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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} + diff --git a/presentations/press_kalman/press.toc b/presentations/press_kalman/press.toc new file mode 100755 index 0000000..229a73b --- /dev/null +++ b/presentations/press_kalman/press.toc @@ -0,0 +1,4 @@ +\babel@toc {english}{} +\beamer@sectionintoc {1}{Introduction}{2}{0}{1} +\beamer@sectionintoc {2}{Application: Parameter recovery}{27}{0}{2} +\beamer@sectionintoc {3}{Summary}{44}{0}{3}