diff --git a/.DS_Store b/.DS_Store
index 8069b1d04c64660dcc0234f0841dd05ea62c2d55..44a5bbbce45dbd57ddae3f986309557f4dc85f51 100644
Binary files a/.DS_Store and b/.DS_Store differ
diff --git a/.gitignore b/.gitignore
index a61e3835856f69dd25e30c045a81b572496c6006..4f9ce6daa25ae01acfbe87616c89c4dd2b40a4ef 100644
--- a/.gitignore
+++ b/.gitignore
@@ -11,4 +11,4 @@ venv
 .DS_Store
 # Ignore LyX temporary files
 *.lyx~
-*.lyx#
+*.lyx#
\ No newline at end of file
diff --git a/Project3/LyX/Abbreviations.lyx b/Project3/LyX/Abbreviations.lyx
new file mode 100644
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--- /dev/null
+++ b/Project3/LyX/Abbreviations.lyx
@@ -0,0 +1,311 @@
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+
+\begin_layout Section*
+List of Abbreviations
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="10" columns="2">
+<features tabularvalignment="middle">
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+
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+
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+
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+
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+
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+
+\begin_layout Plain Layout
+Method of Characteristics
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+
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+\begin_inset Text
+
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+
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+
+\begin_layout Plain Layout
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+\end_layout
+
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+
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+
+\end_inset
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+<cell alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Artificial Diffusion
+\end_layout
+
+\end_inset
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+
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+\end_inset
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+\begin_inset Text
+
+\begin_layout Plain Layout
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+\end_layout
+
+\end_inset
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+<cell alignment="center" valignment="top" usebox="none">
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+
+\begin_layout Plain Layout
+Generalized Least-Squares
+\end_layout
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+
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+
+\end_body
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diff --git a/Project3/LyX/Abstract.lyx b/Project3/LyX/Abstract.lyx
new file mode 100644
index 0000000000000000000000000000000000000000..6bab79e95bea7ba2378130d70cd831b072604f17
--- /dev/null
+++ b/Project3/LyX/Abstract.lyx
@@ -0,0 +1,123 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
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+\begin_header
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+
+\begin_body
+
+\begin_layout Abstract
+Roads are an important part of our lives.
+ We use them to get to work,
+ to our hobbies or just to visit our friends by car,
+ bus or by bike.
+ Too crowded roads can cause traffic jams that stop the flow of traffic.
+ These traffic jams restrict our movement and slow us down in our everyday life.
+\end_layout
+
+\begin_layout Abstract
+Different causes of traffic jams and possibilities to dissolve those were discussed in a previous work.
+\end_layout
+
+\begin_layout Abstract
+This work focus on the uncertainty quantification of mathematical modeling of traffic jams.
+ It assumes that the true density of cars is just known at certain positions,
+ as it can be observed by cameras.
+ With these observations,
+ the uncertainty of the mathematical model will be quantified.
+\end_layout
+
+\begin_layout Keywords
+Scalar Conservation Law,
+ Monte-Carlo Markov-Chain,
+ Uncertainty Quantification
+\end_layout
+
+\end_body
+\end_document
diff --git a/Project3/LyX/Appendix.lyx b/Project3/LyX/Appendix.lyx
new file mode 100644
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--- /dev/null
+++ b/Project3/LyX/Appendix.lyx
@@ -0,0 +1,108 @@
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+
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+
+\begin_layout Section
+\start_of_appendix
+Inserting the Dimensionless Parameter
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Appendix_AInserting-the-Dimensionless"
+
+\end_inset
+
+
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+
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+
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diff --git a/Project3/LyX/Bibliography.bib b/Project3/LyX/Bibliography.bib
new file mode 100644
index 0000000000000000000000000000000000000000..d1e5628b81ab76231c9688aec8a9f39d2f367945
--- /dev/null
+++ b/Project3/LyX/Bibliography.bib
@@ -0,0 +1,64 @@
+@unpublished{SteinerEvje-Lecture,
+    author = {Steinar Evje},
+    note   = {},
+    title  = {Lecture notes: Data-Driven Modeling of Conservation Laws},
+    year   = {2025}
+}
+
+@unpublished{LectureNotes,
+    author = {Siddhartha Mishra, Ulrik Skre Fjordholm and R´emi Abgrall},
+    note   = {},
+    title  = {Lecture notes: Numerical methods for conservation laws
+and related equations},
+    year   = {2025}
+}
+
+@unpublished{Intro_ConsLaw_2021,
+    author = {Steinar Evje},
+    note   = {},
+    title  = {Basic theory for nonlinear conservation laws + exercises},
+    year   = {2021}
+}
+
+
+@article{KUZNETSOV20085695,
+title = {Modeling traffic jams in intracellular transport in axons},
+journal = {International Journal of Heat and Mass Transfer},
+volume = {51},
+number = {23},
+pages = {5695-5699},
+year = {2008},
+note = {Biomedical-Related Special Issue},
+issn = {0017-9310},
+doi = {https://doi.org/10.1016/j.ijheatmasstransfer.2008.04.022},
+url = {https://www.sciencedirect.com/science/article/pii/S0017931008002731},
+author = {A.V. Kuznetsov and K. Hooman},
+keywords = {Molecular motors, Motor-assisted transport, Neurons, Axons and dendrites, Intracellular organelles, Traffic jams},
+abstract = {Irregularities in intracellular traffic in axons caused by mutations of molecular motors may lead to “traffic jams”, which often result in swelling of axons causing various neurodegenerative diseases. The purpose of this paper is to suggest a model of the formation of traffic jams in axons during molecular-motor-assisted transport of intracellular organelles utilizing transport equations developed in Smith and Simmons [D.A. Smith, R.M. Simmons, Models of motor-assisted transport of intracellular particles, Biophys. J. 80 (2001) 45–68], which describe the motion of intracellular particles under the combined action of diffusion and motor-driven transport. According to this model, large intracellular organelles are transported in the cytoplasm by a combined action of diffusion and motor-driven transport. In an axon, organelles are transported away from the neuron’s body toward the axon’s terminal by kinesin-family molecular motors running on tracks composed by microtubules; old and used components are carried back toward neuron’s body by dynein-family molecular motors. Binding/detachment kinetic processes between the organelles and microtubules are specified by first rate reaction constants; these lead to coupling between the three organelle concentrations.}
+}
+
+Péclet number. (2024, May 6). In Wikipedia. https://en.wikipedia.org/wiki/P%C3%A9clet_number
+@misc{Peclet_Wikipedia,
+    author       = {{Péclet number}},
+    month        = {May},
+    day          = {6},
+    url          = {https://en.wikipedia.org/wiki/P%C3%A9clet_number},
+    year         = {2024},
+    howpublished = {In \textit{Wikipedia}},
+}
+
+@unpublished{HabscheidProject1,
+    author = {Jan Habscheid},
+    note = {},
+    title = {Project 1: Data-Driven Modeling of Conservation Laws},
+    url = {https://git.rwth-aachen.de/JanHab/ddm},
+    year = {2024}
+}
+
+@software{Repository,
+    author = {Habscheid, Jan},
+    month = {April},
+    title = {Reproducibility Repository for Data-Driven Modeling of Conservation Laws},
+    URL = {https://git.rwth-aachen.de/JanHab/ddm},
+    year = {2025},
+}
diff --git a/Project3/LyX/DiscussionAndConclusion.lyx b/Project3/LyX/DiscussionAndConclusion.lyx
new file mode 100644
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+Discussion and Conclusion
+\begin_inset CommandInset label
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+name "sec:Discussion-and-Conclusion"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+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 Standard
+The major impact of the resistance function on the solution and the importance to predict this accurately were emphasized.
+ An optimal choice for the different hyperparameter was discussed and the influence of those on the solution.
+ 
+\end_layout
+
+\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
+
+\begin_layout Standard
+For future work it remains interesting to find a computationally 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 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_deeper
+\begin_layout Enumerate
+Another possible modification can be to check the acceptance ratio of changes after each MCMC iteration and adapt the control parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ to accept a certain ratio of changes.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Run several independent MCMC iterations with different random starts and take the one with the smallest loss.
+\end_layout
+
+\begin_layout Enumerate
+Increase the number of true datasets even further,
+ as going from one to two datasets already showed promising results.
+\end_layout
+
+\end_body
+\end_document
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+\begin_header
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+Implementation
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+LatexCommand label
+name "sec:Implementation"
+
+\end_inset
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+
+\end_layout
+
+\begin_layout Standard
+Find all the following examples,
+ visualizations and the belonging code implementation in the reproducability repository for this project (see 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Repository"
+literal "false"
+
+\end_inset
+
+).
+ All the implementations are done with the Python programming language,
+ mainly with the numpy package for vector operations and matplotlib for visualizations.
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+\usepackage{tikz}
+\usetikzlibrary{decorations.pathreplacing}
+\end_preamble
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+\pdf_title "Modeling Traffic Flow"
+\pdf_author "Jan Habscheid"
+\pdf_subject "Data-Driven Modeling of Conservation Laws"
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+Introduction
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+name "sec:Introduction"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Traffic jams are part of everyday life.
+ Most people use the road,
+ either by car,
+ bus or bike,
+ to get to work,
+ to meet friends,
+ to go shopping or to enjoy hobbies.
+ Through this enormous use of the road,
+ traffic jams occur.
+ Traffic jams lead to a smaller speed of vehicles,
+ if too many vehicles are close to each other.
+\end_layout
+
+\begin_layout Standard
+The question that arises is,
+ how to reduce these traffic jams to increase the traffic’s quality.
+ This work assumes that the density of cars is measured at certain parts of the road and that a mathematical model is known,
+ up to an uncertain function.
+ The objective of this work is to present a general methodology to identify this uncertain function,
+ based on the known mathematical model and observation data.
+\end_layout
+
+\end_body
+\end_document
diff --git a/Project3/LyX/ListOfVariables.lyx b/Project3/LyX/ListOfVariables.lyx
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+++ b/Project3/LyX/ListOfVariables.lyx
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+\begin_body
+
+\begin_layout Subsection*
+List of Variables
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:ListOfVariables"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+All the variables,
+ listed in the following table,
+ are in their dimensionless form.
+\end_layout
+
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+\align center
+
+\lang american
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+<lyxtabular version="3" rows="19" columns="3">
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+
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+
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+
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+binding rate in negative direction
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+
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diff --git a/Project3/LyX/Project3Habscheid.lyx b/Project3/LyX/Project3Habscheid.lyx
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+USING MONTE CARLO MARKOV CHAIN (MCMC) TO IDENTIFY UNKNOWN FLUX FUNCTION FROM OBSERVATION DATA
+\begin_inset Foot
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+
+\lang english
+Date:
+ 05.04.2025,
+ 
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+Referee:
+ Steinar Evje
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+Jan Habscheid
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diff --git a/Project3/LyX/Results.lyx b/Project3/LyX/Results.lyx
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+\begin_layout Section
+Results
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Results"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+Find all the following examples,
+ visualizations and the belonging code implementation in the reproducability repository for this project (see 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Repository"
+literal "false"
+
+\end_inset
+
+).
+ All the implementations are done with the Python programming language,
+ mainly with the numpy package for vector operations and matplotlib for visualizations.
+\end_layout
+
+\begin_layout Standard
+Two true solutions are available,
+ 
+\begin_inset Formula $d^{I}$
+\end_inset
+
+ and 
+\begin_inset Formula $d^{II}$
+\end_inset
+
+ with the respective initial data 
+\begin_inset Formula $u_{0}^{I}(x)$
+\end_inset
+
+ and 
+\begin_inset Formula $u_{0}^{II}(x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align}
+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}
+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_inset
+
+If not stated otherwise,
+ the artificial diffusion parameter 
+\begin_inset Formula $M$
+\end_inset
+
+ is set to one.
+\end_layout
+
+\begin_layout Subsection
+Influence of Resistance Function
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Maybe move this to the introduction as a motivation?
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+First,
+ quantify the influence of the resistance function on the solution of the general conservation law for initial data 
+\begin_inset Formula $u_{0}^{I}(x)$
+\end_inset
+
+.
+ The resistance functions are assumed to follow a normal distribution with mean 
+\begin_inset Formula $1$
+\end_inset
+
+ and standard deviation 
+\begin_inset Formula $0.25$
+\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 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-1"
+
+\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_1.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-2"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\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
+
+\begin_layout Plain Layout
+
+\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
+
+\begin_layout Plain Layout
+
+\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 three different,
+ randomly generated,
+ resistance functions (left) and the 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).
+ 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,
+ which emphasizes the huge challenge of the MCMC algorithm.
+\end_layout
+
+\begin_layout Subsection
+Finding the Best Hyperparameter
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:FindingBestHyperparameter"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+As the choice for the parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ and 
+\begin_inset Formula $\sigma^{}$
+\end_inset
+
+ has a huge impact on the accuracy of the MCMC algorithm a hyperparameter optimization is performed.
+ The optimization is based on a bayesian optimization using Gaussian processes.
+ It runs 25 different MCMC algorithms to find the one with the smallest loss function 
+\begin_inset Formula $\Phi(\Theta)$
+\end_inset
+
+.
+ The number of MCMC iterations is fixed to 150 for the optimization.
+ The algorithm typically converges after this number of iterations and further improvements for the loss function are rather small.
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/Optimization/TunerConvergence.pdf
+	lyxscale 30
+	width 45text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+The loss function 
+\begin_inset Formula $\Phi(\Theta)$
+\end_inset
+
+ over the different optimization steps.
