v1

.gitignore

@@ -11,4 +11,4 @@ venv
 .DS_Store
 # Ignore LyX temporary files
 *.lyx~
-*.lyx#
+*.lyx#
\ No newline at end of file

Project3/LyX/DiscussionAndConclusion.lyx

@@ -107,35 +107,50 @@ name "sec:Discussion-and-Conclusion"
 \end_layout
 
 \begin_layout Standard
-How to improve the identification of 
+This work presented an accurate algorithm to quantify the uncertainty 
 \begin_inset Formula $k(x)$
+\end_inset
+
+ in a scalar conservation law of the form 
+\begin_inset Formula $u_{t}+\left(k(x)f(u)\right)_{x}=0$
 \end_inset
 
 .
+ 
 \end_layout
 
-\begin_layout Itemize
-Increase number of iterations
+\begin_layout Standard
+The major impact of the resistance function on the solution and the importance to predict this accurately was emphasized.
+ An optimal choice for the different hyperparameter was discussed and the influence of those on the solution.
+ 
 \end_layout
 
-\begin_layout Itemize
-Run 
-\begin_inset Formula $n$
-\end_inset
+\begin_layout Standard
+Increasing the observations from one dataset to two datasets showed a positive effect on the prediction of the resistance function.
+\end_layout
 
- MCMC iterations with different random starts and take best one
+\begin_layout Standard
+For future work it remains interesting to find a more efficient method,
+ as the MCMC algorithm depends on many iterations and is computationally complex.
+ Furthermore,
+ several efforts can be made to improve the identification of the resistance function.
 \end_layout
 
-\begin_layout Itemize
-Increase number of true data
+\begin_layout Enumerate
+Increase the number of MCMC iterations and adapt the control parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ when the convergence does not show an improvement about a certain number of steps.
 \end_layout
 
-\begin_layout Itemize
-Find best hyperparameter
+\begin_layout Enumerate
+Run several independent MCMC iterations with different random starts and take the one with the smallest loss.
 \end_layout
 
-\begin_layout Itemize
-Use even more true solutions
+\begin_layout Enumerate
+Increase the number of true datasets,
+ as going from one to two datasets showed promising results.
 \end_layout
 
 \end_body

Project3/LyX/TheoryAndMethods.lyx

@@ -742,10 +742,6 @@ If the acceptance rate is above
 \end_inset
 
 
-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
 \begin_inset Note Note
 status open
 
@@ -754,11 +750,7 @@ The different observations are weighted differently to ensure that the resistanc
 \begin_inset Formula $k(x)$
 \end_inset
 
- is accurately calculated on the whole domain
-\end_layout
-
-\begin_layout Enumerate
-Consider the weighting vector 
+ is accurately calculated on the whole domain Consider the weighting vector 
 \begin_inset Formula $w=[w_{1},w_{2},w_{3},w_{4}]$
 \end_inset
 
@@ -770,7 +762,6 @@ Consider the weighting vector
 
 \end_layout
 
-\begin_deeper
 \begin_layout Enumerate
 After each iteration update the weights
 \end_layout
@@ -788,13 +779,13 @@ Calculate
 
 \end_layout
 
-\end_deeper
 \end_deeper
 \end_inset
 
 
 \end_layout
 
+\end_deeper
 \begin_layout Standard
 The randomly generated parameter 
 \begin_inset Formula $\Theta^{(0)}$