%%%%%%%%%%%%%%%%%%%%%definitions%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\input{../../doc/related_pages/header.tex}
\input{../../doc/related_pages/newcommands.tex}
\usepackage{minted}

\renewcommand{\ne}{\ensuremath{n }}
\newcommand{\Ni}{\ensuremath{N}}
\newcommand{\ue}{\ensuremath{u_\parallel}}
\newcommand{\Ui}{\ensuremath{U_\parallel}}
\newcommand{\Apar}{\ensuremath{A_\parallel}}
\newcommand{\bperp}{\ensuremath{ \vec b_\perp}}
\newcommand{\neref}{\ensuremath{n_0}}
\newcommand{\Teref}{\ensuremath{T_{e0}}} %\rhoN

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%DOCUMENT%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\title{Collisionless Reconnection}
\author{ M.~Held and M.~Wiesenberger}
\maketitle

\begin{abstract}
\end{abstract}

\section{Compilation and Usage}
Input file format: json\\
Output file format: NetCDF-4\\
File path: \texttt{feltor/src/reco2D/}
\begin{verbatim}
make reconnection device = <omp gpu>
make reconnection_hpc device = <omp gpu>
make reconnection_mpi device = <omp gpu>
\end{verbatim}
\texttt{reconnection} depends on both GLFW3 and NETCDF, while
\texttt{reconnection\_hpc} and \texttt{reconnection\_mpi} avoids the GLFW3 dependency and only operate with ``netcdf'' output.
Run with
\begin{verbatim}
path/to/feltor/src/reco2D/reconnection(_hpc) input.json <output.nc>
\end{verbatim}
The output file is only needed if you chose ``netcdf'' in the output field
of the input file.
\begin{verbatim}
echo 2 2 | mpirun -n 4 path/to/feltor/src/reco2D/reconnection_mpi input.json output.nc
\end{verbatim}
For the mpi program you need to provide the distribution of processes along each
axis on the command line.

\section{Equations}
%tearing instability <-> reconnection
Collisionless reconnection was studied in~\cite{Stanier2015}, gyro-fluid and gyro-kinetic studies~\cite{Comisso2013,Zacharias2014}.
Our reduced model is based on a recent formulation of a full-F gyro-fluid model~\cite{Madsen2013}.
It consists of the first two moment equations for electrons and ions
\begin{align}
%%%
\frac{\partial}{\partial t} \ne
+ \nc\left( \ne (\vec u_E + \ue \vec b_\perp) \right)=& 0
\\
%%%
\label{firstgyromom}
\frac{\partial}{\partial t} \Ni
+ \nc\left( \Ni (\vec u_E^i + \Ui \vec b_\perp^i) \right)=& 0
 \\
 %%%
    \mu_e \frac{\partial}{\partial t} \left( \ne\ue \right)
+  \mu_e \nc\left( \left(\vec u_E+ \ue\vec b_\perp\right) \ne\ue \right) =&
- \tau_e \vec b_\perp\cn\ne - \ne\left(\vec b_\perp\cn\phi +\frac{\partial}{\partial t}\Apar\right)
      \\
%%%
  \mu_i\frac{\partial}{\partial t} \left( \Ni\Ui \right)
      +  \mu_i\nc\left( \left(\vec u_E^i + \Ui \vec b_\perp^i\right) \Ni\Ui \right) =&
      - \tau_i  \vec b_\perp^i \cn \Ni - \Ni \left(\vec b_\perp \cn \psi + \frac{\partial}{\partial t} \Gamma_1\Apar\right)
\end{align}
where
$\vec u_E := \zhat \times \nabla\phi = [-\partial_y \phi,\ \partial_x \phi]$ and
$\bperp := \nabla \Apar \times \zhat = [\partial_y \Apar,\ -\partial_x \Apar]$.

The equations are coupled by polarisation and induction
\begin{align}
 -\ne + \Gamma_1 \Ni &= -\vec{\nabla}\cdot\left(\Ni \vec{\nabla}_\perp \phi\right) \\
 -\frac{1}{\beta} \vec{\nabla}_\perp^2 \Apar &=  -\ne \ue + \Gamma_1 (\Ni \Ui)
\end{align}
where the signs are such that the elliptic operators are positive definite and
with gyro-averaged parallel electromagnetic vector potential, generalized electric potential and gyro-averaging operator
\begin{align}
 \psi &:= \Gamma_1 \phi - u_E^2 /2  \\
  \Gamma_1 &:= (1-\frac{\tau_i}{2} \vec{\nabla}_\perp^2 )^{-1}
\end{align}
and ion gyro-radius, vacuum permeability and ion gyrofrequency
\begin{align}
  \rho_{i}   := \frac{\sqrt{T_{i} m_i}}{e B} \quad
  \mu_0 := 1/(\epsilon_0 c^2) \quad
  \Omega_i := e B / m_i
\end{align}
After exploiting the gyro-Bohm normalisation, our model is controlled by the dimensionless parameters. These
are the
\begin{align}
 \mu   :=  \frac{m}{Z m_i} \quad
 \tau  :=  \frac{T}{ Z T_i}\quad
 \beta_{e0} :=  \frac{\mu_0 \neref \Teref }{ B_0^2 }
\end{align}

