NuMRI/presentations/press_kalman/press.tex

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\documentclass{beamer}
\usetheme{Boadilla}
\usefonttheme[onlylarge]{serif}
\usebeamercolor{dolphin}
\setbeamerfont*{frametitle}{size=\normalsize,series=\bfseries}
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% 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:
\[
u_{inlet} =
\begin{cases}
U sin(\pi t/T) & \text{if} \ t<T^* \\
\alpha U sin( \pi t/T')e^{- \gamma t} & \text{if} \ t \geq T^*\\
\end{cases}
\]
\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}