+ The best values for both parameter are found after 22 iterations.
+\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
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/Optimization/TunerObjective.pdf
+	lyxscale 30
+	width 45text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+The objective of the optimization showing the tested values for 
+\begin_inset Formula $\beta$
+\end_inset
+
+ and 
+\begin_inset Formula $\sigma^{}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+The gaussian process starts with a default value of 
+\begin_inset Formula $\beta=0.05$
+\end_inset
+
+ and 
+\begin_inset Formula $\sigma^{}=0.25$
+\end_inset
+
+.
+ It searches in the range 
+\begin_inset Formula $\beta\in[0.01,0.1]$
+\end_inset
+
+ and 
+\begin_inset Formula $\sigma^{}\in[0.1,0.4]$
+\end_inset
+
+ and yields the optimal values 
+\begin_inset Formula $\beta_{\text{opt}}=0.0364,\sigma_{\text{opt}}=0.2443$
+\end_inset
+
+.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:HyperparameterOptimization"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Figures 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:HyperparameterOptimization"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ visualizes the results of this hyperparameter optimization for 
+\begin_inset Formula $\beta\in[0.01,0.1],\sigma^{2}\in[0.1,0.4]$
+\end_inset
+
+ with a start value 
+\begin_inset Formula $\beta=0.05$
+\end_inset
+
+ and 
+\begin_inset Formula $\sigma^{2}=0.25$
+\end_inset
+
+.
+ The best parameter 
+\begin_inset Formula $\beta_{\text{opt}}=0.0364,\sigma_{\text{opt}}=0.2443$
+\end_inset
+
+ yield a loss 
+\begin_inset Formula $\Phi(\Theta)=7.9612$
+\end_inset
+
+.
+ Note that the initial guess for the standard deviation was very good,
+ while the parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ shows better results for a reduced value.
+ The optimal parameter will be used for the remaining results.
+\end_layout
+
+\begin_layout Subsection
+Performing MCMC on a Single True Solution
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:MCMC_SingleSample"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Initial distribution 
+\begin_inset Formula $u_{0}^{I}(x)$
+\end_inset
+
+ is considered with the objective to identify the resistance function from this data..
+\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/SingleSample/loss_theta.pdf
+	lyxscale 30
+	width 50text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+The loss function 
+\begin_inset Formula $\Phi(\Theta)$
+\end_inset
+
+ over the iterations plotted in a log-log plot.
+ The loss improves consistent over the first iterations and finds an optimum at which it does not improve further.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_SingleSample_Loss"
+
+\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
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/SingleSample/prediction_kx.pdf
+	lyxscale 30
+	width 50text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Prediction for resistance function 
+\begin_inset Formula $k(x)$
+\end_inset
+
+.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_SingleSample_k"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\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
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/SingleSample/predicted_vs_true_single_sample.pdf
+	lyxscale 30
+	width 60text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Prediction vs true data for 
+\begin_inset Formula $u_{0}^{I}(x)$
+\end_inset
+
+ over time at positions 
+\begin_inset Formula $x\in[0.75,1.5,2.25,3.25]$
+\end_inset
+
+.
+ The predictions (solid lines) at position 
+\begin_inset Formula $x\in[0.75,3.25]$
+\end_inset
+
+ are an almost perfect match with the true data (dots).
+ For 
+\begin_inset Formula $x\in[1.5,2.25]$
+\end_inset
+
+ the prediction underestimates the true solution,
+ partially.
+ 
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_SingleSample_u"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Solution of MCMC for 
+\begin_inset Formula $u_{0}^{I}(x)$
+\end_inset
+
+.
+ The loss function shows a good convergence behavior and the predicted solution at 
+\begin_inset Formula $x\in[0.75,1.5,2.25,3.25]$
+\end_inset
+
+ is close to the true solution at those points.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_SingleSample"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MCMC_SingleSample"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ shows the evolution of the loss function (Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MCMC_SingleSample_Loss"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+),
+ the final estimation for the resistance function (Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MCMC_SingleSample_k"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+) and the final solution for the conservation law at the observation points (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MCMC_SingleSample_u"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+).
+ 
+\end_layout
+
+\begin_layout Standard
+The loss function is displayed in a double logarithmic plot,
+ showing consistent improvements for the prediction until it reaches a plateau.
+ New predictions are accepted on this plateau,
+ but those new predictions do not improve the accuracy.
+ 
+\end_layout
+
+\begin_layout Standard
+The predicted resistance function shows a minimum at 
+\begin_inset Formula $x_{2}$
+\end_inset
+
+ and maxima at 
+\begin_inset Formula $x_{6}$
+\end_inset
+
+ and 
+\begin_inset Formula $x_{8}$
+\end_inset
+
+.
+ Using this resistance function to predict the values for 
+\begin_inset Formula $u$
+\end_inset
+
+ at the four observation points yields very good results,
+ visualized in Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MCMC_SingleSample_u"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+.
+ The x-axis shows the time and the y-axis the value of 
+\begin_inset Formula $u$
+\end_inset
+
+ at positions 
+\begin_inset Formula $x\in[0.75,1.5,2.25,3.25]$
+\end_inset
+
+ (blue,
+ red,
+ green,
+ purple).
+ At positions 
+\begin_inset Formula $x\in[0.75,3.25]$
+\end_inset
+
+ the prediction (solid lines) is an almost perfect match with the true solution (dots).
+ Nevertheless,
+ at positions 
+\begin_inset Formula $x\in[1.5,2.25]$
+\end_inset
+
+ the predicted solution overestimates the true solution,
+ partially.
+ Generally,
+ the solution is very accurate at smooth regions and struggles with steep gradients.
+\end_layout
+
+\begin_layout Subsubsection
+Influence of Artificial Diffusion
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:ArtificialDiffusion"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The amount of numerical diffusion,
+ added to the discretization scheme,
+ influences the solution of the PDE.
+ The original PDE is no longer solved,
+ but a slightly adapted one (compare with project 1 and 2),
+ making the numerical discretization inconsistent.
+ 
+\begin_inset Formula 
+\begin{align}
+u_{t}+\left(k(x)f(u)\right)_{x} & =0\\
+\Rightarrow u_{t}^{\epsilon}+\left(k(x)f(u^{\epsilon})\right)_{x} & =u_{xx}^{\epsilon}
+\end{align}
+
+\end_inset
+
+The higher the number of 
+\begin_inset Formula $M$
+\end_inset
+
+,
+ the higher the amount of artificial diffusion 
+\begin_inset Formula $u_{xx}^{\epsilon}$
+\end_inset
+
+,
+ smoothing the solution,
+ especially the shocks.
+ The shocks are hard to predict with the MCMC algorithm,
+ but nevertheless they are in the true dataset.
+ However,
+ only the shocks in the numerical solution are smoothed,
+ not in the observations.
+ Therefore,
+ adding to much artificial diffusion yields problems in the accuracy of the MCMC algorithm.
+\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
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/ArtificialDiffusion/final_error_ArtDiffComparison.pdf
+	lyxscale 30
+	width 50text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Loss function 
+\begin_inset Formula $\Phi(\Theta)$
+\end_inset
+
+ for different values of artificial diffusion
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:ArtificialDiffusion_Loss"
+
+\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/ArtificialDiffusion/prediction_kx_ArtDiffComparison.pdf
+	lyxscale 30
+	width 50text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Reconstructed resistance function for different values of artificial diffusion.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:ArtificialDiffusion_k"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+The parameter 
+\begin_inset Formula $M$
+\end_inset
+
+,
+ describing the amount of artificial diffusion added to the numerical discretization,
+ has a large impact on the accuracy of the MCMC algorithm.
+ The Figure shows four different values of 
+\begin_inset Formula $M\in[1,2,3,4]$
+\end_inset
+
+ (blue,
+ red,
+ green,
+ purple).
+ The MCMC algorithm performs worse for increasing numbers of artificial diffusion,
+ which results in differences for the resistance function.
+ Increasing values of artificial diffusion result in smoothing of the shocks.
+ However,
+ this smoothing is only present in the numerical scheme,
+ not in the true solution.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:ArtificialDiffusion"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:ArtificialDiffusion"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ shows the influence of the artificial diffusion parameter 
+\begin_inset Formula $M$
+\end_inset
+
+ on the accuracy of the MCMC algorithm.
+ The final loss increases exponentially for increasing numerical diffusion (Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:ArtificialDiffusion_Loss"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+),
+ emphasizing to use a consistent numerical scheme with a small amount of artificial diffusion.
+ The resistance function even yields a complete different solution for different numbers of 
+\begin_inset Formula $M$
+\end_inset
+
+ (Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:ArtificialDiffusion_k"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+).
+ 
+\end_layout
+
+\begin_layout Standard
+As a consequence,
+ the value 
+\begin_inset Formula $M$
+\end_inset
+
+ will be fixed to one for the remaining project.
+\end_layout
+
+\begin_layout Subsection
+Extending the MCMC Algorithm to Two True Solutions
+\end_layout
+
+\begin_layout Standard
+For now,
+ only initial distribution 
+\begin_inset Formula $u_{0}^{I}(x)$
+\end_inset
+
+,
+ with the corresponding true solution 
+\begin_inset Formula $d^{I}$
+\end_inset
+
+,
+ was considered.
+ In this example,
+ both initial distributions 
+\begin_inset Formula $u_{0}^{I}(x)$
+\end_inset
+
+ and 
+\begin_inset Formula $u_{0}^{II}(x)$
+\end_inset
+
+ with corresponding solutions 
+\begin_inset Formula $d^{I}$
+\end_inset
+
+ and 
+\begin_inset Formula $d^{II}$
+\end_inset
+
+ are used to inspect the influence of multiple true solutions on the MCMC algorithm.
+ The calculated loss 
+\begin_inset Formula $\text{LOSS}_{\alpha}$
+\end_inset
+
+ is the combined loss of both distributions.
+\begin_inset Formula 
+\begin{align}
+\text{LOSS}_{\alpha}^{I} & =\Phi^{I}(\alpha)=\frac{1}{2}\frac{\left(h^{I}(\alpha)-d^{I}\right)\left(h^{I}(\alpha)-d^{I}\right)^{T}}{\gamma^{2}}\\
+\text{LOSS}_{\alpha}^{II} & =\Phi^{II}(\alpha)=\frac{1}{2}\frac{\left(h^{II}(\alpha)-d^{II}\right)\left(h^{II}(\alpha)-d^{II}\right)^{T}}{\gamma^{2}}\\
+\text{LOSS}_{\alpha} & =\Phi^{I}(\alpha)+\Phi^{II}(\alpha)
+\end{align}
+
+\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/DoubleSample/loss_theta.pdf
+	lyxscale 30
+	width 50text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+The loss function 
+\begin_inset Formula $\Phi(\Theta)$
+\end_inset
+
+ over the iterations plotted in a log-log plot.
+ The algorithm shows a continuous improvement over the first 70-80 steps and then oscillates around an optimum.
+ 
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_DoubleSample_Loss"
+
+\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/DoubleSample/prediction_kx.pdf
+	lyxscale 30
+	width 50text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Prediction for resistance function 
+\begin_inset Formula $k(x)$
+\end_inset
+
+.
+ The global minimum is again at 
+\begin_inset Formula $x_{2}$
+\end_inset
+
+ but only one global maximum at 
+\begin_inset Formula $x_{6}$
+\end_inset
+
+ is present.
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:MCMC_SingleSample"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+.
+ 
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_DoubleSample_k"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Newline newline
+\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
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/DoubleSample/predicted_vs_true_both_samples_sample_1.pdf
+	lyxscale 30
+	width 100text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Prediction vs true data for 
+\begin_inset Formula $u_{0}^{I}(x)$
+\end_inset
+
+.
+ The solution improved,
+ compared to the one presented in section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:MCMC_SingleSample"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+.
+ Only slight offsets at 
+\begin_inset Formula $x=0.75$
+\end_inset
+
+ for 
+\begin_inset Formula $t\approx0.5$
+\end_inset
+
+ and 
+\begin_inset Formula $x=2.25$
+\end_inset
+
+ for 
+\begin_inset Formula $t\approx1.9$
+\end_inset
+
+ can be identified.
+ At positions 
+\begin_inset Formula $x\in[1.5,3.25]$
+\end_inset
+
+ almost no difference between the predicted and the true solution can be seen.
+ 
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_DoubleSample_u1"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/DoubleSample/predicted_vs_true_both_samples_sample_2.pdf
+	lyxscale 30
+	width 100text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Prediction vs true data for 
+\begin_inset Formula $u_{0}^{II}(x)$
+\end_inset
+
+.
+ The solution at all four observed positions is very accurate,
+ showing only slight perturbances in front of the shock at 
+\begin_inset Formula $x=3.25$
+\end_inset
+
+,
+ with a very accurate reproduction of the shock.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_DoubleSample_u2"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Solution of MCMC for 
+\begin_inset Formula $u_{0}^{I}$
+\end_inset
+
+ and 
+\begin_inset Formula $u_{0}^{II}$
+\end_inset
+
+.
+ Considering two datasets,
+ instead of only a single dataset,
+ improves the accuracy of the MCMC algorithm.
+ It is hard to compare the absolute value of the loss function,
+ compared to the value from section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:MCMC_SingleSample"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+,
+ as it combines the error of both distributions.
+ Nevertheless,
+ the prediction of 
+\begin_inset Formula $u$
+\end_inset
+
+ at the observed positions can be interpreted and shows a clear improvement.
+ The prediction is an almost perfect match for both distributions,
+ at all four positions over the whole time.