Input file format: \href{https://en.wikipedia.org/wiki/JSON}{json} \\
\begin{minted}[texcomments]{js}
"physical" : // physical parameters
{
    "mu"   : -0.000544617, // the electron to ion mass ratio $\mu_e$
    "tau"  :  0.0,    // electron to ion temperature $\tau_i$
    "beta" :  1e-3    // plasma beta $\beta$
}
\end{minted}

\section{Initial and boundary conditions}
The boundary conditions for all variables are Dirichlet in $x$.
The $y$-direction is periodic.

The initial conditions are set in the file \texttt{feltor/src/reco2D/init.h}.

We initialize
\begin{align}
 \Ni&=\ne=1 \\
 \phi &= 0 \\
  \Ui&=0 \\
  \ue &= \frac{1}{\ne \beta }\vec{\nabla}^2_{\perp} \Apar
\end{align}
where we can choose $\Apar$.
\subsection{Harris sheet}
The initial parallel magnetic vector potential is
\begin{align}
    \Apar =\beta\left( A_0 / \cosh{( 4  \pi x / L_x )}^2 +A_1\cos{(2 m_y\pi y/L_y)}\right) \cos( \pi x /L_x)
\end{align}
The harris sheet initial condition can be chosen with the following parameters in the input file
\begin{minted}[texcomments]{js}
"init" : // Parameters for initialization
{
    "type" : "harris", // This choice necessitates the following parameters
    "amplitude0"  : 0.1, // harris amplitude
    "amplitude1"  : 1e-3, // perturbation amplitude (making this negative will
    // shift the perturbation by an angle $\pi$)
    "my"  :  1    // perturbation wave number in y
}
\end{minted}
We analytically compute the Laplacian of $\Apar$ with the help of Mathematica.
\subsection{Island}
The initial parallel magnetic vector potential is
\begin{align}
    \Apar =\beta \left[ A_0\frac{L_x}{4\pi} \ln \left\{ \cosh( 4\pi x/L_x) + \varepsilon \cos( 4\pi y/L_x)\right\} + A_1\cos{(2m_y \pi y/L_y)}\right] \cos( \pi x /L_x)
\end{align}
with $\varepsilon = 0.2$.
The island initial condition can be chosen with the following parameters in the input file
\begin{minted}[texcomments]{js}
"init" : // Parameters for initialization
{
    "type" : "island", // This choice necessitates the following parameters
    "amplitude0"  : 0.1, // island amplitude
    "amplitude1"  : 1e-3, // perturbation amplitude (making this negative will
    // shift the perturbation by an angle $\pi$)
    "my"  :  1    // perturbation wave number in y
}
\end{minted}
We analytically compute the Laplacian of $\Apar$ with the help of Mathematica.
(The island simulation takes approximately a factor 5 times smaller timestep than
the harris simulation).

\section{Invariants}
The mass density and diffusions are
\begin{align}
    \mathcal M &= \ne \\
    \vec j_n &= \ne \vec u_E + \ne\ue\bperp \\
     \Lambda_n &= -\nu_\perp \Delta_\perp^2 \ne
\end{align}
\begin{tcolorbox}[title=Note]
    We already incorporate the artificial diffusion terms defined in
    Section~\ref{sec:regularization}
\end{tcolorbox}

The inherent energy density of our system is:
\begin{align}
 \mathcal{E} := &
                    \ne \ln{(\ne)}
                  + \tau_i \Ni \ln{( \Ni)}
                  \nonumber \\
                 &- \frac{1}{2} \mu_e \ne \ue^2
                  + \frac{1}{2} \Ni \Ui^2
                  \nonumber \\
                 &+\frac{1}{2} \Ni u_E^2
                  + \frac{|\nabla_\perp \Apar|^2}{2 \beta}
\end{align}
The energy current density and diffusion are
\begin{align}
  \vec j_{\mathcal E} =& \sum_s z\left[
  \left(\tau \ln N + \frac{1}{2}\mu U_\parallel^2 + \psi \right)N\left(
  \vec u_E +U_\parallel\bperp  \right) + \tau NU_\parallel \bperp\right]
  , \\
    \Lambda_\mathcal{E} :=  &\left( (1+\ln \ne) + \frac{1}{2} z_e \mu_e \ue^2 - \phi \right) (-\nu_\perp \Delta_\perp^2 \ne)
    \nonumber\\
    &+ \left( \tau_i ( 1+\ln \Ni) + \frac{1}{2} \Ui^2 + \psi_i\right)(-\nu_\perp \Delta_\perp^2 \Ni)
    \nonumber\\
    &+ z_e\mu_e \ue\ne (-\nu_\perp \Delta_\perp^2 \ue)
    \nonumber\\
    &+ \Ui\Ni (-\nu_\perp \Delta_\perp^2 \Ui)
    \label{eq:energy_diffusion}
\end{align}
where in the energy flux $\vec j_{\mathcal E}$
we neglect terms  containing time derivatives
of the eletric and magnetic potentials and we sum over all species.

