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Commit 73c2a608 authored by Jan Habscheid's avatar Jan Habscheid
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move influence of k to introduction

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Project3/LyX/Introduction.lyx
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......@@ -232,5 +232,378 @@ The question that arises is,
based on the known mathematical model and observation data.
\end_layout
\begin_layout Subsection
The Mathematical Model
\end_layout
\begin_layout Standard
Consider the general mathematical description of a traffic flow (see project 2 for more reference)
\begin_inset Formula
\begin{align}
u_{t}+f(u)_{x} & =0,\quad x\in[a,b]\\
u(x,t=0) & =u_{0}(x)\\
u_{x}\big|_{x=x_{\text{in}}} & =u_{\text{in}}\\
u_{x}\big|_{x=x_{\text{out}}} & =u_{\text{out}}
\end{align}
\end_inset
with
\begin_inset Formula $a=0$
\end_inset
,
\begin_inset Formula $b=4$
\end_inset
,
the flux function
\begin_inset Formula $f(u)=u(1-u)$
\end_inset
.
In reality,
this model is influenced by some uncertainty,
quantified by the
\begin_inset Quotes eld
\end_inset
resistance
\begin_inset Quotes erd
\end_inset
function
\begin_inset Formula $k(x)$
\end_inset
\begin_inset Formula
\begin{align}
u_{t}+\left(k(x)f(u)\right)_{x} & =0,\quad x\in[a,b]\label{eq:GenConsLaw_k}\\
u(x,t=0) & =u_{0}(x)\\
u_{x}\big|_{x=a}=u_{x}\big|_{x=b} & =0
\end{align}
\end_inset
The boundaries are described by Neumann boundary conditions.
Purpose of this work is to identify the resistance function
\begin_inset Formula $k(x)$
\end_inset
for a known solution
\begin_inset Formula $u(x_{i},t)$
\end_inset
at specified positions
\begin_inset Formula $x_{i}$
\end_inset
.
Therefore,
an inverse problem will be formulated,
which utilizes Monte-Carlo Markov-Chains for the uncertainty quantification.
\end_layout
\begin_layout Subsection
Major Influence of the Resistance Function
\end_layout
\begin_layout Standard
Consider three different,
randomly generated,
resistance functions following a normal distribution with mean
\begin_inset Formula $1$
\end_inset
and standard deviations
\begin_inset Formula $0.25$
\end_inset
.
The form of the resistance function contributes to the solution of equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:GenConsLaw_k"
plural "false"
caps "false"
noprefix "false"
nolink "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement document
alignment document
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Float figure
placement document
alignment document
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Graphics
filename Figures/InfluenceResistance/sample_0.png
lyxscale 30
width 48text%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:InfluenceResistanceFunction-1"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\begin_inset Float figure
placement document
alignment document
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Graphics
filename Figures/InfluenceResistance/sample_1.png
lyxscale 30
width 48text%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:InfluenceResistanceFunction-2"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset Float figure
placement document
alignment document
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Graphics
filename Figures/InfluenceResistance/sample_2.png
lyxscale 30
width 60text%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:InfluenceResistanceFunction-3"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\begin_inset Caption Standard
\begin_layout Plain Layout
The resistance function
\begin_inset Formula $k(x)$
\end_inset
has a major impact on the solution of the general conservation law
\begin_inset Formula $u_{t}+\left(k(x)f(u)\right)_{x}=0$
\end_inset
.
Randomly generated,
normal distributed resistance functions are shown on the left,
while the predicted solution (solid lines) and the true solution (dotted lines) at observation points
\begin_inset Formula $x_{i}\in[0.75,1.5,2.25,3.25]$
\end_inset
for initial data
\begin_inset Formula $u_{0}^{I}(x)$
\end_inset
are shown on the right.
\begin_inset CommandInset label
LatexCommand label
name "fig:InfluenceResistanceFunction"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:InfluenceResistanceFunction"
plural "false"
caps "false"
noprefix "false"
nolink "false"
\end_inset
shows the resistance functions (left) and corresponding solution
\begin_inset Formula $u(x_{i},t)$
\end_inset
at
\begin_inset Formula $x_{i}\in[0.75,1.5,2.25,3.25]$
\end_inset
(right) for initial distribution
\begin_inset Formula $u_{0}^{I}(x)$
\end_inset
.
\begin_inset Formula
\begin{equation}
u_{0}^{I}(x)=\begin{cases}
0.2 & ,x\in[0,0.5]\\
0.4 & ,x\in(0.5,1.5]\\
0.6 & ,x\in(1.5,2.5]\\
0.7 & ,x\in(2.5,3.5]\\
0.4 & ,x\in(3.5,4]
\end{cases}
\end{equation}
\end_inset
The resistance function is visualized over the spatial domain while the solution
\begin_inset Formula $u(x_{i},t)$
\end_inset
is visualized for the different positions over time.
The solid line is the predicted solution,
while the dotted points show the true observations.