+ It only shows slight perturbances at sharp gradients.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:MCMC_DoubleSample"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MCMC_DoubleSample"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ visualizes the solution,
+ when considering both initial distributions.
+ The loss function shows a clear convergence (see Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MCMC_DoubleSample_Loss"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+) with a good initial improvement until it oscillates around a minimum.
+\end_layout
+
+\begin_layout Standard
+The found resistance function (Figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:MCMC_DoubleSample_k"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+) is similar to the one from section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:MCMC_SingleSample"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ for 
+\begin_inset Formula $x\in[0,2.5)$
+\end_inset
+
+ and shows differences for 
+\begin_inset Formula $x\in[2.5,4]$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Where the solution from section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:MCMC_SingleSample"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ shows two critical regions,
+ the newly identified function predicts the true solution in these regions very accurate.
+ Furthermore,
+ the observation points for the second dataset are reproduced very accurate,
+ too.
+ The prediction only shows slight problems at sharp gradients,
+ which probably occurs because of the remaining numerical diffusion.
+ This numerical diffusion changes the numerical solution,
+ compared to the true solution,
+ even for a perfectly identified resistance function (compare with section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:ArtificialDiffusion"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+).
+\end_layout
+
+\end_body
+\end_document
diff --git a/Project3/LyX/TheoryAndMethods.lyx b/Project3/LyX/TheoryAndMethods.lyx
new file mode 100644
index 0000000000000000000000000000000000000000..958fc5a143e750428f7ee8a1f2f15df8f6067acb
--- /dev/null
+++ b/Project3/LyX/TheoryAndMethods.lyx
@@ -0,0 +1,868 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass IEEEtran-CompSoc
+\begin_preamble
+\usepackage{tikz}
+\usetikzlibrary{decorations.pathreplacing}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children no
+\language american
+\language_package default
+\inputencoding utf8
+\fontencoding auto
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+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\float_placement class
+\float_alignment center
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\spellchecker_ignore american Sobolev
+\spellchecker_ignore american OpenMP
+\spellchecker_ignore american scalable
+\spellchecker_ignore american discretized
+\spellchecker_ignore american splitted
+\spellchecker_ignore american bilinear
+\leftmargin 2.5cm
+\topmargin 2.5cm
+\rightmargin 2.5cm
+\bottommargin 2.5cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style english
+\dynamic_quotes 0
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+\tablestyle default
+\tracking_changes false
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+\change_bars false
+\postpone_fragile_content true
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\docbook_table_output 0
+\docbook_mathml_prefix 1
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Theory and Methods
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Theory-and-Methods"
+
+\end_inset
+
+
+\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
+
+\begin_layout Standard
+The general conservation law will be solved numerically with an explicit Euler for time integration and an upwind method for the spatial derivatives.
+ Discretizing the equation results in
+\begin_inset Formula 
+\begin{equation}
+\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+\frac{1}{\Delta x}\left(k(x_{i+\frac{1}{2}})F_{i+\frac{1}{2}}^{n}-k(x_{i-\frac{1}{2}})F_{i-\frac{1}{2}}^{n}\right)=0
+\end{equation}
+
+\end_inset
+
+with the numerical flux 
+\begin_inset Formula 
+\begin{equation}
+F_{i+\frac{1}{2}}=\frac{f(u_{i}^{n})+f(u_{i+1}^{n})}{2}-\frac{M}{2}\left(u_{i+1}^{n}-u_{i}^{n}\right)
+\end{equation}
+
+\end_inset
+
+and the time step size 
+\begin_inset Formula $\Delta t$
+\end_inset
+
+,
+ 
+\begin_inset Formula $\Delta x$
+\end_inset
+
+ the spatial discretization,
+ the upper index 
+\begin_inset Formula $\cdot^{n}$
+\end_inset
+
+ identify the time and the lower index 
+\begin_inset Formula $\cdot_{i}$
+\end_inset
+
+ identifying the space.
+ 
+\begin_inset Formula $M$
+\end_inset
+
+ adds a certain amount of artificial diffusion.
+ For more reference on the numerical discretization scheme see project 1 and project 2.
+\end_layout
+
+\begin_layout Subsection
+The Resistance Function 
+\begin_inset Formula $k(x)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Assume that the resistance function is represented as a piecewise linear function over the spatial domain.
+ The function is defined at 11 uniformly spaced points 
+\begin_inset Formula $\{x_{i}\}_{i=0}^{i=10}$
+\end_inset
+
+ including the boundary 
+\begin_inset Formula 
+\begin{align}
+x_{i} & \in\left[a,a+\frac{b-a}{11},2\frac{b-a}{11},\dots,b\right]\\
+k_{i} & =k(x_{i})
+\end{align}
+
+\end_inset
+
+The points 
+\begin_inset Formula $x_{i}$
+\end_inset
+
+ are known.
+ However,
+ the respective values of the resistance function are unknown and the objective is to identify these 11 values.
+ Assume that the resistance function is close to one
+\begin_inset Formula 
+\begin{equation}
+k(x)\in[0.25,1.75]
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+The Observer
+\end_layout
+
+\begin_layout Standard
+Assume that the density of cars is observed by some cameras.
+ These cameras are placed at four positions 
+\begin_inset Formula $\tilde{x}_{i}\in[0.75,1.5,2.25,3.25]$
+\end_inset
+
+.
+ The density of cars is observed at some uniformly sampled,
+ discrete timeframes
+\begin_inset Formula 
+\begin{equation}
+t_{i}\in[t_{0},t_{1},\dots t_{N}]
+\end{equation}
+
+\end_inset
+
+with the first timeframe observing the initial distribution 
+\begin_inset Formula $t_{0}=0$
+\end_inset
+
+ and the last timeframe observing the final distribution 
+\begin_inset Formula $t_{N}=10=T_{\text{end}}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $N=40$
+\end_inset
+
+ different observations are made in total.
+\end_layout
+
+\begin_layout Subsection
+Monte-Carlo Marcov-Chains
+\end_layout
+
+\begin_layout Standard
+The general idea of a Monte-Carlo Marcov-Chain (MCMC) is to identify an unknown parameter 
+\begin_inset Formula $\Theta$
+\end_inset
+
+,
+ based on true solutions.
+ For this work,
+ the Metropolis-Hastings algorithm will be utilized to identify the resistance function 
+\begin_inset Formula $k(x)$
+\end_inset
+
+ (see 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "SteinerEvje-Lecture"
+literal "false"
+
+\end_inset
+
+).
+ The algorithm iteratively minimizes the loss function 
+\begin_inset Formula $\Phi$
+\end_inset
+
+ describing the mismatch between the prediction 
+\begin_inset Formula $\Theta$
+\end_inset
+
+ and true solution 
+\begin_inset Formula $d$
+\end_inset
+
+.
+\begin_inset Formula 
+\begin{equation}
+\Phi^{k}\sim\parallel h(\Theta^{k})-d\parallel^{2}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The main assumption is,
+ that the unknown parameter vector can be described by a normal distribution.
+ This assumption can be reformulated in a way to describe the true solution 
+\begin_inset Formula $d$
+\end_inset
+
+ by the current solution 
+\begin_inset Formula $h(\Theta)$
+\end_inset
+
+ and some offset 
+\begin_inset Formula $\eta$
+\end_inset
+
+.
+\begin_inset Formula 
+\begin{align}
+d & =h(\Theta)+\eta\\
+\eta & =d-h(\Theta)\sim N(0,\gamma^{2}\boldsymbol{I})
+\end{align}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+with the probability for the random variable 
+\begin_inset Formula $\eta$
+\end_inset
+
+ given by 
+\begin_inset Formula $P(\eta;0,\gamma^{2}\boldsymbol{I})$
+\end_inset
+
+ and the identity matrix 
+\begin_inset Formula $\boldsymbol{I}$
+\end_inset
+
+.
+\begin_inset Formula 
+\begin{align}
+P\left(\eta=d-h(\Theta);0,\gamma^{2}\boldsymbol{I}\right) & =\frac{1}{\sqrt{2\pi\gamma^{2}}}\exp\left[-\frac{1}{2}\frac{\left(d-h(\Theta)\right)\left(d-h(\Theta)\right)^{2}}{\gamma^{2}}\right]\\
+ & =\frac{1}{\sqrt{2\pi\gamma^{2}}}\exp\left[-\frac{1}{2}\frac{\parallel d-h(\Theta)\parallel^{2}}{\gamma^{2}}\right]\\
+ & =\frac{1}{\sqrt{2\pi\gamma^{2}}}\exp\left[-\Phi(\Theta)\right]\\
+ & =P_{\Theta}\\
+\text{with: }\Phi(\Theta) & =\frac{1}{2}\frac{\parallel d-h(\Theta)\parallel^{2}}{\gamma^{2}}
+\end{align}
+
+\end_inset
+
+Also consider a slightly perturbed version 
+\begin_inset Formula $\phi$
+\end_inset
+
+ of 
+\begin_inset Formula $\Theta$
+\end_inset
+
+,
+ which is randomly generated.
+ The probability for the random variable 
+\begin_inset Formula $\eta$
+\end_inset
+
+ can be calculated according to
+\begin_inset Formula 
+\begin{align}
+p\left(\eta=d-h(\phi);0,\gamma^{2}\boldsymbol{I}\right) & =\frac{1}{\sqrt{2\pi\gamma^{2}}}\exp\left[-\Phi(\phi)\right]\\
+ & =P_{\phi}\\
+\text{with: }\Phi(\phi) & =\frac{1}{2}\frac{\parallel d-h(\phi)\parallel^{2}}{\gamma^{2}}
+\end{align}
+
+\end_inset
+
+The arising question is if either 
+\begin_inset Formula $\Theta$
+\end_inset
+
+ or 
+\begin_inset Formula $\phi$
+\end_inset
+
+ describes the unknown function 
+\begin_inset Formula $k(x)$
+\end_inset
+
+ better.
+ The smaller the value of 
+\begin_inset Formula $\Phi(\cdot)\sim\parallel d-h(\cdot)\parallel$
+\end_inset
+
+ the better does 
+\begin_inset Formula $h(\cdot)$
+\end_inset
+
+ describes the unknown function.
+ Rewriting this ends in a rule to choose the property with the higher probability.
+\begin_inset Formula 
+\begin{equation}
+\max\{P_{\Theta},P_{\phi}\}
+\end{equation}
+
+\end_inset
+
+If 
+\begin_inset Formula $\phi$
+\end_inset
+
+ has the higher probability (
+\begin_inset Formula $\frac{P_{\phi}}{P_{\Theta}}\geq1$
+\end_inset
+
+),
+ 
+\begin_inset Formula $\Theta$
+\end_inset
+
+ should be updated to match 
+\begin_inset Formula $\phi$
+\end_inset
+
+,
+ otherwise it can be continued with the current 
+\begin_inset Formula $\Theta$
+\end_inset
+
+ and tried again with another randomly generated perturbation 
+\begin_inset Formula $\phi$
+\end_inset
+
+.
+ To avoid ending in local minima,
+ an element of randomness is introduced.
+ The decision function 
+\begin_inset Formula $a(\Theta,\phi)$
+\end_inset
+
+ is defined for this.
+\begin_inset Formula 
+\begin{equation}
+a(\Theta,\phi)=\min\{1,\frac{P_{\phi}}{P_{\Theta}}\}\qquad\text{with: }\frac{P_{\phi}}{P_{\Theta}}=\exp\left[\Phi(\Theta)-\Phi(\phi)\right]
+\end{equation}
+
+\end_inset
+
+The current prediction 
+\begin_inset Formula $\Theta$
+\end_inset
+
+ will be updated,
+ if a randomly,
+ uniformly sampled number 
+\begin_inset Formula $i$
+\end_inset
+
+ (
+\begin_inset Formula $i\sim U([0,1])$
+\end_inset
+
+ is smaller than a,
+ otherwise the algorithm continues with the current guess.
+\end_layout
+
+\begin_layout Subsubsection
+The Metropolis-Hastings Algorithm
+\end_layout
+
+\begin_layout Standard
+Define
+\begin_inset Formula 
+\begin{align}
+a(\Theta,\phi) & =\min\{1,\exp(\Phi(\Theta)-\Phi(\phi)\}\\
+\text{LOSS}_{\alpha} & =\Phi(\alpha)=\frac{1}{2}\frac{\left(h(\alpha)-d\right)\left(h(\alpha)-d\right)^{T}}{\gamma^{2}}\\
+\eta\sim N(0,\gamma^{2}\boldsymbol{I}),\quad\gamma & =0.03\\
+\mu & =1,\\
+\beta=\beta,\quad\sigma & =\sigma\\
+\tau & =100
+\end{align}
+
+\end_inset
+
+where the newly introduced parameter 
+\begin_inset Formula $\mu$
+\end_inset
+
+ describes the mean of the assumed normal distribution and 
+\begin_inset Formula $\tau$
+\end_inset
+
+ is the number of iteration steps performed before the MCMC algorithm starts,
+ as a burn-in period.
+ Note that the parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ and 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ are not fixed,
+ yet.
+ Their optimal values will be calculated in a hyperparameter optimization in section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:FindingBestHyperparameter"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+.
+ Index 
+\begin_inset Formula $k$
+\end_inset
+
+ describes the current iteration index.