With our choice of boundary conditions both the mass as well as the energy flux
vanishes on the boundary. In the absence of artificial viscosity the volume integrated
mass and energy density are thus exact invariants of the system.
\begin{tcolorbox}[title=Note]
    For the canonical-viscosity scheme the viscosity terms in the last two
    terms of Eq.~\eqref{eq:energy_diffusion} change to the canonical velocity.
\end{tcolorbox}

\section{Reconnection rate}
 A consistent definition of the reconnection rate is not trivial~\cite{Comisso2016}.
 We adopt a similar definiton as in Ref.~\cite{Comisso2013}
\begin{align}
    Q_X&:= \Apar(\vec{x}_X,t) - \Apar( \vec{x}_X,0) \\
 \gamma&:= \frac{1}{Q_X}\frac{\d Q_X}{\d t} = \frac{\d \ln{|Q_X|}}{\d t}
\end{align}
where $\vec{x}_X = [0,0]$.
This can be easily evaluated in post-processing using a python script.

\section{Numerical methods}

\subsection{Spatial grid}
The spatial grid is a two-dimensional Cartesian product-grid $[-L_x/2, L_x/2]\times [-L_y/2, L_y/2]$ adaptable with the following parameters
\begin{minted}[texcomments]{js}
"grid" :
{
    "n"  :  3, // The number of polynomial coefficients
    "Nx"  : 48, // Number of cells in x
    "Ny"  : 48, // Number of cells in y
    "lxhalf"  : 80.0, // Half box length in x
    "lyhalf"  : 80.0, // Half box length in y
}
\end{minted}
\subsection{Timestepper}
The time stepper can be an explicit multistep method where you can chose the
tableau to use
\begin{minted}[texcomments]{js}
"timestepper" :
{
    "type"    : "multistep", //Choose an explicit multistep method
    "tableau" : "TVB-3-3", //  Any explicit multistep tableau *
    "dt"      : 20.0, // Fixed timestep
}
\end{minted}
*See the \href{https://feltor-dev.github.io/doc/dg/html/structdg_1_1_explicit_multistep.html}{dg documentation} for what tableaus are available.
The second option is an adaptive explicit embedded Runge-Kutta scheme
\begin{minted}[texcomments]{js}
"timestepper":
{
    "type"    : "adaptive", //Choose an explicit adaptive RK scheme
    "tableau" : "Tsitouras09-7-4-5", // Any explicit embedded RK tableau *
    "rtol"    : 1e-7, // The relative tolerance in the timestep
    "atol"    : 1e-10 // The absolute tolerance in the timestep
}
\end{minted}
*See the \href{https://feltor-dev.github.io/doc/dg/html/structdg_1_1_e_r_k_step.html}{dg documentation} for what tableaus are available.

\subsection{Advection scheme}
\begin{minted}[texcomments]{js}
"advection":
{
    "type" : "arakawa" // The Arakawa bracket scheme
    //note that this should only be used in connection with artificial diffusion
}
\end{minted}
We implemented the advection in terms of Arakawa brackets
\begin{align}
%%%
\frac{\partial}{\partial t} \ne =&
 - \left[\phi, \ne\right]
+ \left[\Apar,\ne \ue  \right]
\\
%%%
\label{firstgyromom}
\frac{\partial}{\partial t} \Ni =&
 - \left[\psi, \Ni\right]
+ \left[\Gamma_1 \Apar ,\Ni \Ui  \right]
 \\
 %%%
\frac{\partial}{\partial t} \left( \ue+ \frac{1}{\mu_e} \Apar \right) =&
      -  \left[ \phi, \ue+ \frac{1}{\mu_e} \Apar  \right]% \nonumber \\
    +   \left[\Apar,\ue^2/2   \right]% \nonumber \\
      - \frac{1}{\mu_e}  \left[\Apar,\ln{\ne}   \right]
      \\
      \frac{\partial}{\partial t} \left( \Ui+ \Gamma_1 \Apar  \right) =&
      -  \left[ \psi, \Ui+ \Gamma_1 \Apar  \right]% \nonumber \\
     +   \left[\Gamma_1 \Apar,\Ui^2/2   \right]% \nonumber \\
      + \tau_i  \left[\Gamma_1 \Apar,\ln{\Ni}   \right]
\end{align}

