{exampleblock}

\end{itemize}
\end{frame}


\begin{frame}{Non-linear problems}

\begin{itemize}
\item[1.]<1-> \emph{Extended Kalman Filter (EKF)}

\begin{itemize}
\item[a.]<2-> Taylor's expansion on the non-linear operator (tangent operators)
\item[b.]<3-> High cost if the Jacobian can be found numerically
\item[c.]<4-> Not optimal when the system is highly non-linear
\end{itemize}

\item[2.]<5-> \emph{Unscented Kalman Filter (UKF)}

\begin{itemize}
\item[a.]<6-> Approximate propagation of vectors by propagating suitable particles
\item[b.]<7-> Could be shown that by computing mean and covariance of the particles, a better approx could be reached. 
\end{itemize}

\item[3.]<8-> \emph{Reduced Order Unscented Kalman Filter (ROUKF)}

\begin{itemize}
\item[a.]<9-> LU factorization could be performed on the covariance matrix $P_n^-$
\end{itemize}


\end{itemize}


\end{frame}






\begin{frame}{Graphical Picture: Initial State}
\begin{figure}
\includegraphics[width=\textwidth]{pictures/kalman1.png}
\end{figure}
\end{frame}

\begin{frame}{Graphical Picture: Initial State}
\begin{figure}
\includegraphics[width=\textwidth]{pictures/kalman2.png}
\end{figure}
\end{frame}

\begin{frame}{Graphical Picture: Prediction}
\begin{figure}
\includegraphics[width=\textwidth]{pictures/kalman3.png}
\end{figure}
\end{frame}


\begin{frame}{Graphical Picture: Updating Measurements}
\begin{figure}
\includegraphics[width=\textwidth]{pictures/kalman4.png}
\end{figure}
\end{frame}

\begin{frame}{Graphical Picture: Correction}
\begin{figure}
\includegraphics[width=\textwidth]{pictures/kalman5.png}
\end{figure}
\end{frame}



\section{Application: Parameter recovery}

\begin{frame}
\begin{center}
\large{Application: Parameter recovery}
\end{center}
\end{frame}




\begin{frame}{Application: Parameter recovery}
\onslide<1-> Consider a Poiseuille flow in a cylinder coming from a simulation. Assume we have:

  \begin{columns}[t]
    \column{.4\textwidth}
    

\vspace{0.3cm}

\column{.4\textwidth}

\onslide<2->

\begin{figure}
\includegraphics[width=1.3\textwidth]{pictures/u_ref.png}
\vspace{1.5cm}
\end{figure}

  \end{columns}
\end{frame}




\begin{frame}{Application: Parameter recovery}
 Consider a Poiseuille flow in a cylinder coming from a simulation. Assume we have:

  \begin{columns}[t]
    \column{.4\textwidth}
    
\begin{itemize}
\item[1.]The velocity measurements with the addition of some noise
\item[2.]<2-> The measurement's mesh
\end{itemize} 
\vspace{0.3cm}

\onslide<3-> We want to estimate the amplitude of the inlet flow

\onslide<4-> $$ u_{inlet} = \alert{U} \ (R^2-r^2) \ sin(\pi t / T) $$

\column{.4\textwidth}
\onslide<1->
\begin{figure}
\includegraphics[width=1.3\textwidth]{pictures/u_noi.png}
\vspace{1.5cm}
\end{figure}

  \end{columns}
\end{frame}




\begin{frame}{Application: Parameter recovery}

\begin{figure}
\includegraphics[width=0.8\textwidth]{pictures/channel_inlet.png}
\end{figure}

\begin{itemize}
\item[] Reparametrized value: $\theta_0 \cdot 2^\theta$
\end{itemize}

\end{frame}




\begin{frame}{Application: More complex scenario}
 
 \onslide<1-> Aortic velocity data with reduced order boundary condition:

  \begin{columns}[t]
    \column{.55\textwidth}
    
\begin{itemize}
\item[1.]<2-> Navier-Stokes simulation with a \emph{plug-flow} at the intlet:
\[   
u_{inlet} = 
     \begin{cases}
        U sin(\pi t/T) & \text{if} \ t<T^* \\
        \alpha U sin( \pi t/T')e^{- \gamma t} & \text{if} \ t \geq T^*\\
     \end{cases}
\]
\item[2.]<3-> A 1-element Windkessel boundary condition is defined in every inlet.

\end{itemize} 

\vspace{0.3cm}

\onslide<4-> We want to recover the proximal resistances $R_i$, $i=1,2,3,4$ and the amplitude $U$ from noisy velocity measurements.



\column{.4\textwidth}

\onslide<1->
\begin{figure}
\includegraphics[width=1.0\textwidth]{pictures/windk_model.png}
\end{figure}
  \end{columns}

\end{frame}


\begin{frame}{Application: Parameter recovery $\theta_0 \ 2^\theta$}

\begin{figure}
\includegraphics[width=0.9\textwidth]{pictures/windk_res.png}
\end{figure}
\end{frame}



\begin{frame}{Application: Parameter recovery}

\begin{tabular}{ l c r }
      & \emph{true} & \emph{recovered} \\[0.1cm]
      \hline
  $R_1 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ & $242.14$ \\
  $R_2 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ & $249.16$ \\
  $R_3 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ & $246.03$ \\
  $R_4 \ (dyn\cdot s \cdot cm^{-5})$ & $10$ & $9.87$ \\
  $U \ (cm/s)$ & $30$ & $29.94$ \\
    
\end{tabular}

\end{frame}


\begin{frame}{Application: Parameter recovery (\alert{only using 1 vel. component})}

\onslide<2->
\begin{figure}
\includegraphics[width=0.9\textwidth]{pictures/windk_res2.png}
\end{figure}
\end{frame}



\begin{frame}{Application: Parameter recovery}

\begin{tabular}{ l c c c }
      & \emph{true} & \emph{recovered} & \emph{recovered with reduced vel}\\[0.1cm]
      \hline
  $R_1 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ &  $242.14$ & $247.31$ \\
  $R_2 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ &  $249.16$ & $255.56$ \\
  $R_3 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ & $246.03$ & $277.37$ \\
  $R_4 \ (dyn\cdot s \cdot cm^{-5})$ & $10$ & $9.87$ & $8.03$ \\
  $U \ (cm/s)$ & $30$ & $29.94$ & $29.80$ \\
    
\end{tabular}

\end{frame}












\section{Summary}



\begin{frame}
\begin{center}
\large{Summary}
\end{center}
\end{frame}


\begin{frame}
  \frametitle<presentation>{Summary}

  \begin{itemize}
  \item<1->
    Kalman's filter uses a series of measurements and produce an estimate in two steps: Prediction and Correction
   \item<2-> 
   The Reduced Order Kalman Filter (ROUKF) its a simplification for non-linear problems which generally run faster than others methods. (no derivatives are need it)
   \item<3-> Parameter recovery its a straightforward application.

  \end{itemize}
\end{frame}



\end{document}