+\end_layout
+
+\begin_layout Standard
+The algorithm is as follows:
+\end_layout
+
+\begin_layout Enumerate
+Start with initial guess 
+\begin_inset Formula $\Theta^{(0)}\sim N(\mu,\sigma^{2}\boldsymbol{I})$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+Avoid to much noise by clipping the boundaries of 
+\begin_inset Formula $\Theta^{(0)}\in[0.8,1.2]$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Calculate the perturbation:
+ 
+\begin_inset Formula $\phi^{(k)}=\sqrt{1-\beta^{2}}\Theta^{(k)}+\beta\xi^{(k)},\xi^{(k)}\sim N(0,\sigma^{2}\boldsymbol{I})$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Calculate the decision function 
+\begin_inset Formula $a(\Theta,\phi)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Draw a random number 
+\begin_inset Formula $i^{(k)}$
+\end_inset
+
+ from a uniform distribution and determine 
+\begin_inset Formula $\Theta^{(k+1)}$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+If 
+\begin_inset Formula $i\leq a(\Theta,\phi)$
+\end_inset
+
+ update 
+\begin_inset Formula $\Theta^{(k+1)}=\phi^{(k)}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Else update 
+\begin_inset Formula $\Theta^{(k+1)}=\Theta^{(k)}$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+For the burn-in period,
+ steps 2-4 are repeated for 
+\begin_inset Formula $k\in[0,\tau]$
+\end_inset
+
+ to precondition the prediction vector 
+\begin_inset Formula $\Theta$
+\end_inset
+
+.
+ Then,
+ the MCMC algorithm is performed,
+ where steps 2-4 are repeated for 
+\begin_inset Formula $k\in[0,n_{\text{MCMC}}]$
+\end_inset
+
+ and for each MCMC step the steps are repeated 20 times,
+ too.
+\end_layout
+
+\begin_layout Subsubsection
+Discussing the Parameter of the MCMC Algorithm
+\end_layout
+
+\begin_layout Standard
+The parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ is of high interest for the algorithm,
+ as it controls the difference between the guess 
+\begin_inset Formula $\Theta^{(k)}$
+\end_inset
+
+ and the perturbation 
+\begin_inset Formula $\phi^{(k)}$
+\end_inset
+
+.
+ An increased value for 
+\begin_inset Formula $\beta$
+\end_inset
+
+ will increase this difference,
+ giving a higher weight to the random value 
+\begin_inset Formula $\xi^{(k)}$
+\end_inset
+
+,
+ which may lead to a faster improvement of the loss function but comes with the danger to get stuck in a local minimum.
+ A good rule of thumb is to start with large steps and a large 
+\begin_inset Formula $\beta$
+\end_inset
+
+ in the beginning to then iteratively reduce this parameter for a better convergence.
+ Nevertheless,
+ this work utilizes a constant parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+,
+ for the sake of simplicity.
+ 
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+In this work,
+ the parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ is updated after each MCMC iteration,
+ based on the acceptance rate to update the guess 
+\begin_inset Formula $\Theta^{(k+1)}=\phi^{(k)}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+The parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ is updated after each MCMC iteration to push the algorithm to an acceptance rate of 
+\begin_inset Formula $\omega\in[\omega_{\min},\omega_{\max}]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+If the acceptance rate is below 
+\begin_inset Formula $\omega_{\min}$
+\end_inset
+
+ update 
+\begin_inset Formula $\beta=\frac{9}{10}\beta$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+If the acceptance rate is above 
+\begin_inset Formula $\omega_{\max}$
+\end_inset
+
+ update 
+\begin_inset Formula $\beta=\frac{11}{10}\beta$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+The different observations are weighted differently to ensure that the resistance function 
+\begin_inset Formula $k(x)$
+\end_inset
+
+ 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
+
+,
+ with 
+\begin_inset Formula $w_{i}=1\forall i\in[0,4]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+After each iteration update the weights
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+Calculate 
+\begin_inset Formula $\tilde{w}_{i}=L_{2}(h(\Theta_{i}),d_{i})$
+\end_inset
+
+ and calculate 
+\begin_inset Formula $w_{i}=n\frac{\tilde{w}_{i}}{\sum_{i=1}^{n}\tilde{w}_{i}}$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The randomly generated parameter 
+\begin_inset Formula $\Theta^{(0)}$
+\end_inset
+
+ influences the initial convergence behavior,
+ as it determines the starting point of the optimization problem.
+ If 
+\begin_inset Formula $\Theta^{(0)}$
+\end_inset
+
+ is chosen very close to the true solution the algorithm will converge fast.
+ If,
+ on the other hand,
+ 
+\begin_inset Formula $\Theta^{(0)}$
+\end_inset
+
+ is chosen poorly,
+ it will take more iterations for the algorithm to come to a good solution.
+ It may even happen that the algorithm will get stuck in some local minimum,
+ instead of finding the global minimum.
+ To reduce randomness in the algorithm,
+ it is a good choice to either reduce the standard deviation 
+\begin_inset Formula $\sigma^{(0)}$
+\end_inset
+
+ for the initial guess or to clip the range of 
+\begin_inset Formula $\Theta^{(0)}$
+\end_inset
+
+.
+ Both will reduce the range,
+ and therefore the variance,
+ of the initial guess,
+ reducing the impact of randomness.
+ It is unknown if this will improve the convergence behavior,
+ as the variance of the true solution may differ,
+ but it will balance the convergence behavior for different initial guesses.
+\end_layout
+
+\end_body
+\end_document
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diff --git a/Project3/src/GivenCode/mcmc_task2_u0X1_final_student_mars21.py b/Project3/src/GivenCode/mcmc_task2_u0X1_final_student_mars21.py
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--- /dev/null
+++ b/Project3/src/GivenCode/mcmc_task2_u0X1_final_student_mars21.py
@@ -0,0 +1,422 @@
+import matplotlib.pyplot as plt
+import numpy as np
+#import scipy.io as sio
+import scipy.sparse as sparse
+
+try:
+    from simulations_Feb20 import simulate_g
+except ImportError:
+     from python_scripts.simulations_Feb20 import simulate_g
+
+
+##############################################
+# read observation data provided by teacher  #
+##############################################
+
+# use only the first data set
+observation = np.load("data/True_Observation_data_1_upd.npz")
+#print( list(observation.keys()) )
+
+
+######################################################################################
+#  In the following different parameters required for solving conslaw are extracted  #
+######################################################################################
+
+
+# number of observation data set 
+Ant_Init = 1
+
+#y_k_true = observation["y_k_true"]
+#y_k_true = np.zeros_like(y_k_true)
+
+# provide information about grid, 
+x    = observation["x"]
+xh   = observation["xh"]
+
+# grid cells for cons law
+N = observation["M"].item()
+
+# Final time 
+T = observation["T"].item()
+
+# describe time points for which we have observations
+time_obs = observation["t_obspoint"]
+JX = time_obs <= T
+
+Ldom = observation["Ldom"].item()
+dx = Ldom/N
+
+# positions of point values for k(x)
+x_k = observation["x_k"]
+
+# index vector for numbering positions in space and time
+Nt = np.sum(JX)
+Ix = np.arange(0, N)
+It = np.arange(0, Nt)
+
+# number of unknown points to be determined in k(x) which is assumed piecewise linear
+Mr = observation["Mr"].item() 
+
+###############################
+# Observation data  h(theta)  #
+###############################
+u_true_X1 = np.zeros((Nt))
+u_true_X2 = np.zeros((Nt))
+u_true_X3 = np.zeros((Nt))
+u_true_X4 = np.zeros((Nt))
+
+# correct when Ant_Init = 1
+u_true_X1 = observation["u_true_X1"] 
+u_true_X2 = observation["u_true_X2"] 
+u_true_X3 = observation["u_true_X3"] 
+u_true_X4 = observation["u_true_X4"]   
+
+
+# provide initial data set: each initialdata (of total=Ant_Init) is a column 
+initial_true = np.zeros((N,2))
+initial_true[:,0] = observation["u0"]
+
+
+# provide position of observation points X_1, X_2, X_3, X_4 
+X_position = observation["X_position"]
+X_1 = X_position[0]
+X_2 = X_position[1]
+X_3 = X_position[2]
+X_4 = X_position[3]
+
+
+####################################
+# define parameters used in MCMC   #
+####################################
+add_unc = 0.03            # noise in observation data
+error_val = 100           # lower limit in Loss function
+kk_tolerance = 1000       # number of steps in MCMC algorithm
+teller = 1
+tau = 20                  # extract "theta^k" with step size "tau" to represent a posteriori "theta" 
+
+N_ens = int(kk_tolerance/tau)
+
+# function to create one long column vector to represent the matrix of 
+# 4 column vectors representing observation data at X1, X2, X3, X4
+def gen_measurements(u, weights):
+    measurement = []
+    for i, u in enumerate(u):
+        m = u + add_unc * np.random.normal(size=Nt * Ant_Init)
+        m[m < 0] = 0
+        m[m > 1] = 1
+        measurement.append(weights[i] * m)
+    return np.concatenate(measurement)
+
+
+# weight factor used to adjust the importance of the 4 different time series
+weights = [1, 1, 1, 1]
+
+# compute one single column vector to represent all observation data
+measurement = gen_measurements([u_true_X1, u_true_X2, u_true_X3, u_true_X4], weights)
+
+
+# parametervector "theta" which is updated by the mcmc algorithm
+# theta is composed of both k(x) and f(u) - however, only k(x) unknown
+initial_ensemble_kx = np.zeros((Mr, 1))
+
+# we generate an ensemble parameter vector by extracting every 20th member from "initial_ensemble_F"
+updated_ensemble_kx = np.zeros((Mr, N_ens))
+
+# mcmc parameter 
+#beta = 0.05 * 4  #   
+beta = 0.05   # 
+#beta = 0.05 * 0.25  # 
+
+#  we generate initial k(x) ~ {k_i}_i=1^9 as random normal distribution N(mu,sigma)
+mu = 1
+sigma = 0.25
+
+
+
+#########################################################
+#  Choose what should be the initial parameter vector   #
+#########################################################
+
+
+
+mcmc_start = True
+#mcmc_start = False
+
+if mcmc_start:
+      #####################################
+      #  choice 1:  generate theta^{(0)}  #
+      #####################################
+
+       print("Initial guess of k(x)")
+       kk_x = np.random.normal(loc=mu, scale=sigma, size=Mr)
+       kk_x[kk_x < 0.95] = 0.95
+       kk_x[kk_x > 1.3] = 1.3
+       kkx_u = kk_x
+else:
+      #######################################
+      #  choice 2:  start with theta^{(k)}  #
+      #######################################
+
+      with np.load('data/mcmc_u0X1_sample100_it1_student_Mars21.npz') as data_kx:
+      #with np.load('data/mcmc_u0X1_sample100_it2_student_Mars21.npz') as data_kx: 
+      #with np.load('data/mcmc_u0X1_sample100_it3_student_Mars21.npz') as data_kx:
+      #with np.load('data/mcmc_u0X1_sample100_it4_student_Mars21.npz') as data_kx:     
+       kkx_u = data_kx["theta_1"]
+
+
+# Create the y_k vector with padding 1s at the beginning and the end
+y_k = np.concatenate(([1], kkx_u, [1]))
+
+
+# number of parameter vectors (in case of more than one ensemble)
+K = 0
+
+
+# Create the plot
+plt.figure(1)
+plt.plot(x_k, y_k)
+plt.axis([-0.1, Ldom + 0.1, 0.1, 1.9])
+plt.grid(True)
+plt.xlabel("X")
+plt.ylabel("k(x)")
+plt.title("Resistance k(x)")
+
+# plt.ioff()
+
+# Show the plot and pause execution to view it
+plt.show(block=False)
+plt.pause(2)  # Pause for a short period of time; adjust as needed
+
+# Close the plot after pausing
+plt.close(1)
+
+# "initial_ensemble_kx" is composed of vector k(x) (M:M+Mr) 
+#  which is hidden and we seek to recover 
+initial_ensemble_kx[0:Mr, K] = kkx_u
+
+#######################################################
+# Run simulation to generate h(kkx_u)
+#######################################################
+
+###############################
+#  Step S3: Compute h(theta)  #
+###############################
+
+# h ~ simulate forward model
+# theta ~ initial_ensemble_kx
+
+# u_X =
+# ensembleOfPredictedObservations = 
+
+ensembleOfPredictedObservations = simulate_g(
+    N, x, xh, Ldom, T,
+    initial_true,
+    Ant_Init,
+    Nt,
+    time_obs,
+    It,
+    JX,
+    K,
+    weights,
+    initial_ensemble_kx,
+    X_1, X_2, X_3, X_4,
+    x_k, kkx_u,
+    u_true_X1, u_true_X2, u_true_X3, u_true_X4,
+    #to_plot=False,
+    #to_plot=True,
+)
+
+
+# do a numbering of the column vector of predicted observation = 1 long column vector
+# in case we want to throw away some observation data
+measIndex = np.arange(0, ensembleOfPredictedObservations.shape[0]) 
+weight_obs = sparse.diags(1 / (add_unc * np.ones(measIndex.shape[0])) ** 2)
+
+
+# Compute loss function
+meas_diff = ensembleOfPredictedObservations[measIndex, K] - measurement[measIndex]
+Phi_kkx = meas_diff.T @ weight_obs @ meas_diff
+
+
+# Initialization
+theta_1 = kkx_u.copy()
+Phi_theta_1 = Phi_kkx.copy()
+
+Loss = -1 * np.ones(kk_tolerance + 1)
+Loss[0] = Phi_theta_1.copy()
+
+input("Now entering main loop: \n Press enter to continue!")