\begin{minted}[texcomments]{js}
"advection":
{
    "type" : "upwind"  // The upwind scheme
}
\end{minted}
For the upwind scheme we first rewrite the equations in terms of
$\vec u_E := \zhat \times \nabla\phi = [-\partial_y \phi,\ \partial_x \phi]$ and
$\bperp := \nabla \Apar \times \zhat = [\partial_y \Apar,\ -\partial_x \Apar]$. We use
that $\nc \vec u_E = \nc \bperp =0$
\begin{align}
%%%
\frac{\partial}{\partial t} \ne =&
- ( \vec u_E + \ue \bperp)\cn \ne - \ne \bperp \cn  \ue
\\
%%%
\frac{\partial}{\partial t} \Ni =&
- ( \vec u_E^i + \Ui \bperp^i)\cn \Ni - \Ni \bperp^i\cn  \Ui
 \\
 %%%
\frac{\partial}{\partial t} \left( \ue+ \frac{1}{\mu_e} \Apar \right) =&
-  (\vec u_E + \ue \bperp ) \cn \ue
+ \frac{1}{\mu_e} \bperp \cn \ln \ne
- \frac{1}{\mu_e} \bperp \cn \phi
      \\
      \frac{\partial}{\partial t} \left( \Ui+ \Gamma_1 \Apar  \right) =&
    - (\vec u_E^i + \Ui \bperp^i ) \cn \Ui
    - \tau_i \bperp^i \cn \ln \Ni
    - \bperp^i \cn \psi
\end{align}
where the $i$ index signifies that $\vec u_E$ respectively $\bperp$ are to
be evaluated using the ion potentials $\psi$ and $\Apar$.
\begin{tcolorbox}[title=Note]
    We also tried to use the conservative upwind scheme, where the density equation
    is discretized in divergence form together with an interpolated multiplication
    scheme. This however was completely unstable for the harris sheet.
\end{tcolorbox}
\subsection{Regularization} \label{sec:regularization}
In order to prevent shocks and regularize the advection scheme
we implement an artificial viscosity of order 2.
\begin{align}
%%%
    \frac{\partial}{\partial t} \ne =& \ldots -\nu_\perp \Delta_\perp^2 \ne
\\
%%%
\frac{\partial}{\partial t} \Ni =& \ldots -\nu_\perp \Delta_\perp^2 \Ni
 \\
 %%%
\frac{\partial}{\partial t} \left( \ue+ \frac{1}{\mu_e} \Apar \right) =&
\ldots -\nu_\perp \Delta_\perp^2 \ue
      \\
      \frac{\partial}{\partial t} \left( \Ui+ \Gamma_1 \Apar  \right) =& \ldots -\nu_\perp \Delta_\perp^2 \Ui
\end{align}
Regularly we apply the artificial viscosity to the parallel velocities. However,
we can also apply it to the canonical velocity:
\begin{align}
 %%%
\frac{\partial}{\partial t} \left( \ue+ \frac{1}{\mu_e} \Apar \right) =&
\ldots -\nu_\perp \Delta_\perp^2 \left( \ue+ \frac{1}{\mu_e} \Apar \right)
      \\
      \frac{\partial}{\partial t} \left( \Ui+ \Gamma_1 \Apar  \right) =& \ldots -\nu_\perp \Delta_\perp^2 \left( \Ui+ \Gamma_1 \Apar  \right)
\end{align}

\begin{minted}[texcomments]{js}
"regularization",
{
    "type" : "velocity-viscosity",  // Apply viscosity on density and
    // parel velocity
    "type" : "canonical-viscosity",  // Apply viscosity on density and
    // parallel canonical velocity
    "direction" : "centered",  // The direction of the Laplacian
    "nu_perp" : 1e-6 // The strength of the diffusion
}
\end{minted}
Simply set the diffusion to 0 if you do not want any regularization.
\begin{tcolorbox}[title=Note]
    If you set nu\_perp too large the timestep in the explicit timestepper will
    have to be very small due to the restrictive CFL condition. At the same
    time in our experience having to make the timestep smaller due to viscosity
    is a clear warning that nu\_perp is too large for the chosen resolution.
    Chose nu\_perp as large as you can without decreasing the timestep.
\end{tcolorbox}

\subsection{Elliptic solvers}
In order to solve the elliptic equations we chose a multigrid scheme (nested
iterations). The accuaracies for the polarization equation can be chosen for
each stage separately, while for the Helmholtz type equations (The gamma
operators and the Ampere equation) only one accuracy can be set:
\begin{minted}[texcomments]{js}
"elliptic",
{
    "stages"    : 3,  // Number of stages (3 is best in virtually all cases)
    "eps_pol"   : [1e-6,10,10],
    // Accuracy requirement on each stage of the
    // multigrid scheme. $\eps_0 = \eps_{pol,0}$, $\eps_i = \eps_{pol,i} \eps_{pol,0}$  for $i>1$. \\
    "eps_gamma" : 1e-10, // Accuracy requirement of Gamma operator on density
    "eps_maxwell": 1e-7, //Accuracy requirement of Ampere equation
    "direction" : "forward", // Direction of the Laplacian: forward or centered
    "jumpfactor" : 1.0
    // Jump factor of Laplacian in polarization, Gamma and Ampere equations
}
\end{minted}