\end_layout
\begin_layout Standard
Although all resistance functions follow the same random distribution,
the solution for the conservation law differs strongly.
Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:InfluenceResistanceFunction-1"
plural "false"
caps "false"
noprefix "false"
nolink "false"
\end_inset
shows a shock for all observations,
while Figures
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:InfluenceResistanceFunction-2"
plural "false"
caps "false"
noprefix "false"
nolink "false"
\end_inset
and
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:InfluenceResistanceFunction-3"
plural "false"
caps "false"
noprefix "false"
nolink "false"
\end_inset
only show shocks for certain observation points.
Additionally,
the value of
\begin_inset Formula $u$
\end_inset
at some observation points increase over time for certain resistance functions,
while it decreases for others.
\end_layout
\begin_layout Standard
Therefore,
the solution of the PDE is strongly sensitive to the chosen resistance function.
This further emphasizes the importance to find an algorithm to identify the resistance function.
\end_layout
\end_body
\end_document
Project3/LyX/Results.lyx
+ 50
− 82
View file @ 73c2a608
......@@ -146,25 +146,24 @@ Two true solutions are available,
\begin_layout Standard
\begin_inset Formula
\begin{align}
u_{0}^{I}(x) & =\begin{cases}
\begin{equation}
u_{0}^{I}(x)=\begin{cases}
0.2 & ,x\in[0,0.5]\\
0.4 & ,x\in(0.5,1.5]\\
0.6 & ,x\in(1.5,2.5]\\
0.7 & ,x\in(2.5,3.5]\\
0.4 & ,x\in(3.5,4]
\end{cases}\\
u_{0}^{II}(x) & =\begin{cases}
\end{cases},\qquad u_{0}^{II}(x)=\begin{cases}
0.1 & ,x\in[0,0.5]\\
0.3 & ,x\in(0.5,1.5]\\
0.7 & ,x\in(1.5,2.5]\\
0.2 & ,x\in(2.5,4]
\end{cases}
\end{align}
\end{equation}
\end_inset
If not stated otherwise,
If not stated otherwise,
the artificial diffusion parameter
\begin_inset Formula $M$
\end_inset
......@@ -172,6 +171,10 @@ If not stated otherwise,
is set to one.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Subsection
Influence of Resistance Function
\begin_inset Note Note
......@@ -186,7 +189,7 @@ Maybe move this to the introduction as a motivation?
\end_layout
\begin_layout Standard
\begin_layout Plain Layout
First,
quantify the influence of the resistance function on the solution of the general conservation law for initial data
\begin_inset Formula $u_{0}^{I}(x)$
......@@ -205,7 +208,7 @@ First,
\end_layout
\begin_layout Standard
\begin_layout Plain Layout
\begin_inset Float figure
placement document
alignment document
......@@ -225,7 +228,7 @@ status open
\begin_inset Graphics
filename Figures/InfluenceResistance/sample_0.png
lyxscale 30
width 60text%
width 48text%
\end_inset
......@@ -253,10 +256,6 @@ name "fig:InfluenceResistanceFunction-1"
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset Float figure
placement document
alignment document
......@@ -268,7 +267,7 @@ status open
\begin_inset Graphics
filename Figures/InfluenceResistance/sample_1.png
lyxscale 30
width 60text%
width 48text%
\end_inset
......@@ -291,10 +290,6 @@ name "fig:InfluenceResistanceFunction-2"
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
......@@ -338,10 +333,6 @@ name "fig:InfluenceResistanceFunction-3"
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
......@@ -382,10 +373,6 @@ name "fig:InfluenceResistanceFunction"
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
......@@ -421,7 +408,7 @@ nolink "false"
while the dotted points show the true observations.
\end_layout
\begin_layout Standard
\begin_layout Plain Layout
Although all resistance functions follow the same random distribution,
the solution for the conservation law differs strongly.
Figure
......@@ -468,12 +455,17 @@ nolink "false"
while it decreases for others.
\end_layout
\begin_layout Standard
\begin_layout Plain Layout
Therefore,
the solution of the PDE is strongly sensitive to the chosen resistance function,
which emphasizes the huge challenge of the MCMC algorithm.