+
+# counter of steps in MCMC
+kk = 0
+
+while Loss[kk] > error_val and kk < kk_tolerance:
+    
+    
+    #####################################################################
+    # Step S2:  compute "phi=gu" as a perturbation of "theta=fu_kp1"    #
+    #####################################################################
+    
+    phi_1 = np.sqrt((1 - beta**2)) * theta_1 + beta * sigma * np.random.normal(size = Mr)
+    
+    phi_1[phi_1 < 0.25] = 0.25
+    phi_1[phi_1 > 1.75] = 1.75
+
+    # prepare for solving the conservation law to obtain h(phi)
+    initial_ensemble_kx[np.arange(0, Mr), K] = phi_1
+
+    # Run simulation to generate G(gu)
+    # u_X is the generated time series to be compared with data
+    
+    
+    ###############################
+    #  Step S3: Compute h(phi)    #
+    ###############################
+    ensembleOfPredictedObservations = simulate_g(
+        N, x, xh, Ldom, T, 
+        initial_true,
+        Ant_Init,
+        Nt,
+        time_obs,
+        It,
+        JX,
+        K,
+        weights,
+        initial_ensemble_kx,
+        X_1, X_2, X_3, X_4,
+        x_k, theta_1,
+        u_true_X1, u_true_X2, u_true_X3, u_true_X4
+    )
+
+    # Compute Loss
+    meas_diff = ensembleOfPredictedObservations[measIndex, K] - measurement[measIndex]
+    Phi_phi_1 = meas_diff.T @ weight_obs @ meas_diff
+ 
+    ########################################
+    # Step S3:  Compute a(theta_1, phi_1)  #
+    ########################################
+  
+    # MCMC a(theta,phi)
+    a_theta_1_phi_1 = np.min([1, np.exp(Phi_theta_1 - Phi_phi_1)] )
+
+    ##########################################################
+    # Step S4 and S5:  compare "choice" and "a(theta, phi)"  #
+    #########################################################
+    
+    choice = np.random.uniform(0, 1)
+     
+    if choice <= a_theta_1_phi_1:
+        theta_1      = phi_1.copy()
+        Phi_theta_1  = Phi_phi_1.copy()
+        
+    else:
+        theta_1      = theta_1.copy()
+        Phi_theta_1  = Phi_theta_1.copy()   
+
+    #####################################################
+    #  check if we should extract this parameter vector #
+    #####################################################
+    
+    check = kk / tau
+
+    if check == teller:
+        updated_ensemble_kx[0 : Mr, teller] = theta_1
+        teller += 1
+        
+        fig, axs = plt.subplots(1, 2, figsize=(15, 5))
+        
+        #axs[0].plt.figure()
+        axs[0].to_plot = Loss[Loss >= 0]
+        axs[0].plot(Loss)
+        axs[0].set_title("Loss function")
+        axs[0].set_ylabel("Loss")
+        axs[0].set_xlabel("Iteration number")
+        axs[0].grid(True)
+        #axs[0].show()
+        
+        y_k_ident = np.concatenate([[1], theta_1, [1]])
+        
+        #axs[1].plt.figure()
+        #axs[1].plot(x_k, y_k_true, '-r')
+        axs[1].plot(x_k, y_k_ident,'-b')
+        axs[1].set_title("k(x) function")
+        axs[1].set_ylabel("k(x)")
+        axs[1].set_xlabel("x")
+        #axs[1].legend(['True', 'Identified'])
+        axs[1].legend(['Identified'])
+        plt.axis([0, Ldom, 0.25, 1.75])
+        axs[1].grid(True)
+        #axs[1].show()
+
+        # Update the plot
+        #plt.draw()
+        
+        # Show the plots
+        plt.show()        
+        #plt.pause(2)          
+        #input("Extracting this parameter vector: \n Press enter to continue!")
+
+    #############################################
+    # Step S6:  move to next iteration in MCMC  #
+    #############################################
+
+    kk += 1
+    Loss[kk] = Phi_theta_1.copy()
+
+    y_k_ident = np.concatenate([[1], theta_1, [1]])
+    
+    print("Iteration:", kk, "   Loss:", Loss[kk])
+
+
+######################################################
+#  Save to file variables from this MCMC compuation  #
+######################################################
+
+np.savez(
+   "data/mcmc_u0X1_sample100_it1_student_Mars21.npz",
+   #"data/mcmc_u0X1_sample100_it2_student_Mars21.npz",
+   #"data/mcmc_u0X1_sample100_it3_student_Mars21.npz",
+   # "data/mcmc_u0X1_sample100_it4_student_Mars21.npz",
+   # "data/mcmc_u0X1_sample100_it5_student_Mars21.npz",
+   #"data/mcmc_u0X1_sample100_it6_student_Mars21.npz",
+    Loss=Loss,
+    #fu_kp1=fu_kp1,
+    #Phi_fu_kp1_y=Phi_fu_kp1_y,
+    theta_1=theta_1,
+    Phi_theta_1=Phi_theta_1,
+    updatedEnsemble_kx = updated_ensemble_kx,
+    kk=kk,
+    teller=teller,
+    x_k=x_k,
+    #y_k_true=y_k_true,
+    y_k_ident=y_k_ident,
+    beta=beta,
+)
+print("Sampling of 100 members done")
+print("Final loss was:", Loss)
+print("And the lower error val was set to:", error_val)
+
+
+# Plot the loss function
+plt.figure()
+to_plot = Loss[Loss >= 0]
+plt.plot(Loss)
+plt.title("Loss function")
+plt.ylabel("Loss")
+plt.xlabel("Iteration number")
+#plt.axis([0, Ldom, 0.5, 2.0])
+plt.grid(True)
+plt.show()
+
+plt.figure()
+#plt.plot(x_k, y_k_true, '-r')
+plt.plot(x_k, y_k_ident, '-b')
+plt.title("k(x) function")
+plt.ylabel("k(x)")
+plt.xlabel("x")
+plt.axis([0, Ldom, 0.25, 1.75])
+#plt.legend(['True', 'Identified'])
+plt.legend(['Identified'])
+plt.grid(True)
+plt.show()
+
diff --git a/Project3/src/GivenCode/simulations_Feb20.py b/Project3/src/GivenCode/simulations_Feb20.py
new file mode 100644
index 0000000000000000000000000000000000000000..571b913b11db935248d42ab17dd2d28e2dd0b2cd
--- /dev/null
+++ b/Project3/src/GivenCode/simulations_Feb20.py
@@ -0,0 +1,75 @@
+#import matplotlib.pyplot as plt
+import numpy as np
+#import scipy.sparse as sparse
+
+try:
+    from python_scripts.solver_Feb20 import my_solver
+except ImportError:
+    from solver_Feb20 import my_solver
+
+
+def simulate_g(
+    N, x, xh, Ldom, Tend, 
+    initial_true,
+    Ant_Init,
+    Nt,
+    time_obs,
+    It,
+    JX,
+    K,
+    weights,
+    initial_ensemble_kx,
+    X_1, X_2, X_3, X_4,
+    x_k, kx_old,
+    u_true_X1, u_true_X2, u_true_X3, u_true_X4,
+    #to_plot=False,
+    #to_plot = True,
+):
+    numEns = 1
+    
+    # vector that can store u(x1,t) for "Ant_Init" different initial data in one column vector 
+    ensembleOfPredictedObs_X1_0toT3 = np.zeros((Nt * Ant_Init, numEns))
+    ensembleOfPredictedObs_X2_0toT3 = np.zeros((Nt * Ant_Init, numEns))
+    ensembleOfPredictedObs_X3_0toT3 = np.zeros((Nt * Ant_Init, numEns))
+    ensembleOfPredictedObs_X4_0toT3 = np.zeros((Nt * Ant_Init, numEns))
+    
+    # list that can store u(x1,t),  u(x2,t),  u(x3,t),  u(x4,t) in column vector for each initial_data #K0
+    u_X = [np.zeros((len(time_obs), Ant_Init)) for _ in range(4)]
+    
+    #####################################
+    # Loop to simulate Ant_Init cases   #
+    #####################################
+    
+    for K_0 in range(0, Ant_Init):
+        
+        u_X[0][:, K_0], u_X[1][:, K_0], u_X[2][:, K_0], u_X[3][:, K_0], y_k = my_solver(
+            N, x, xh, Ldom, Tend, 
+            initial_ensemble_kx[:, K],  
+            initial_true[:, K_0],  
+            time_obs,
+            [X_1, X_2, X_3, X_4],
+            x_k,
+            #to_plot=to_plot,
+            #to_plot = False,
+            #to_plot = True,
+        )
+
+ # Store the obtained 4 time series "u_X" in the "ensembleOfPredictedObs_X1_0toT3"
+
+        ensembleOfPredictedObs_X1_0toT3[It + (K_0) * Nt, K] = u_X[0][JX, K_0]
+        ensembleOfPredictedObs_X2_0toT3[It + (K_0) * Nt, K] = u_X[1][JX, K_0]
+        ensembleOfPredictedObs_X3_0toT3[It + (K_0) * Nt, K] = u_X[2][JX, K_0]
+        ensembleOfPredictedObs_X4_0toT3[It + (K_0) * Nt, K] = u_X[3][JX, K_0]
+
+
+ # generate "ensembleOfPredictedObservations" which possess same structure as "measurement" 
+    ensembleOfPredictedObservations = np.concatenate(
+        [
+            weights[0] * ensembleOfPredictedObs_X1_0toT3,
+            weights[1] * ensembleOfPredictedObs_X2_0toT3,
+            weights[2] * ensembleOfPredictedObs_X3_0toT3,
+            weights[3] * ensembleOfPredictedObs_X4_0toT3,
+        ]
+    )
+    # return u_X, ensembleOfPredictedObservations
+    return ensembleOfPredictedObservations
diff --git a/Project3/src/GivenCode/solver_Feb20.py b/Project3/src/GivenCode/solver_Feb20.py
new file mode 100644
index 0000000000000000000000000000000000000000..5721f0dc1d794a7a94fcb5e2d3216b457f15a2b5
--- /dev/null
+++ b/Project3/src/GivenCode/solver_Feb20.py
@@ -0,0 +1,112 @@
+import matplotlib.pyplot as plt
+import numpy as np
+
+
+def my_solver(
+    #M,
+    N,
+    x,
+    xh,
+    Ldom,
+    Tend,
+    Fluxkx_ensem,
+    initial_true,
+    time_obs,
+    X,
+    x_k,
+    #to_plot=False,
+    #to_plot=True,
+):
+    """
+    Solution of u_t + (k(x)f(u))_x = 0
+    """
+
+    # Final time
+    T = Tend
+
+    # Delta x
+    dx = Ldom / N
+
+    # observation points in spatial domain
+    s = [ (np.sum(x <= X[i]) - 1) for i in range(4)]
+
+    # output predicted observations
+    u_X_out = [np.zeros(len(time_obs)) for _ in range(4)]
+
+    # terrain function
+    k_val = Fluxkx_ensem[:]
+    #y_k = np.concatenate((np.array([1]), k_val, np.array([1])))
+    y_k = np.concatenate(([1.0], k_val, [1.0] ))
+
+    # Define the flux function f(u)
+    def fun_flux(u):
+        f = np.zeros_like(u)
+        f =  u * (1 - u)
+        return f
+    
+    
+    # estimate  the time step dt based om CFL condition
+    dt_estimat = dx * (3/7)    # based on max |f'| = 1, max k(x) < 4/3
+    NTime = round(T / dt_estimat)
+
+    dt = T / NTime
+
+    #MA = max(maxfu_prime, 0.1)
+    
+    MA  = 1         # Rusanov flux is used
+    lmd = dt / dx
+
+    t_obs = np.linspace(0, T, NTime + 1)
+
+    J1 = np.arange(1, N - 1)
+    J2 = np.arange(0, N - 1)
+
+    u0 = initial_true[0:N]
+
+    u = np.zeros(N)
+
+  #  teller = 1
+
+    u_old = u0.copy()
+
+    # Compute predicted time serie state at t=0 
+    u_X = [np.zeros(NTime + 1) for i in range(4)]
+    for ind in range(4):
+        u_X[ind][0] = u0[s[ind]]
+
+   
+
+    for j in range(NTime):
+        
+        # fluxes at grid interface
+        F_half = np.zeros(N - 1)
+        Flux   = np.zeros(N)   # compute flux for all cells
+        
+        Flux = fun_flux(u_old)
+        F_half[J2] = 0.5 * (Flux[J2] + Flux[J2 + 1]) - 0.5 * MA * (u_old[J2 + 1] - u_old[J2] )
+        
+        k_half = np.zeros(N - 1)
+        k_half[J2] = np.interp(xh[J2 + 1], x_k, y_k)
+
+        # The numerical scheme
+
+        u[J1] = u_old[J1] - lmd * (k_half[J1] * F_half[J1] - k_half[J1 - 1] * F_half[J1 - 1])
+
+        u[u > 1] = 1.0
+        u[u < 0] = 0.0
+
+        u[0]  = u[1]
+        u[-1] = u[-2]
+
+       # Extract predicted observation data  
+        for ind in range(4):
+               u_X[ind][j+1] = u[s[ind]]  
+
+       
+        u_old = u.copy()
+
+   # transform predicted observation over to same time grid as used for true observation
+    for ind in range(4):
+      u_X_out[ind][:] = np.interp(time_obs[:], t_obs[:], u_X[ind][:])
+
+    return *u_X_out, y_k
\ No newline at end of file
diff --git a/Project3/src/HyperParameterOptimization.py b/Project3/src/HyperParameterOptimization.py
new file mode 100644
index 0000000000000000000000000000000000000000..235b2bfa36b0768e9d2c89f34712e953108a269f
--- /dev/null