%..................................................................
\section{Output}
Our program can either write results directly to screen using the glfw library
or write results to disc using netcdf.
This can be controlled via
\begin{minted}[texcomments]{js}
"output":
{
    // Use glfw to display results in a window while computing (requires to
    // compile with the glfw3 library)
    "type"  : "glfw"
    "itstp"  : 4, // The number of steps between outputs of 2d fields

    // Use netcdf to write results into a file (filename given on command line)
    // (see next section for information about what is written in there)
    "type"  : "netcdf"
    "itstp"  : 4,    // The number of steps between outputs of 2d fields
    "maxout"  : 500, // The total number of field outputs. The endtime is
    // T=itstp*maxout*dt
    "n"  : 3 , // The number of polynomial coefficients in the output file
    "Nx" : 48, // Number of cells in x in the output file
    "Ny" : 48  // Number of cells in y in the output file
}
\end{minted}
The number of points in the output file can be lower (or higher) than the number of
grid points used for the calculation. The points will be interpolated from the
computational grid.
\subsection{Netcdf file}
Output file format: netcdf-4/hdf5

\begin{longtable}{lll>{\RaggedRight}p{7cm}}
\toprule
\rowcolor{gray!50}\textbf{Name} &  \textbf{Type} & \textbf{Dimension} & \textbf{Description}  \\ \midrule
inputfile        & text attribute & 1 & verbose input file as a string \\
time             & Coord. Var. & 1 (time) & time at which fields are written \\
x                & Coord. Var. & 1 (x) & x-coordinate  \\
y                & Coord. Var. & 1 (y) & y-coordinate \\
xc               & Dataset & 2 (y,x) & Cartesian x-coordinate  \\
yc               & Dataset & 2 (y,x) & Cartesian y-coordinate \\
X                & Dataset & 3 (time, y, x) & 2d outputs \\
X\_1d            & Dataset & 1 (time) & 1d volume integrals $\int \dV X$ \\
time\_per\_step  & Dataset & 1 (time) & Average computation time for one step \\
\bottomrule
\end{longtable}
The output fields X are determined in the file \texttt{feltor/src/reco2D/diag.h}.

%..................................................................
\bibliography{../../doc/related_pages/references, references}
%..................................................................
\end{document}


\documentclass[a4paper,twocolumn,10pt]{article}
\input{preamble}

\title{Notes for M.A.~Armstrong's \textit{Groups and Symmetry}}
\author{Christian Stigen Larsen}
\date{March 2016}

\begin{document}
  \maketitle
  \section{Symmetries of the Tetrahedron}
  \paragraph{Symmetry group} Captures the rules of how symmetries combine for a
  given object.

  \paragraph{Order of operations} In the \textit{product}\footnote{Rotations,
  flips, multiplications, additions, etc. Same order as functional
  composition.} $xyz$, do $z$ first, then $y$ and finally $x$. If order doesn't
  matter in $G$, it's commutative (or \textbf{\textit{abelian}}). Remember to
  label geometric vertices.

  \section{Axioms}
  \paragraph{Group}  Set $G$ with \textit{multiplication} (addition,
  rotation, etc.) satisfying
  \begin{itemize}
    \item \textbf{\textit{associativity}}, i.e.~$(xy)z = x(yz)$
    \item \textbf{\textit{identity element}} $e \in G$ such that $xe=x=ex$
    \item \textbf{\textit{inverse}} $e \in G$ such that $x^{-1}x=e=xx^{-1}$
  \end{itemize}

  \paragraph{Properties common to all groups}
  \begin{itemize}
    \item The identify element of a group is unique.
    \item The inverse of each element of a group is unique.
  \end{itemize}

  \section{Numbers}
  \paragraph{Addition of $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$}
  \begin{itemize}
    \item Identity is zero
    \item $-x$ is the inverse
  \end{itemize}
  \paragraph{Multiplication}
  \begin{itemize}
  \item For $\mathbb{Q}-\{0\}$, $\mathbb{Q}^{\textbf{pos}}$,
    $\mathbb{R}-\{0\}$, $\mathbb{R}^{\textbf{pos}}$, $\{+1,-1\}$,
    $\mathbb{C}-\{0\}$, $\mathcal{C}$\footnote{Complex numbers of modulus 1.},
    $\{\pm 1, \pm i\}$: $e=1$ and $x^{-1}=\nicefrac{1}{x}$.
  \end{itemize}

  \paragraph{$\mathbb{Z}$ under addition modulus $n$} $e=0$,
  $x^{-1}=n-x$ for $x\ne0$, finite \textit{abelian} group and denoted
  $\mathbb{Z}_n$.

  \paragraph{$\mathbb{Z}$ under multiplication modulus $n$} Requires $n$ to be
  prime.

  \section{Dihedral Groups}
  When $n\geqslant3$ we can manufacture a plate whish has $n$ equal sides. These are
  the non-commutative \textit{dihedral rotational symmetry groups} $D_n$. E.g.
  $D_3 = \{e,r,r^2,s,rs,r^2s\}$. $x^mx^n=x^{m+n}$ and $(x^m)^n=x^{mn}$ provided
  we interpret $x^0=e$. For any multiplication table, each element in $G$
  appears only once in every given column or row.

