adding press

kalman/aorta.yaml

@@ -25,28 +25,28 @@ boundary_conditions:
         type: 'dirichlet'
         value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th<t)*(-3.67949466208*U*sin(9*DOLFIN_PI*t/Th)*exp(-t*10))']
         parameters:
-            U: 60
+            U: 30
             Th: 0.35
             t: 0   
     - 
         id: 3
         type: 'windkessel'
-        value: [20,0,0]
+        value: [10,0,0]
         p0: [0,1333.223874]
     - 
         id: 4
         type: 'windkessel'
-        value: [500,0,0]
+        value: [250,0,0]
         p0: [0,1333.223874]
     - 
         id: 5
         type: 'windkessel'
-        value: [500,0,0]
+        value: [250,0,0]
         p0: [0,1333.223874]
     - 
         id: 6
         type: 'windkessel'
-        value: [500,0,0]
+        value: [250,0,0]
         p0: [0,1333.223874]
 
 timemarching:
@@ -143,7 +143,7 @@ estimation:
             file_root: 'results/aorta/measurements/u{i}.h5'
             indices: 0 # indices of checkpoints to be processed. 0 == all
             velocity_direction: ~
-            noise_stddev: 25     # standard deviation of Gaussian noise
+            noise_stddev: 5     # standard deviation of Gaussian noise
     
     roukf:
         particles: 'simplex' # unique or simplex

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presentations/press_kalman/press.out

@@ -0,0 +1,3 @@
+\BOOKMARK [2][]{Outline0.1}{Introduction}{}% 1
+\BOOKMARK [2][]{Outline0.2}{Application: Parameter recovery}{}% 2
+\BOOKMARK [2][]{Outline0.3}{Summary}{}% 3

presentations/press_kalman/press.tex

@@ -0,0 +1,418 @@
+
+\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:
+\[   
+u_{inlet} = 
+     \begin{cases}
+        U sin(\pi t/T) & \text{if} \ t<T^* \\
+        \alpha U sin( \pi t/T')e^{- \gamma t} & \text{if} \ t \geq T^*\\
+     \end{cases}
+\]
+\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}
+

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