+++ b/Project3/src/HyperParameterOptimization.py
@@ -0,0 +1,300 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+import skopt
+from skopt import gp_minimize # Import the skopt library for hyperparameter optimization
+from skopt.plots import plot_convergence, plot_objective
+from skopt.space import Real
+from skopt.utils import use_named_args
+
+from MCMC import *
+
+####################################################################################################
+# Setup
+####################################################################################################
+n_calls = 25 
+
+# MCMC and solver parameters
+MCMC_STEPS = 150 #!500
+MCMC_SUBSTEPS = 20
+NUMBER_SAMPLES = 1
+if NUMBER_SAMPLES == 1:
+    FigureFolder = 'Figures/Optimization/SingleSample'
+else:
+    FigureFolder = 'Figures/Optimization/DoubleSample'
+
+x0, x1 = 0, 4
+
+NTime = 500
+T_end = 10
+
+# Upwind viscosity parameter (used in flux computation)
+M = 1  
+
+k_i = 9
+k_mu, k_sigma_default = 1., 0.25
+
+beta_default = 0.05
+
+
+# Define hyperparameters for optimization
+dim_beta = Real(low=0.01, high=0.1, name='dim_beta')
+dim_k_sigma = Real(low=0.1, high=0.4, name='dim_k_sigma')
+
+dimensions = [
+    dim_beta,
+    dim_k_sigma,
+]
+
+# start parameter for optimization
+default_parameters = [
+    beta_default, 
+    k_sigma_default,
+]
+
+
+####################################################################################################
+# Hyperparameter Optimization
+####################################################################################################
+@use_named_args(dimensions=dimensions)
+def fitness(dim_beta, dim_k_sigma):
+    global ITERATION
+
+    print(f'ITERATION: {ITERATION}')
+    # Print the hyper-parameter configuration
+    print('beta:', dim_beta)
+    print('k_sigma:', dim_k_sigma)
+    print()
+
+    # MCMC
+    Theta, x_Theta, LOSS_THETA, LOSS_PHI, weights_historical = MCMC(
+        MCMC_STEPS=MCMC_STEPS,
+        MCMC_SUBSTEPS=MCMC_SUBSTEPS,
+        tau=tau,
+        beta=dim_beta,
+        gamma=gamma,
+        k_i=k_i,
+        k_mu=k_mu,
+        k_sigma=dim_k_sigma,
+        u_true=u_true,
+        number_samples=NUMBER_SAMPLES
+    )
+
+    # Get the error
+    error = LOSS_THETA[-1]
+
+    print(f'Error: {error}')
+
+    if np.isnan(error):
+        error = 1e10
+
+    # plotting
+    # Combine both loss functions
+    plt.figure()
+    plt.title('Losses')
+    plt.plot(LOSS_THETA[:], '--o', color='red', label='$\Phi(\Theta)$')
+    plt.plot(LOSS_PHI[:], '--o', color='green', label='$\Phi(\phi$)')
+    plt.xlabel('Iteration')
+    plt.ylabel('Loss')
+    plt.legend()
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/losses_ITERATION_{ITERATION}.pdf', bbox_inches='tight')
+    # plt.show()
+    plt.close()
+
+    # Only the phi(theta) loss is plotted
+    plt.figure()
+    plt.title('Evolution of entropy loss')
+    plt.loglog(LOSS_THETA[:], '--o', color='red', label='$\Phi(\Theta)$')
+    plt.xlabel('log(Iteration)')
+    plt.ylabel('log($\Phi(\Theta)$)')
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/loss_theta_ITERATION_{ITERATION}.pdf', bbox_inches='tight')
+    # plt.show()
+    plt.close()
+
+    # Solve with the final Theta
+    t_vec, h_final = solve_PDE(Theta, T_end, NTime, x, dx, x_Theta, NUMBER_SAMPLES)
+    h_final = [extract_positions(t_vec[i], h_final[i], x) for i in range(NUMBER_SAMPLES)]
+    h_final = np.array([h_final[i].T for i in range(NUMBER_SAMPLES)])
+
+    colors = ['blue', 'red', 'green', 'purple']
+    for sample in range(NUMBER_SAMPLES):
+        plt.figure()
+        plt.title(f'Predicted vs True - sample {sample+1}')
+        for i in range(4):
+            plt.plot([], [], color=colors[i], label=f'x = {x_observed[i]}')
+            plt.plot(times_observed, h_final[sample, i, :], color=colors[i])
+            plt.plot(times_observed, u_true[i, :, sample], '--o', color=colors[i])
+        plt.plot([], [], color='black', label='Predicted')
+        plt.plot([], [], '--o', color='black', label='True')
+        plt.xlabel('Time')
+        plt.ylabel('u(x,t)')
+        plt.legend(ncol=4, loc='upper center', bbox_to_anchor=(0.5, -0.1))
+        plt.grid()
+        if NUMBER_SAMPLES == 1:
+            plt.savefig(f'{FigureFolder}/predicted_vs_true_single_sample_ITERATION_{ITERATION}.pdf', bbox_inches='tight')
+        else:
+            plt.savefig(f'{FigureFolder}/predicted_vs_true_both_samples_sample_{sample+1}_ITERATION_{ITERATION}.pdf', bbox_inches='tight')
+        # plt.show()
+        plt.close()
+
+    # Plot the final theta -> prediction for k(x)
+    plt.figure()
+    plt.title('Prediction for k(x)')
+    plt.plot(x_Theta, Theta, '--o', color='blue')
+    plt.xlabel('$x$')
+    plt.ylabel('$k(x)$')
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/prediction_kx_ITERATION_{ITERATION}.pdf', bbox_inches='tight')
+    # plt.show()
+    plt.close()
+
+    # next iteration
+    ITERATION += 1
+
+    return error
+
+
+
+
+####################################################################################################
+# Optimization
+####################################################################################################
+ITERATION = 0
+
+# Run the hyperparameter optimization
+search_result = gp_minimize(
+    func=fitness,
+    dimensions=dimensions,
+    base_estimator='GP',#DUMMY', # ! 'GP', 'RF', 'ET', 'GBRT', 'DUMMY'
+    acq_func='EI', # Expected Improvement.
+    n_calls=n_calls,
+    x0=default_parameters,
+    random_state=42,
+)
+
+# print result
+print(search_result.x)
+
+# Plot convergence
+plt.figure()
+_ = plot_convergence(search_result)
+plt.savefig('Figures/Optimization/TunerConvergence.pdf')
+plt.savefig('Figures/Optimization/TunerConvergence.png')
+# plt.show()
+plt.close()
+
+# Plot objective
+plt.figure()
+_ = plot_objective(search_result, show_points=True, size=3.8)
+plt.savefig('Figures/Optimization/TunerObjective.pdf')
+plt.savefig('Figures/Optimization/TunerObjective.png')
+# plt.show()
+plt.close()
+
+
+# Train best one
+# MCMC
+Theta, x_Theta, LOSS_THETA, LOSS_PHI, weights_historical = MCMC(
+    MCMC_STEPS=MCMC_STEPS,
+    MCMC_SUBSTEPS=MCMC_SUBSTEPS,
+    tau=tau,
+    beta=search_result.x[0],
+    gamma=gamma,
+    k_i=k_i,
+    k_mu=k_mu,
+    k_sigma=search_result.x[1],
+    u_true=u_true,
+    number_samples=NUMBER_SAMPLES
+)
+
+# Save results + statistics of Theta
+if NUMBER_SAMPLES == 1:
+    DataResultsFile = 'data/TunerResultsSingleSample.txt'
+else:
+    DataResultsFile = 'data/TunerResultsDoubleSample.txt'
+with open(f'{DataResultsFile}.txt', 'w') as f:
+    f.write('... Tuner Results and Timings ...\n')
+    f.write(f'... Tuner results ...\n')
+    f.write(f'n_calls: {n_calls}\n')
+    f.write(f'Best error: {search_result.fun}\n')
+    f.write(f'Best parameters: {search_result.x}\n')
+    f.write(f'Best beta: {search_result.x[0]}\n')
+    f.write(f'Best k_sigma: {search_result.x[1]}\n')
+    f.write(f'... Statistics of Theta ...\n')
+    f.write(f'Mean of Theta: {Theta.mean()}\n')
+    f.write(f'Standard deviation of Theta: {Theta.std()}\n')
+    
+# Save to another file
+np.savez(
+    f'{DataResultsFile}.npz', 
+    n_calls=n_calls, best_error=search_result.fun, 
+    best_parameters=search_result.x, 
+    best_beta=search_result.x[0], 
+    beta_k_sigma=search_result.x[1], 
+    Theta_mean=Theta.mean(),
+    Theta_std=Theta.std()
+)
+
+# plotting
+# Combine both loss functions
+plt.figure()
+plt.title('Losses')
+plt.plot(LOSS_THETA[:], '--o', color='red', label='$\Phi(\Theta)$')
+plt.plot(LOSS_PHI[:], '--o', color='green', label='$\Phi(\phi$)')
+plt.xlabel('Iteration')
+plt.ylabel('Loss')
+plt.legend()
+plt.grid()
+plt.savefig(f'{FigureFolder}/best_losses.pdf', bbox_inches='tight')
+# plt.show()
+plt.close()
+
+# Only the phi(theta) loss is plotted
+plt.figure()
+plt.title('Evolution of entropy loss')
+plt.loglog(LOSS_THETA[:], '--o', color='red', label='$\Phi(\Theta)$')
+plt.xlabel('log(Iteration)')
+plt.ylabel('log($\Phi(\Theta)$)')
+plt.grid()
+plt.savefig(f'{FigureFolder}/best_loss_theta.pdf', bbox_inches='tight')
+# plt.show()
+plt.close()
+
+# Solve with the final Theta
+t_vec, h_final = solve_PDE(Theta, T_end, NTime, x, dx, x_Theta, NUMBER_SAMPLES)
+h_final = [extract_positions(t_vec[i], h_final[i], x) for i in range(NUMBER_SAMPLES)]
+h_final = np.array([h_final[i].T for i in range(NUMBER_SAMPLES)])
+
+colors = ['blue', 'red', 'green', 'purple']
+for sample in range(NUMBER_SAMPLES):
+    plt.figure()
+    plt.title(f'Predicted vs True - sample {sample+1}')
+    for i in range(4):
+        plt.plot([], [], color=colors[i], label=f'x = {x_observed[i]}')
+        plt.plot(h_final[sample, i, :], color=colors[i])
+        plt.plot(u_true[i, :, sample], '--o', color=colors[i])
+    plt.plot([], [], color='black', label='Predicted')
+    plt.plot([], [], '--o', color='black', label='True')
+    plt.xlabel('Time')
+    plt.ylabel('u(x,t)')
+    plt.legend(ncol=4, loc='upper center', bbox_to_anchor=(0.5, -0.1))
+    plt.grid()
+    if NUMBER_SAMPLES == 1:
+        plt.savefig(f'{FigureFolder}/best_predicted_vs_true_single_sample.pdf', bbox_inches='tight')
+    else:
+        plt.savefig(f'{FigureFolder}/best_predicted_vs_true_both_samples_sample_{sample+1}.pdf', bbox_inches='tight')
+    # plt.show()
+    plt.close()
+
+# Plot the final theta -> prediction for k(x)
+plt.figure()
+plt.title('Prediction for k(x)')
+plt.plot(x_Theta, Theta, '--o', color='blue')
+plt.xlabel('$x$')
+plt.ylabel('$k(x)$')
+plt.grid()
+plt.savefig(f'{FigureFolder}/best_prediction_kx_ITERATION.pdf', bbox_inches='tight')
+# plt.show()
+plt.close()
\ No newline at end of file
diff --git a/Project3/src/InfluenceArtificialDiffusion.py b/Project3/src/InfluenceArtificialDiffusion.py
new file mode 100644
index 0000000000000000000000000000000000000000..6977aedbd32addf3b170ea68836859386ab55f6f
--- /dev/null
+++ b/Project3/src/InfluenceArtificialDiffusion.py
@@ -0,0 +1,147 @@
+import numpy as np
+import matplotlib.pyplot as plt
+from tqdm import tqdm
+
+from MCMC import *
+
+ArtificialDiffusion = [1,2,3,4]
+
+# MCMC and solver parameters
+MCMC_STEPS = 500
+MCMC_SUBSTEPS = 20
+NUMBER_SAMPLES = 1
+if NUMBER_SAMPLES == 1:
+    FigureFolder = 'Figures/ArtificialDiffusion'
+else:
+    raise ValueError('NUMBER_SAMPLES must be 1 for this example')
+
+# solution vector
+Theta_sol = []
+x_Theta_sol = []
+Loss_Theta_sol = []
+Loss_Phi_sol = []
+
+
+for art_diff in tqdm(ArtificialDiffusion):
+    print(f'Solving for M = {art_diff}')
+    Theta, x_Theta, LOSS_THETA, LOSS_PHI, weights_historical = MCMC(
+        MCMC_STEPS=MCMC_STEPS,
+        MCMC_SUBSTEPS=MCMC_SUBSTEPS,
+        tau=tau,
+        beta=beta,
+        gamma=gamma,
+        k_i=k_i,
+        k_mu=k_mu,
+        k_sigma=k_sigma,
+        u_true=u_true,
+        number_samples=NUMBER_SAMPLES,
+        M=art_diff,
+    )
+
+    # Save the solution
+    Theta_sol.append(Theta)
+    x_Theta_sol.append(x_Theta)
+    Loss_Theta_sol.append(LOSS_THETA)
+    Loss_Phi_sol.append(LOSS_PHI)
+
+for i, art_diff in enumerate(ArtificialDiffusion):
+    Theta = Theta_sol[i]
+    x_Theta = x_Theta_sol[i]
+    LOSS_THETA = Loss_Theta_sol[i]
+    LOSS_PHI = Loss_Phi_sol[i]
+