  $r^n=e$, $s^2=e$, $sr=r^{n-1}s$, $r^{n-1}=r^{-1}$, etc.

  Each element is of form $r^a$, $r^as$ where $0\leqslant a\leqslant n-1$.

  For $k=a+_nb$, $r^ar^b=r^k$ and $r^a(r^bs)=r^ks$.  For $l=a+_n(n-b)$,
  $(r^as)r^b=r^ls$ and $(r^as)(r^bs)=r^l$ --- thus $r$ and $s$
  \textbf{\textit{generate}} $D_n$.

  The \textbf{\textit{order}} $|G|$ is the number of elements in the group. If
  $x^n=e$, then the \textit{element} $x$ has \textit{finite} order $n$ when $n$
  is the smallest such $n$.

  \section{Subgroups and Generators}
  A \textbf{\textit{subgroup}} of $G$ is a subset of $G$ which itself forms a
  group under the multiplication of $G$. For $H$ to be a subgroup of $G$,
  $H<G$:
  \begin{itemize}
    \item $xy \in G$ for any $x,y \in H$
    \item $e_H \in G$
    \item For any $x \in H$, $x^{-1} \in G$
    \item Associativity in $G$ implies the same for $H$.
  \end{itemize}

  \paragraph{Subgroup generated by $x$, or $\langle x \rangle$}
  For an element $x$ in $G$, the set of all $x^n$ is a subgroup of $G$
  (remember $x^0=e$). Finite order $m$ means $x^0=e, x^1, \ldots, x^{m-1}$.
  So order of $x\in G$ is precisely the order of $\langle x\rangle$. If
  $\langle x\rangle=G$, i.e., generates all of $G$, then $G$ is a
  \textbf{\textit{cyclic group}}.

  \paragraph{Subgroup generated by $X$} If $X<G$\footnote{$X$ is a subgroup of
  $G$.} and, for example, $r$,$s$,$r^2$,$sr$ (called \textit{words} of $X$).

  \paragraph{Theorems}
  \begin{itemize}[leftmargin=0.45in]
    \item[\theorem{5.1}] A non-empty subset $H$ of a group $G$ is a subgroup of
      $G$ if and only if $xy^{-1}$ belongs to $H$ whenever $x$ and $y$ belong
      to $H$.

    \item[\theorem{5.2}] The intersection of two subgroups of a group is itself
      a subgroup.

    \item[\theorem{5.3}] Every subgroup of $\mathbb{Z}$ is cyclic. Every
      subgroup of a cyclic group is cyclic.
  \end{itemize}

  \section{Permutations}
  A \textit{permutation} is a bijection\footnote{A one-to-one mapping between
  the elements of two sets, meaning you can always go backwards as well.} from a
  set $X$ to itself (e.g.,~replace all $3$s with $1$s).  The collecticon of
  \textit{all} permutations of $X$ forms a group $S_x$ under composition of
  functions (who each perform one specific permutation). When $X$ consists of
  the first $n$ positive integers, we get the \textbf{\textit{symmetric group}}
  $S_n$ of degree $n$ and order $n!$. $S_3$ is not abelian

  $(a_1a_2\ldots a_k)$ is called a \textbf{\textit{cyclic permutation}},
  sending $a_1$ to $a_2$, $\ldots$ , $a_k$ to $a_1$. Its length is $k$ and a
  cyclic permutation of length $k$ is called a \textbf{\textit{k-cycle}}. A
  2-cycle is called a \textbf{\textit{transposition}}. Every element of $S_n$
  can be written as many such \textbf{\textit{disjoint}}, meaning no integer is
  moved by more than one of them. Therefore they are \textit{commutative}.

  \paragraph{A few tricks}
  \begin{itemize}
    \item Each \textit{element} of $S_n$ can be written as a product of cyclic
      permutations, and any cyclic permutation can be written as a product of
      transpositions: $(a_1a_2 \ldots a_k) = (a_1a_k) \ldots (a_1a_3)(a_1)(a_2)$.
      Therefore, each \textit{element} of $S_n$ can be written as a product of
      transpositions.
    \item $(ab) = (1a)(1b)(1a)$
    \item $(1k) = (k-1,k)$$\ldots$$(34) (23) (12) (23) (34)$$\ldots$$(k-1,k)$
  \end{itemize}

  \paragraph{Theorems}
  \begin{itemize}[leftmargin=0.45in]
    \item[\theorem{6.1}] The transpositions in $S_n$ together generate
      $S_n$.

    \item[\theorem{6.2a}] The transpositions $(12), (13), \ldots, ({1n)}$
      together generate $S_n$.

    \item[\theorem{6.2b}] The transpositions $(12), (23), \ldots, (n-1,n)$
      together generate $S_n$.

