diff --git a/Project3/LyX/Introduction.lyx b/Project3/LyX/Introduction.lyx
index ec830f0c361622b63c463b5ba038dd6a2cf92d5f..f6c448f7c3b9bb9e5cd054b66a3c8e0fb5ede6c6 100644
--- a/Project3/LyX/Introduction.lyx
+++ b/Project3/LyX/Introduction.lyx
@@ -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
diff --git a/Project3/LyX/Results.lyx b/Project3/LyX/Results.lyx
index 411bf73e01625519ade654c858b7c85e53e619cc..03223a81a12a6430bb18e99853c2d81da80e18dc 100644
--- a/Project3/LyX/Results.lyx
+++ b/Project3/LyX/Results.lyx
@@ -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
diff --git a/Project3/LyX/TheoryAndMethods.lyx b/Project3/LyX/TheoryAndMethods.lyx
index 958fc5a143e750428f7ee8a1f2f15df8f6067acb..9a7e5f4b1c885e3b2dee97393d04ed15f1180e77 100644
--- a/Project3/LyX/TheoryAndMethods.lyx
+++ b/Project3/LyX/TheoryAndMethods.lyx
@@ -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