+    # Plot the loss function
+    plt.figure()
+    plt.title(f'Losses - M = {art_diff}')
+    plt.plot(LOSS_THETA[:], '--o', color='red', label='$\Phi(\Theta)$')
+    plt.plot(LOSS_PHI[:], '--o', color='green', label='$\Phi(\phi$)')
+    plt.xlabel('Iteration')
+    plt.ylabel('Loss')
+    plt.legend()
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/losses_ArtDiff{art_diff}.pdf', bbox_inches='tight')
+    plt.show()
+
+    # Only the phi(theta) loss is plotted
+    plt.figure()
+    plt.title(f'Evolution of entropy loss - M = {art_diff}')
+    plt.loglog(LOSS_THETA[:], '--o', color='red', label='$\Phi(\Theta)$')
+    plt.xlabel('log(Iteration)')
+    plt.ylabel('log($\Phi(\Theta)$)')
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/loss_theta_ArtDiff{art_diff}.pdf', bbox_inches='tight')
+    plt.show()
+
+    # Solve with the final Theta
+    t_vec, h_final = solve_PDE(Theta, T_end, NTime, x, dx, x_Theta, NUMBER_SAMPLES, M=M)
+    h_final = [extract_positions(t_vec[i], h_final[i], x) for i in range(NUMBER_SAMPLES)]
+    h_final = np.array([h_final[i].T for i in range(NUMBER_SAMPLES)])
+
+    colors = ['blue', 'red', 'green', 'purple']
+    for sample in range(NUMBER_SAMPLES):
+        plt.figure()
+        plt.title(f'Predicted vs True - M = {art_diff}')
+        for i in range(4):
+            plt.plot([], [], color=colors[i], label=f'x = {x_observed[i]}')
+            plt.plot(times_observed, h_final[sample, i, :], color=colors[i])
+            plt.plot(times_observed, u_true[i, :, sample], '--o', color=colors[i])
+        plt.plot([], [], color='black', label='Predicted')
+        plt.plot([], [], '--o', color='black', label='True')
+        plt.xlabel('Time')
+        plt.ylabel('u(x,t)')
+        plt.legend(ncol=4, loc='upper center', bbox_to_anchor=(0.5, -0.1))
+        plt.grid()
+        plt.savefig(f'{FigureFolder}/predicted_vs_true_single_sample_ArtDiff{art_diff}.pdf', bbox_inches='tight')
+        plt.show()
+
+    # Plot the final theta -> prediction for k(x)
+    plt.figure()
+    plt.title(f'Prediction for k(x) - M = {art_diff}')
+    plt.plot(x_Theta, Theta, '--o', color='blue')
+    plt.xlabel('$x$')
+    plt.ylabel('$k(x)$')
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/prediction_kx_ArtDiff{art_diff}.pdf', bbox_inches='tight')
+    plt.show()
+
+# Compare the loss function for different M
+plt.figure()
+plt.title('Comparison of the errors for different M')
+[plt.plot(Loss_Theta_sol[i][:], '--o', color=colors[i], label=f'M = {ArtificialDiffusion[i]}') for i in range(len(ArtificialDiffusion))]
+plt.xlabel('Iteration')
+plt.ylabel('Loss')
+plt.legend()
+plt.grid()
+plt.savefig(f'{FigureFolder}/losses_ArtDiffComparison.pdf', bbox_inches='tight')
+plt.show()
+
+# Compare the resistance fucntion k(x) for different M
+plt.figure()
+plt.title('Comparison of the resistance function k(x) for different M')
+[plt.plot(x_Theta_sol[i], Theta_sol[i], '--o', color=colors[i], label=f'M = {ArtificialDiffusion[i]}') for i in range(len(ArtificialDiffusion))]
+plt.xlabel('$x$')
+plt.ylabel('$k(x)$')
+plt.legend()
+plt.grid()
+plt.savefig(f'{FigureFolder}/prediction_kx_ArtDiffComparison.pdf', bbox_inches='tight')
+plt.show()
+
+# Compare final error over M
+plt.figure()
+plt.title('Comparison of the final error for different M')
+plt.plot(ArtificialDiffusion, np.array(Loss_Theta_sol)[:,-1], '--o', color='black')
+plt.xlabel('M')
+plt.ylabel('$\Phi(\Theta)$')
+plt.legend()
+plt.grid()
+plt.savefig(f'{FigureFolder}/final_error_ArtDiffComparison.pdf', bbox_inches='tight')
+plt.show()
+
+# Save the results
+np.savez(
+    f'{FigureFolder}/results_ArtDiffComparison.npz', Theta_sol=Theta_sol,
+    x_Theta_sol=x_Theta_sol,
+    Loss_Theta_sol=Loss_Theta_sol,
+    Loss_Phi_sol=Loss_Phi_sol,
+    ArtificialDiffusion=ArtificialDiffusion
+)
\ No newline at end of file
diff --git a/Project3/src/InfluenceResistance.py b/Project3/src/InfluenceResistance.py
new file mode 100644
index 0000000000000000000000000000000000000000..174635173f0e8c07f17b6c6a354840ce8128623c
--- /dev/null
+++ b/Project3/src/InfluenceResistance.py
@@ -0,0 +1,62 @@
+import numpy as np
+import scipy.io
+import matplotlib.pyplot as plt
+
+from MCMC import Upwind, solve_PDE, extract_positions
+
+# Parameters
+x0, x1 = 0, 4
+
+NTime = 500
+T_end = 10
+
+# Upwind viscosity parameter (used in flux computation)
+M = 1  
+
+k_i = 9
+k_mu, k_sigma = 1., 0.25
+
+# Load solution
+datafile = 'data/TrueData_FluxF.mat'
+mat = scipy.io.loadmat(datafile)
+
+Nx = int(mat['N'].item())+2
+T_end = float(mat['Tend'].item())
+times_observed = mat['time_obs'][0]
+n_extraction_points = 4
+x_observed = [float(mat[f'X_{i}'].item()) for i in range(1, n_extraction_points+1)]
+u_true = np.array([mat[f'u_true_X{i}'] for i in range(1, n_extraction_points+1)])
+
+x = np.linspace(x0, x1, Nx)
+dx = x[1] - x[0]
+
+
+# Create 5 different resistance functions to inspect the influence of the resistance function
+n_resistance_functions = 15
+for iter_function in range(n_resistance_functions):
+    Theta = np.random.normal(k_mu, k_sigma, k_i+2)
+    Theta[0], Theta[-1] = k_mu, k_mu
+    x_Theta = np.linspace(x0, x1, k_i+2)
+    t_vec, u_vec = solve_PDE(Theta, T_end, NTime, x, dx, x_Theta)
+    u_vec = np.array([extract_positions(t_vec[i], u_vec[i], x) for i in range(1)]).T.reshape(n_extraction_points,-1)
+
+    # Plot
+    colors = ['blue', 'red', 'green', 'purple']
+    fig, axs = plt.subplots(ncols=2, figsize=(10, 5))
+    axs[0].plot(x_Theta, Theta, 'o-', color='black')
+    axs[0].set_xlabel('x')
+    axs[0].set_ylabel('$k(x)$')
+    for i in range(n_extraction_points):
+        axs[1].plot([], [], color=colors[i], label=f'x = {x_observed[i]}')
+        axs[1].plot(times_observed, u_vec[i,:], color=colors[i])
+        axs[1].plot(times_observed, u_true[i, :, 0], '--o', color=colors[i])
+    plt.plot([], [], color='black', label='Predicted')
+    plt.plot([], [], '--o', color='black', label='True')
+    axs[1].set_xlabel('Time')
+    axs[1].set_ylabel('$u(x,t)$')
+    axs[1].legend(ncol=4, loc='upper center', bbox_to_anchor=(0.48, -0.1))
+    [ax.grid() for ax in axs]
+
+    fig.tight_layout()
+    fig.savefig(f'figures/InfluenceResistance/sample_{iter_function}.png')
+    fig.show()
diff --git a/Project3/src/MCMC.py b/Project3/src/MCMC.py
new file mode 100644
index 0000000000000000000000000000000000000000..6994be3cc7f6df12b82ab00313ce0670d7cfe605
--- /dev/null
+++ b/Project3/src/MCMC.py
@@ -0,0 +1,403 @@
+import numpy as np
+import scipy.io
+import matplotlib.pyplot as plt
+from tqdm import tqdm
+
+# MCMC and solver parameters
+MCMC_STEPS = 500
+MCMC_SUBSTEPS = 20
+NUMBER_SAMPLES = 1
+
+x0, x1 = 0, 4
+
+NTime = 500
+T_end = 10
+
+# Upwind viscosity parameter (used in flux computation)
+M = 1  
+
+k_i = 9
+k_mu, k_sigma = 1., 0.25
+
+# Load solution
+datafile = 'data/TrueData_FluxF.mat'
+mat = scipy.io.loadmat(datafile)
+
+Nx = int(mat['N'].item())+2
+T_end = float(mat['Tend'].item())
+NT_obs = int(mat['NT_obs'].item())
+times_observed = mat['time_obs'][0]
+n_extraction_points = 4
+x_observed = [float(mat[f'X_{i}'].item()) for i in range(1, n_extraction_points+1)]
+u_true = np.array([mat[f'u_true_X{i}'] for i in range(1, n_extraction_points+1)])#[:,:,0]
+u_true_flatten = u_true.flatten()
+
+x = np.linspace(x0, x1, Nx)
+dx = x[1] - x[0]
+
+# MCMC tuning parameters
+beta = 0.036442897496557186 # ! best value from optimization 0.05  # initial proposal scale
+gamma = 0.03
+tau = 100  # burn in iterations
+
+# Diagnostics storage for adaptive beta
+beta_history = []
+acceptance_history = []
+
+# Define f(u) and its derivative
+f = lambda u: u * (1 - u)
+df = lambda u: 1 - 2*u
+
+def Upwind(u, M=M):
+    """
+    Compute the numerical flux F_{j+1/2} using:
+        F_{j+1/2} = 0.5*(f(u_j)+f(u_{j+1})) - (M/2)*(u_{j+1}-u_j)
+    """
+    F = np.zeros_like(u)
+    F[:-1] = 0.5 * (f(u[:-1]) + f(u[1:])) - (M/2)*(u[1:] - u[:-1])
+    F[-1] = f(u[-1])
+    return F
+
+def solve_PDE(Theta, T_end, NTime, x, dx, x_Theta, number_samples=1, CFL=0.5, time_scheme='Euler', M=1):
+    """
+    Solve the PDE 
+        u_t + (k(x) f(u))_x = 0,
+    using a conservative update:
+        (u_i^{n+1} - u_i^n)/dt + (1/dx)*(k_{i+1/2} F_{i+1/2} - k_{i-1/2} F_{i-1/2}) = 0.
+    
+    k(x) is defined via linear interpolation of Theta at x_Theta.
+    An adaptive dt is computed using the CFL condition based on the maximum
+    characteristic speed |k(x)*f'(u)|.
+    
+    The option time_scheme can be:
+        - 'Euler': standard forward Euler (first-order)
+        - 'Heun': Heun's method (second-order predictor-corrector)
+    """
+    # Define k(x) via interpolation
+    k = lambda x_val: np.interp(x_val, x_Theta, Theta)
+    
+    # Initial condition
+    u_init_1 = np.where(
+            x <= 0.5, 
+            0.2,
+            np.where(
+                x <= 1.5, 
+                0.4,
+                np.where(
+                    x <= 2.5, 
+                    0.6,
+                    np.where(
+                        x <= 3.5, 
+                        0.7, 
+                        0.4
+                    )
+                )
+            )
+        )
+    
+    u_init_2 = np.where(
+        x <= 0.5,
+        0.1,
+        np.where(
+            x <= 1.5,
+            0.3,
+            np.where(
+                x <= 2.5,
+                0.7,
+                0.2
+            )
+        )
+    )
+    
+    def solver(u):
+        t = 0.0
+        u_vec = [u.copy()]
+        t_vec = [t]
+        while t < T_end:
+            # Compute dt using the CFL condition: dt <= CFL*dx / max|k(x) * f'(u)|
+            speeds = np.abs(k(x) * df(u))
+            max_speed = np.max(speeds)
+            dt_CFL = CFL * dx / (max_speed + 1e-6)
+            dt = min(dt_CFL, T_end - t)
+            
+            # Compute fluxes at interfaces
+            F = Upwind(u, M)
+            # Spatial locations of cell interfaces
+            x_half = (x[:-1] + x[1:]) / 2.0
+            k_half = k(x_half)
+            
+            if time_scheme == 'Euler':
+                # Euler update
+                u_new = u.copy()
+                # Update interior cells (i = 1,..., len(u)-2)
+                u_new[1:-1] = u[1:-1] - dt/dx * ( k_half[1:] * F[1:-1] - k_half[:-1] * F[:-2] )
+            elif time_scheme == 'Heun':
+                # Predictor: Euler step
+                u_predictor = u.copy()
+                u_predictor[1:-1] = u[1:-1] - dt/dx * ( k_half[1:] * F[1:-1] - k_half[:-1] * F[:-2] )
+                # Compute fluxes for the predictor
+                F_predictor = Upwind(u_predictor, M)
+                # Average fluxes for a second order update
+                u_new = u.copy()
+                # Compute the slope at u and at u_predictor
+                slope1 = np.zeros_like(u)
+                slope2 = np.zeros_like(u)
+                slope1[1:-1] = -1/dx * ( k_half[1:] * F[1:-1] - k_half[:-1] * F[:-2] )
+                slope2[1:-1] = -1/dx * ( k_half[1:] * F_predictor[1:-1] - k_half[:-1] * F_predictor[:-2] )
+                u_new[1:-1] = u[1:-1] + dt/2.0*(slope1[1:-1] + slope2[1:-1])
+            else:
+                raise ValueError("Unknown time integration scheme.")