    \item[\theorem{6.3}] The transposition $(12)$ and the $n$-cycle $(12
      \ldots n)$ together generate $S_n$.
  \end{itemize}

  Any \textit{element} $\alpha$ of $S_n$ can be written as a product of
  \textit{transpositions} in many different ways. But the number of
  transpositions is always even or always odd. If $\alpha$ \textit{can} be
  written as the product of an even number of transpositions, then its sign
  must be $+1$; for odd, it is $-1$. Therefore, by the first trick above, a
  \textit{cyclic permutation} is even precisely when its length is odd.

  \paragraph{Theorems}
  \begin{itemize}[leftmargin=0.45in]
    \item[\theorem{6.4}] The even permutations in $S_n$ form a subgroup of
      order $n!/2$ called the \textbf{\textit{alternating group $A_n$}} of
      degree $n$.

    \item[\theorem{6.5}] For $n\geqslant 3$ the 3-cycles generate $A_n$.
  \end{itemize}

  \section{Isomorphisms}
  If two multiplication tables have corresponding elements and products, they
  are \textbf{\textit{isomorphic}}.

  Two groups $G$ and $G'$ are \textbf{\textit{isomorphic}} if there is a
  bijection $\varphi$ from $G$ to $G'$ which satisfies $\varphi(xy) =
  \varphi{(x)}\varphi{(y)}$ for all $x,y \in G$. The function $\varphi$ is called an
  \textbf{\textit{isomorphism}} between $G$ and $G'$.
  This is written $G \cong G'$.

  \paragraph{Notes}
  \begin{itemize}
    \item $G$ and $G'$ have the same order.
    \item $\varphi{(x)}^{-1} = \varphi{(x^{-1})}$ for all $x \in G$.
    \item If $G$ is abelian, then so is $G'$.
    \item If $H$ is a subgroup of $G$ then $\varphi{(H)}$ a subgroup of $G'$.
    \item An isomorphism preserves the order of each element.
    \item If $\varphi\colon G \rightarrow G'$ and $\psi\colon G' \rightarrow G''$ are both
      isomorphisms, then the composition $\psi\varphi\colon G \rightarrow G''$ is also
      an isomorphism.
  \end{itemize}
  \paragraph{Examples}
  \begin{itemize}
    \item $\varphi\colon \mathbb{R} \rightarrow \mathbb{R}^{\textbf{pos}}$ by
      $\varphi{(x)} = e^x$ and $\varphi{(x+y)}=e^{x+y}=e^xe^y=\varphi{(x)}\varphi{(y)}$.
    \item The non-abelian, rotational group $G$ for the tetrahedron is isomorphic to $A_4$.
    \item Any infinite cyclic group $G$ is isomorphic to $\mathbb{Z}$ by
      $\varphi{(x^m)} = m$ and $\varphi{(x^mx^n)} =
      \varphi{(x^{m+n})}=m+n=\varphi{(x^m)}+\varphi{(x^n)}$.
    \item Any finite cyclic group of order $n$ is isomorphic to $\mathbb{Z}_n$
      by $\varphi{(x^m)} = m\Mod{n}$.
    \item The numbers $1$, $-1$, $i$, $-i$ form a group under complex
      multiplication. It is cyclic, and $i$, $-i$ are both generators. It gives
      two isomorphisms between this group and $\mathbb{Z}_4$.
    \item $D_3$ and $S_3$ are isomorphic.
    \item There is no isomorphism between $\mathbb{Q}$ and
      $\mathbb{Q}^{\textbf{pos}}$.
  \end{itemize}

  \section{Plato's Solids and Cayley's Theorem}
  \textit{Remember:} A surjection between two finite sets which have the same
  number of elements must be a bijection.

  \begin{itemize}
    \item The rotational symmetry group of the tetrahedron is isomorphic to
      $A_4$.

    \item The cube and octehedron both have rotational symmetry groups which
      are isomorphic to $S_4$.

    \item The dodecahedron and icosahedron both have rotational symmetry groups
      which are isomorphic to $A_5$.

    \item If two solids are \textbf{\textit{dual}} to one another, their
      rotational symmetry groups are isomorphic.
  \end{itemize}

  \paragraph{Theorems}
  \textit{Every} group is isomorphic to a subgroup of permutations:
  \begin{itemize}[leftmargin=0.45in]
    \item[\theorem{8.1}] \textbf{Cayley's Theorem.} Let $G$ be a group, then
      $G$ is isomorphic to a subgroup of $S_G$.

    \item[\theorem{8.2}] If $G$ is a finite group of order $n$, then $G$ is
      isomorphic to a subgroup of $S_n$.
  \end{itemize}

  \section{Matrix Groups}
  The set of all invertible $n \times n$ matrices with real numbers as entries
  forms a group under matrix multiplication: Matrix multiplication is
  associative, the $n \times n$ identity matrix $I_n=\epsilon$ and the inverse
  of $AB$ is $B^{-1}A^{-1}$. This group is called the \textbf{\textit{General
  Linear Group}}, $GL_n$.

