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   \documentclass{beamer}      \usetheme{Boadilla}  \usefonttheme[onlylarge]{serif}  \usebeamercolor{dolphin}  \setbeamerfont*{frametitle}{size=\normalsize,series=\bfseries}  \setbeamertemplate{navigation symbols}{}      % Standard packages    \usepackage[english]{babel}  \usepackage[latin1]{inputenc}  \usepackage{times}  \usepackage[T1]{fontenc}      % Setup TikZ    \usepackage{tikz}  \usetikzlibrary{arrows}  \tikzstyle{block}=[draw opacity=0.7,line width=1.4cm]      % 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 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{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}