+            
+            # Boundary conditions: here, we keep the boundary values fixed (Dirichlet)
+            u_new[0] = u_new[1] # !u[0]
+            u_new[-1] = u_new[-2] # !u[-1]
+            
+            u = u_new.copy()
+            t += dt
+            u_vec.append(u.copy())
+            t_vec.append(t)
+        
+        return np.array(t_vec), np.array(u_vec)
+    
+    t_vec_1, u_vec_1 = solver(
+        u=u_init_1
+    )
+    if number_samples == 1:
+        return [t_vec_1], [u_vec_1]
+    
+    # number_samples = 2
+    t_vec_2, u_vec_2 = solver(
+        u=u_init_2
+    )
+    t_vec = [t_vec_1, t_vec_2]
+    u_vec = [u_vec_1, u_vec_2]
+
+    return t_vec, u_vec
+
+def extract_positions(t_vec, h_, x):
+    """
+    Extract the solution at the observation times and spatial positions.
+    """
+    h_extracted = []
+    for t_obs in times_observed:
+        idx_t = np.argmin(np.abs(t_vec - t_obs))
+        h_extracted.append(h_[idx_t, :])
+    # h_extracted = np.array(h_extracted)[:, [np.argmin(np.abs(x - x_obs)) for x_obs in x_observed]]
+    h_extracted = np.array(h_extracted)[:, [np.argmin(np.abs(x_obs - x)) for x_obs in x_observed]]
+    return h_extracted
+
+def MCMC(
+    MCMC_STEPS=MCMC_STEPS,
+    MCMC_SUBSTEPS=MCMC_SUBSTEPS,
+    tau=tau,
+    beta=beta,
+    gamma=gamma,
+    k_i=k_i,
+    k_mu=k_mu,
+    k_sigma=k_sigma,
+    u_true=u_true,
+    number_samples=1,
+    M=1
+):
+    if number_samples == 1:
+        u_true = u_true[:,:,0]
+        u_true_flatten = u_true.flatten()
+    else:
+        u_true_flatten = u_true.flatten()
+    # Initial guess for Theta (parameters for k(x))
+    Theta = np.random.normal(k_mu, k_sigma, k_i+2).clip(0.8, 1.2)#!.clip(0.95, 1.3)
+    Theta[0], Theta[-1] = k_mu, k_mu
+    x_Theta = np.linspace(x0, x1, k_i+2)
+
+    # Solve the PDE with the initial Theta
+    t_vec, h_Theta = solve_PDE(Theta, T_end, NTime, x, dx, x_Theta, number_samples, M=M)
+    h_Theta = np.array([extract_positions(t_vec[i], h_Theta[i], x) for i in range(number_samples)]).T
+    h_Theta = h_Theta.flatten()
+    dist_Theta = h_Theta - u_true_flatten
+
+    # Initialize weights
+    weights = np.ones_like(dist_Theta) # 4 is the number of observed points
+    weights /= np.sum(weights) * (number_samples*4)
+    nx_observed = u_true.shape[1]
+    weights_historical = [np.ones(4*number_samples)]
+
+    # Burn-in phase (without adaptive beta for simplicity)
+    for i in range(tau):
+        xi = np.random.normal(0, k_sigma, size=Theta.shape)
+        Phi = np.sqrt(1 - beta**2) * Theta + beta * xi
+        Phi[0], Phi[-1] = k_mu, k_mu
+        
+        t_vec, h_Phi = solve_PDE(Phi, T_end, NTime, x, dx, x_Theta, number_samples, M=M)
+        h_Phi = np.array([extract_positions(t_vec[i], h_Phi[i], x) for i in range(number_samples)]).T
+        h_Phi = h_Phi.flatten()
+        dist_Phi = h_Phi - u_true_flatten
+        
+        # loss_Theta = 0.5 * (weights * dist_Theta @ dist_Theta.T) / gamma**2
+        # loss_Phi   = 0.5 * (weights * dist_Phi @ dist_Phi.T) / gamma**2
+        loss_Theta = 0.5 * (dist_Theta @ dist_Theta.T) / gamma**2
+        loss_Phi   = 0.5 * (dist_Phi @ dist_Phi.T) / gamma**2
+        
+        a = min(1, np.exp(loss_Theta - loss_Phi))
+        if np.random.uniform(0, 1) <= a:
+            Theta = Phi
+            h_Theta = h_Phi
+            dist_Theta = dist_Phi
+    
+    # Update weights
+    # L2 = []
+    # for i in range(number_samples*4):
+    #     L2.append(np.linalg.norm(dist_Theta[i*nx_observed:(i+1)*nx_observed]))
+    # L2 = np.array(L2)
+    # weights = L2/np.sum(L2) *(number_samples*4)
+    # weights_historical.append(weights)
+    # weights = np.repeat(weights, nx_observed)
+
+    LOSS_THETA = []
+    LOSS_PHI = []
+
+    print('Starting MCMC with adaptive beta')
+    # For adaptive beta, track the number of accepted proposals in each block.
+    target_accept_low = 0.1#0.2 # ! 0.1
+    target_accept_high = 0.3#0.5 # ! 0.3
+
+    for mcmc_iter in tqdm(range(MCMC_STEPS)):
+        accepted = 0
+        for _ in range(MCMC_SUBSTEPS):
+            xi = np.random.normal(0, k_sigma, size=Theta.shape)
+            Phi = np.sqrt(1 - beta**2) * Theta + beta * xi
+            Phi[0], Phi[-1] = k_mu, k_mu
+            
+            t_vec, h_Phi = solve_PDE(Phi, T_end, NTime, x, dx, x_Theta, number_samples, M=M)
+            h_Phi = np.array([extract_positions(t_vec[i], h_Phi[i], x) for i in range(number_samples)]).T
+            h_Phi = h_Phi.flatten()
+            dist_Phi = h_Phi - u_true_flatten
+            
+            # loss_Theta = 0.5 * (weights * dist_Theta @ dist_Theta.T) / gamma**2
+            # loss_Phi   = 0.5 * (weights * dist_Phi @ dist_Phi.T) / gamma**2
+            loss_Theta = 0.5 * (dist_Theta @ dist_Theta.T) / gamma**2
+            loss_Phi   = 0.5 * (dist_Phi @ dist_Phi.T) / gamma**2
+            
+            a = min(1, np.exp(loss_Theta - loss_Phi))
+            if np.random.uniform(0, 1) <= a:
+                Theta = Phi
+                h_Theta = h_Phi
+                dist_Theta = dist_Phi
+                accepted += 1
+
+        # Save losses at the end of each outer step
+        LOSS_THETA.append(loss_Theta)
+        LOSS_PHI.append(loss_Phi)
+        
+        # Compute acceptance ratio for this block
+        acc_ratio = accepted / MCMC_SUBSTEPS
+        acceptance_history.append(acc_ratio)
+        beta_history.append(beta)
+        
+        # # Adapt beta based on acceptance ratio:
+        # if acc_ratio < target_accept_low:
+        #     beta *= 0.9
+        # elif acc_ratio > target_accept_high:
+        #     beta *= 1.1
+
+        # # Update weights
+        # L2 = []
+        # for i in range(number_samples*4):
+        #     L2.append(np.linalg.norm(dist_Theta[i*nx_observed:(i+1)*nx_observed]))
+        # L2 = np.array(L2)
+        # weights = L2/np.sum(L2)*(number_samples*4)
+        # weights_historical.append(weights)
+        # weights = np.repeat(weights, nx_observed)
+
+    return Theta, x_Theta, LOSS_THETA, LOSS_PHI, weights_historical
+
+if __name__ == '__main__':
+    Theta, x_Theta, LOSS_THETA, LOSS_PHI, weights_historical = MCMC(
+        MCMC_STEPS=MCMC_STEPS,
+        MCMC_SUBSTEPS=MCMC_SUBSTEPS,
+        tau=tau,
+        beta=beta,
+        gamma=gamma,
+        k_i=k_i,
+        k_mu=k_mu,
+        k_sigma=k_sigma,
+        u_true=u_true,
+        number_samples=NUMBER_SAMPLES
+    )
+    if NUMBER_SAMPLES == 1:
+        FigureFolder = 'Figures/SingleSample'
+    else:
+        FigureFolder = 'Figures/DoubleSample'
+    # Combine both loss functions
+    plt.figure()
+    plt.title('Losses')
+    plt.plot(LOSS_THETA[:], '--o', color='red', label='$\Phi(\Theta)$')
+    plt.plot(LOSS_PHI[:], '--o', color='green', label='$\Phi(\phi$)')
+    plt.xlabel('Iteration')
+    plt.ylabel('Loss')
+    plt.legend()
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/losses.pdf', bbox_inches='tight')
+    plt.show()
+
+    # Only the phi(theta) loss is plotted
+    plt.figure()
+    plt.title('Evolution of entropy loss')
+    plt.loglog(LOSS_THETA[:], '--o', color='red', label='$\Phi(\Theta)$')
+    plt.xlabel('log(Iteration)')
+    plt.ylabel('log($\Phi(\Theta)$)')
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/loss_theta.pdf', bbox_inches='tight')
+    plt.show()
+
+    # Solve with the final Theta
+    t_vec, h_final = solve_PDE(Theta, T_end, NTime, x, dx, x_Theta, NUMBER_SAMPLES, M=M)
+    h_final = [extract_positions(t_vec[i], h_final[i], x) for i in range(NUMBER_SAMPLES)]
+    h_final = np.array([h_final[i].T for i in range(NUMBER_SAMPLES)])
+
+    colors = ['blue', 'red', 'green', 'purple']
+    for sample in range(NUMBER_SAMPLES):
+        plt.figure()
+        plt.title(f'Predicted vs True - sample {sample+1}')
+        for i in range(4):
+            plt.plot([], [], color=colors[i], label=f'x = {x_observed[i]}')
+            plt.plot(times_observed, h_final[sample, i, :], color=colors[i])
+            plt.plot(times_observed, u_true[i, :, sample], '--o', color=colors[i])
+        plt.plot([], [], color='black', label='Predicted')
+        plt.plot([], [], '--o', color='black', label='True')
+        plt.xlabel('Time')
+        plt.ylabel('u(x,t)')
+        plt.legend(ncol=4, loc='upper center', bbox_to_anchor=(0.5, -0.1))
+        plt.grid()
+        if NUMBER_SAMPLES == 1:
+            plt.savefig(f'{FigureFolder}/predicted_vs_true_single_sample.pdf', bbox_inches='tight')
+        else:
+            plt.savefig(f'{FigureFolder}/predicted_vs_true_both_samples_sample_{sample+1}.pdf', bbox_inches='tight')
+        plt.show()
+
+    # Plot the final theta -> prediction for k(x)
+    plt.figure()
+    plt.title('Prediction for k(x)')
+    plt.plot(x_Theta, Theta, '--o', color='blue')
+    plt.xlabel('$x$')
+    plt.ylabel('$k(x)$')
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/prediction_kx.pdf', bbox_inches='tight')
+    plt.show()
+
+    # History of weights
+    plt.figure()
+    plt.title('History of weights')
+    marker = [':', '--']
+    weights_historical = np.array(weights_historical)
+    for sample in range(NUMBER_SAMPLES):
+        for i in range(4*NUMBER_SAMPLES):
+            plt.plot(weights_historical[:,i], marker[sample], color=colors[i%4])
+    plt.plot([], [], ':', color='black', label='Sample 1')
+    plt.plot([], [], '--', color='black', label='Sample 2')
+    [plt.plot([], [], color=colors[i], label=f'x = {x_observed[i%4]}') for i in range(4)]
+    plt.xlabel('Update Iteration')
+    plt.ylabel('Weight')
+    plt.legend()
+    plt.grid()
+    plt.savefig(f'{FigureFolder}/weights.pdf', bbox_inches='tight')
+    plt.show()
\ No newline at end of file
diff --git a/Project3/src/data/TrueData_FluxF.mat b/Project3/src/data/TrueData_FluxF.mat
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diff --git a/Project3/src/data/TunerResultsSingleSample.txt.npz b/Project3/src/data/TunerResultsSingleSample.txt.npz
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diff --git a/Project3/src/data/TunerResultsSingleSample.txt.txt b/Project3/src/data/TunerResultsSingleSample.txt.txt
new file mode 100644
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+++ b/Project3/src/data/TunerResultsSingleSample.txt.txt
@@ -0,0 +1,10 @@
+... Tuner Results and Timings ...
+... Tuner results ...
+n_calls: 25
+Best error: 7.961210545809955
+Best parameters: [0.036442897496557186, 0.24429936306439973]
+Best beta: 0.036442897496557186
+Best k_sigma: 0.24429936306439973
+... Statistics of Theta ...
+Mean of Theta: 1.013517780964137
+Standard deviation of Theta: 0.19961814808942516