Browse Source

adding press

master
jeremias 2 years ago
parent
commit
292d997a33
  1. 12
      kalman/aorta.yaml
  2. 0
      presentations/press_4dflow/images/4dflow.png
  3. 0
      presentations/press_4dflow/images/aorta_blender.png
  4. 0
      presentations/press_4dflow/images/aorta_perturbation.png
  5. 0
      presentations/press_4dflow/images/channel_aliasing.png
  6. 0
      presentations/press_4dflow/images/channel_curves_SNR10.png
  7. 0
      presentations/press_4dflow/images/channel_curves_SNRinf.png
  8. 0
      presentations/press_4dflow/images/channel_noise.png
  9. 0
      presentations/press_4dflow/images/channel_ppt_1.png
  10. 0
      presentations/press_4dflow/images/channel_ppt_2.png
  11. 0
      presentations/press_4dflow/images/channel_ppt_3.png
  12. 0
      presentations/press_4dflow/images/channel_ppt_4.png
  13. 0
      presentations/press_4dflow/images/channel_under.png
  14. 0
      presentations/press_4dflow/images/cilinder.png
  15. 0
      presentations/press_4dflow/images/cilinder_2.png
  16. 0
      presentations/press_4dflow/images/histo_blender.png
  17. 0
      presentations/press_4dflow/images/histo_channel.png
  18. 0
      presentations/press_4dflow/images/norms_blender.png
  19. 0
      presentations/press_4dflow/images/norms_channel.png
  20. 0
      presentations/press_4dflow/images/perturbation_pres.png
  21. 0
      presentations/press_4dflow/images/phantom.jpg
  22. 0
      presentations/press_4dflow/images/phantom.mp4
  23. 0
      presentations/press_4dflow/images/phantom_cib.png
  24. 0
      presentations/press_4dflow/images/undersampling_blender.png
  25. 0
      presentations/press_4dflow/images/undersampling_final.png
  26. 0
      presentations/press_4dflow/images/undersampling_press.png
  27. 0
      presentations/press_4dflow/pres03.tex
  28. BIN
      presentations/press_kalman/pictures/aorta_meas.png
  29. BIN
      presentations/press_kalman/pictures/channel_inlet.png
  30. BIN
      presentations/press_kalman/pictures/kalman1.png
  31. BIN
      presentations/press_kalman/pictures/kalman2.png
  32. BIN
      presentations/press_kalman/pictures/kalman3.png
  33. BIN
      presentations/press_kalman/pictures/kalman4.png
  34. BIN
      presentations/press_kalman/pictures/kalman5.png
  35. BIN
      presentations/press_kalman/pictures/u_noi.png
  36. BIN
      presentations/press_kalman/pictures/u_ref.png
  37. BIN
      presentations/press_kalman/pictures/windk_model.png
  38. BIN
      presentations/press_kalman/pictures/windk_res.png
  39. BIN
      presentations/press_kalman/pictures/windk_res2.png
  40. 82
      presentations/press_kalman/press.aux
  41. 1263
      presentations/press_kalman/press.log
  42. 58
      presentations/press_kalman/press.nav
  43. 3
      presentations/press_kalman/press.out
  44. BIN
      presentations/press_kalman/press.pdf
  45. 0
      presentations/press_kalman/press.snm
  46. BIN
      presentations/press_kalman/press.synctex.gz
  47. 418
      presentations/press_kalman/press.tex
  48. 4
      presentations/press_kalman/press.toc

12
kalman/aorta.yaml

@ -25,28 +25,28 @@ boundary_conditions: @@ -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: @@ -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.aux

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\documentclass{beamer}
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% Standard packages
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% Setup TikZ
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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}

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presentations/press_kalman/press.toc

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\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}
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