forked from surenkum/uq_gaussian_processes
-
Notifications
You must be signed in to change notification settings - Fork 0
/
motivation.tex
85 lines (75 loc) · 2.76 KB
/
motivation.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
\section{Introduction}
\begin{frame}{Introduction}
\tikzstyle{block} = [draw, fill=blue!20, rectangle,
minimum height=3em, minimum width=6em]
\tikzstyle{sum} = [draw, fill=blue!20, circle, node distance=1cm]
\tikzstyle{input} = [coordinate]
\tikzstyle{output} = [coordinate]
\tikzstyle{pinstyle} = [pin edge={to-,thin,black}]
% The block diagram code is probably more verbose than necessary
\begin{center}
\begin{tikzpicture}[auto, node distance=2cm,>=latex']
% We start by placing the blocks
\node [input, name=input] {};
\node [block, right of=input, pin={[pinstyle]above:Disturbances},
node distance=2cm] (system) {System};
\node [output, right of=system] (output) {};
% Once the nodes are placed, connecting them is easy.
\draw [draw,->] (input) -- node {$x$} (system);
\draw [->] (system) -- node [name=y] {$y^e(x)$} (output) ;
\end{tikzpicture}
\end{center}
\pause
\begin{center}
% The block diagram code is probably more verbose than necessary
\begin{tikzpicture}[auto, node distance=2cm,>=latex']
% We start by placing the blocks
\node [input, name=input] {};
\node [block, right of=input, pin={[pinstyle]above:Disturbances},
node distance=2cm] (system) {System};
\node [block, below of=system] (model) {Model};
\node [sum, right of=output] (sum) {};
\node [output, right of=sum] (final_output) {};
% Once the nodes are placed, connecting them is easy.
\draw [draw,->] (input) -- node [name=x] {$x$} (system);
\draw [->] (system) -- node [name=y] {$y^e(x)$} (sum) ;
\draw [->] (x) |- (model);
\draw [->] (model) -| node [near end] {$y^m(x)$} (sum);
\draw [->] (sum) -- node [name=y_final] {$\delta(x)$} (final_output) ;
\end{tikzpicture}
\end{center}
\end{frame}
\begin{frame}{Motivation}{Inverse Uncertainty Quantification}
\begin{block}{Bias Correction}
\begin{align}
y^e(x) = y^m(x)+\delta(x)+\epsilon \nonumber
\end{align}
where $\epsilon$ is the experimental uncertainty.
\end{block}
% Model Inadequacy
\pause
\begin{block}{Parameter Calibration}
\begin{align}
y^e(x) = y^m(x,\theta^*)+\epsilon \nonumber
\end{align}
\end{block}
\pause
\begin{block}{Bias Correction and Parameter Calibration}
\begin{align}
y^e(x) = y^m(x,\theta^*)+\delta(x)+\epsilon \nonumber
\end{align}
\end{block}
\end{frame}
\begin{frame}{Need for Function Approximation}
\begin{itemize}
\item Lack of simulation results $y^m(x,\theta)$
\item Parameterizing discrepancy function $\delta(x)$
\end{itemize}
\begin{block}{Why Bayesian}
\begin{itemize}
\item Integration of prior knowledge
\end{itemize}
\end{block}
\vskip0pt plus 1filll
Bayesian calibration of computer models, Marc C. Kennedy, Anthony O'Hagan in Journal of the Royal Statistical Society, 2001
\end{frame}