report innit 3

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Project3/LyX/Bibliography.bib

+@unpublished{SteinerEvje-Lecture,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+@unpublished{LectureNotes,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+@unpublished{Intro_ConsLaw_2021,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 = {March},
+    title = {Reproducibility Repository for Data-Driven Modeling of Conservation Laws},
+    URL = {https://git.rwth-aachen.de/JanHab/ddm},
+    year = {2025},
+}

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+\branch Appendix
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+\end_branch
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+\shortcut idx
+\color #008000
+\end_index
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+\end_header
+
+\begin_body
+
+\begin_layout Section
+Introduction
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Introduction"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\end_body
+\end_document

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
+
+\begin_layout Standard
+\align center
+
+\lang american
+\begin_inset Tabular
+<lyxtabular version="3" rows="19" columns="3">
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+\begin_inset Text
+
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+
+\series bold
+DIMENSIONLESS VARIABLES
+\end_layout
+
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+
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+
+\begin_layout Plain Layout
+
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+Variable
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+
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+\begin_inset Text
+
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+Default Value
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+\begin_inset Text
+
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+\series bold
+Property
+\end_layout
+
+\end_inset
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+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $L$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $20$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
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+Axon length
+\end_layout
+
+\end_inset
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+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $D_{0}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $0.4$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
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+Diffusion coefficient
+\end_layout
+
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+<row>
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+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $N_{0}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $0.1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\lang american
+Concentration of free particles at left boundary
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $N_{L}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $0.01$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\lang american
+Concentration of free particles at right boundary
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $k_{-}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
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+1
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+
+\end_inset
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+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\lang american
+binding rate in negative direction
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sigma_{0},\sigma_{L}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $0.1,0.1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\lang american
+Degree of loading at left and right boundary
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $k_{p},k_{n}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $0.5$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
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+
+\lang american
+Detachment rate for particles in positive and negative direction
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
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+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
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+\begin_inset Text
+
+\begin_layout Plain Layout
+0.2
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+
+\end_inset
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+\begin_inset Text
+
+\begin_layout Plain Layout
+Cell size
+\end_layout
+
+\end_inset
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+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $t_{\text{end}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
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+10
+\end_layout
+
+\end_inset
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+\begin_inset Text
+
+\begin_layout Plain Layout
+Final time
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $N_{\text{time}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+200
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Number of time steps
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
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+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $0.05$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Time step size
+\end_layout
+
+\end_inset
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+<row>
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+\begin_inset Text
+
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+
+\end_layout
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+
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+
+\end_layout
+
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+\begin_inset Text
+
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+
+\end_layout
+
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+<row>
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+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
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+
+\begin_layout Plain Layout
+
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+
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+<row>
+<cell multicolumn="1" alignment="center" valignment="top" usebox="none">
+\begin_inset Text
+
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+
+\series bold
+INDEXING
+\end_layout
+
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+
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+
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+
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+
+\end_layout
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+<row>
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+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Variable
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+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\series bold
+Meaning
+\end_layout
+
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+
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+
+\end_layout
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+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\cdot^{j}$
+\end_inset
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+
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+<cell multicolumn="1" alignment="center" valignment="top" topline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
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+\end_inset
+
+-th time step
+\end_layout
+
+\end_inset
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+<cell multicolumn="2" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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+
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+
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+
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+\end_inset
+
+-th grid point
+\end_layout
+
+\end_inset
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+\end_body
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Project3/LyX/Project3Habscheid.lyx

+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
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+\usetikzlibrary{decorations.pathreplacing}
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+\use_hyperref true
+\pdf_title "Modeling Traffic Flow"
+\pdf_author "Jan Habscheid"
+\pdf_subject "Data-Driven Modeling of Conservation Laws"
+\pdf_keywords "Traffic Flow, Advection, Riemann Problem, Lax-Entropy"
+\pdf_bookmarks true
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+\selected 0
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+\index Index
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+\color #008000
+\end_index
+\leftmargin 2cm
+\topmargin 1.5cm
+\rightmargin 2cm
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+\tocdepth 3
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+\end_header
+
+\begin_body
+
+\begin_layout Title
+
+\lang english
+USING MONTE CARLO MARKOV CHAIN (MCMC) TO IDENTIFY UNKNOWN FLUX FUNCTION FROM OBSERVATION DATA
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+
+\lang english
+Date:
+ 05.04.2025,
+ 
+\begin_inset Newline newline
+\end_inset
+
+Referee:
+ Steinar Evje
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Author
+
+\lang english
+Jan Habscheid
+\begin_inset Foot
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+
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+
+\lang english
+Author:
+ Jan Habscheid,
+ 
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+,
+ Student ID:
+ 287338
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+
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Project3/LyX/Results.lyx

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Project3/LyX/TheoryAndMethods.lyx

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