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Finding the Best Hyperparameter
\begin_inset CommandInset label
......@@ -558,6 +550,10 @@ The loss function
\end_inset
\begin_inset space \quad{}
\end_inset
\begin_inset Float figure
placement document
alignment document
......@@ -734,7 +730,7 @@ status open
\begin_inset Graphics
filename Figures/SingleSample/loss_theta.pdf
lyxscale 30
width 50text%
width 45text%
\end_inset
......@@ -765,6 +761,10 @@ name "fig:MCMC_SingleSample_Loss"
\end_inset
\begin_inset space \quad{}
\end_inset
\begin_inset Float figure
placement document
alignment document
......@@ -780,7 +780,7 @@ status open
\begin_inset Graphics
filename Figures/SingleSample/prediction_kx.pdf
lyxscale 30
width 50text%
width 45text%
\end_inset
......@@ -833,7 +833,7 @@ status open
\begin_inset Graphics
filename Figures/SingleSample/predicted_vs_true_single_sample.pdf
lyxscale 30
width 60text%
width 45text%
\end_inset
......@@ -1104,7 +1104,7 @@ status open
\begin_inset Graphics
filename Figures/ArtificialDiffusion/final_error_ArtDiffComparison.pdf
lyxscale 30
width 50text%
width 45text%
\end_inset
......@@ -1134,6 +1134,10 @@ name "fig:ArtificialDiffusion_Loss"
\end_inset
\begin_inset space \quad{}
\end_inset
\begin_inset Float figure
placement document
alignment document
......@@ -1145,7 +1149,7 @@ status open
\begin_inset Graphics
filename Figures/ArtificialDiffusion/prediction_kx_ArtDiffComparison.pdf
lyxscale 30
width 50text%
width 45text%
\end_inset
......@@ -1344,7 +1348,7 @@ status open
\begin_inset Graphics
filename Figures/DoubleSample/loss_theta.pdf
lyxscale 30
width 50text%
width 45text%
\end_inset
......@@ -1376,6 +1380,10 @@ name "fig:MCMC_DoubleSample_Loss"
\end_inset
\begin_inset space \quad{}
\end_inset
\begin_inset Float figure
placement document
alignment document
......@@ -1383,11 +1391,15 @@ wide false
sideways false
status open
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\begin_inset Graphics
filename Figures/DoubleSample/prediction_kx.pdf
lyxscale 30
width 50text%
width 45text%
\end_inset
......@@ -1443,25 +1455,6 @@ name "fig:MCMC_DoubleSample_k"
\end_inset
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "40text%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "foreground"
backgroundcolor "none"
status open
\begin_layout Plain Layout
\begin_inset Float figure
placement document
alignment document
......@@ -1473,7 +1466,7 @@ status open
\begin_inset Graphics
filename Figures/DoubleSample/predicted_vs_true_both_samples_sample_1.pdf
lyxscale 30
width 100text%
width 45text%
\end_inset
......@@ -1539,30 +1532,10 @@ name "fig:MCMC_DoubleSample_u1"
\end_inset
\end_layout
\begin_inset space \quad{}
\end_inset
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "40text%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "foreground"
backgroundcolor "none"
status open
\begin_layout Plain Layout
\begin_inset Float figure
placement document
alignment document
......@@ -1574,7 +1547,7 @@ status open
\begin_inset Graphics
filename Figures/DoubleSample/predicted_vs_true_both_samples_sample_2.pdf
lyxscale 30
width 100text%
width 45text%
\end_inset
......@@ -1611,11 +1584,6 @@ name "fig:MCMC_DoubleSample_u2"
\end_inset
\end_layout
\end_inset
\begin_inset Caption Standard
\begin_layout Plain Layout
......
Project3/LyX/TheoryAndMethods.lyx
+ 0
− 70
View file @ 73c2a608
......@@ -113,76 +113,6 @@ name "sec:Theory-and-Methods"
\end_layout
\begin_layout Standard
Consider the general mathematical description of a traffic flow (see project 2 for more reference)
\begin_inset Formula
\begin{align}
u_{t}+f(u)_{x} & =0,\quad x\in[a,b]\\
u(x,t=0) & =u_{0}(x)\\
u_{x}\big|_{x=x_{\text{in}}} & =u_{\text{in}}\\
u_{x}\big|_{x=x_{\text{out}}} & =u_{\text{out}}
\end{align}
\end_inset
with
\begin_inset Formula $a=0$
\end_inset
,
\begin_inset Formula $b=4$
\end_inset
,
the flux function
\begin_inset Formula $f(u)=u(1-u)$
\end_inset
.
In reality,
this model is influenced by some uncertainty,
quantified by the
\begin_inset Quotes eld
\end_inset
resistance
\begin_inset Quotes erd
\end_inset
function
\begin_inset Formula $k(x)$
\end_inset
\begin_inset Formula
\begin{align}
u_{t}+\left(k(x)f(u)\right)_{x} & =0,\quad x\in[a,b]\\
u(x,t=0) & =u_{0}(x)\\
u_{x}\big|_{x=a}=u_{x}\big|_{x=b} & =0
\end{align}
\end_inset
The boundaries are described by Neumann boundary conditions.
Purpose of this work is to identify the resistance function
\begin_inset Formula $k(x)$
\end_inset
for a known solution
\begin_inset Formula $u(x_{i},t)$
\end_inset
at specified positions
\begin_inset Formula $x_{i}$
\end_inset
.
Therefore,
an inverse problem will be formulated,
which utilizes Monte-Carlo Markov-Chains for the uncertainty quantification.
\end_layout
\begin_layout Subsection
Numerical Discretization of the General Conservation Law
\end_layout
......
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