  Matrix multiplication is not commutative for $n \geqslant 2$, so we have a
  family of \textit{infinite non-abelian} groups $GL_2$, $GL_3$, etc. For $n=1$
  the single entry must be a non-zero number (the matrix is invertible), and
  reduces to ordinary multiplication of numbers. Hence, $GL_1 \cong
  \mathbb{R}-\{0\}$.

  $AB^{-1}$ is orthogonal and by theorem (\ref{theorem-5.1}.1) the collection
  of all $n \times n$ orthogonal matrices is a subgroup of $GL_n$. This
  subgroup is called the \textbf{\textit{Orthogonal Group}}, $O_n$. Those
  elements of $O_n$ which have determinant equal to $+1$ form a subgroup of
  $O_n$ called the \textbf{\textit{Special Orthogonal Group}}, $SO_n$.

  \textit{No further notes here, at the moment.}

  \section{Products}
  The \textbf{\textit{direct product}} $G \times H$ of two groups $G$ and $H$
  is constructed by $(g,h)(g',h') = (gg',hh')$, where $g, g' \in G$ and $h, h'
  \in H$. Thus, $(gg',hh') \in G \times H$ and $G \times H$ is a group. The
  correspondence $(g,h) \rightarrow (h,g)$ means that $G \times H$ is
  isomorphic to $H \times G$. Unless either of $G$ or $H$ are of infinite
  order, $|G \times H| = |G|\cdot|H|$. If both $G$ and $H$ are abelian, so is
  $G \times H$. In reverse, if $G \times H$ is abelian, so are both $G$ and
  $H$.

  E.g., the elements of $\mathbb{Z}_2 \times \mathbb{Z}_3$ are
  $\{0,1\}\times\{0,1,2\} = \{(0,0), (0,1), (0,2), (1,0), (1,1), (1,2)\}$ and
  their elements are combined by $(x,y) + (x',y') = (x \,+_2\, x', y \,+_3\, y')$.
  We follow the convention of using $+$ for the group structure whenever we
  have products of cyclic groups. As continually adding $(1,1)$ to itself, we
  can fill out the whole group, and therefore $\mathbb{Z}_2 \times
  \mathbb{Z}_3$ is cyclic and isomorphic to $\mathbb{Z}_6$.

  \paragraph{Klein's group} $\mathbb{Z}_2 \times \mathbb{Z}_2$ is non-cyclic
  and isomorphic to the group of plane symmetries of a chessboard.

  We write $\mathbb{R}^n$ for the direct product of $n$ copies of $\mathbb{R}$.

  \paragraph{Theorem \theorem{10.1}} $\mathbb{Z}_m \times \mathbb{Z}_n$ is
  cyclic if and only if the highest common factor of $m$ and $n$ is 1.

  \paragraph{Theorem \theorem{10.2}} If $H$ and $K$ are subgroups of $G$ for
  which $HK=G$, if they have only the identity element in common, and if every
  element of $H$ commutes with every element of $K$, then $G$ is isomorphic to
  $H \times K$.

  The linear transformation $f_J\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$
  sends each vector $\bm{x}$ to $\bm{-x}$ and is called
  \textbf{\textit{central inversion}}.

  \textit{Some important notions at the end of the chapter have been left out,
  currently.}

  \section{Lagrange's Theorem}
  Let $H<G$ and break it up as the union of the $k+1$ pieces $H, g_1H, \ldots,
  g_kH$, then $|G|=(k+1)|H|$.

  \theorem{11.1} The order of a subgroup of a finite group is always a divisor
  of the order of the group.

  \textbf{Note:} The opposite is not true; the existence of a divisor $m$
  of $|G|$ does \textit{not} imply the existence of a subgroup of $G$.

  \paragraph{Corrolaries}
  \begin{itemize}[leftmargin=0.45in]
    \item[\theorem{11.2}] The order of every element of $G$ is a divisor of the
      order of $G$.

    \item[\theorem{11.3}] If $G$ has prime order, then $G$ is cyclic.

    \item[\theorem{11.4}] If $x$ is an element of $G$ then $x^{|G|}=e$.

    \item[\theorem{11.5}] \textbf{Euler's Theorem.} If the highest common
      factor of $x$ and $n$ is $1$, then $x^{\phi{(n)}}$ is congruent to $1$
      modulo $n$.

    \item[\theorem{11.6}] \textbf{Fermat's Little Theorem.} If $p$ is prime and
      if $x$ is not a multiple of $p$, then $x^{p-1}$ is congruent to $1$
      modulo $p$.
  \end{itemize}

  \section{Partitions}
  Let $X$ be a set and let $\mathscr{R}$ be a subset of the cartesian product
  $X\times X$. Given two points $x$ and $y$ of $X$, we say that $x$ is
  \textbf{\textit{related}} to $y$ if the ordered pair $(x,y)$ happen to lie in
  $\mathscr{R}$. If (a) each $x\in X$ is related to itself, (b) if $x$ is related
  to $y$, then $y$ is related to $x$, for any two points $x,y \in X$, (c) if
  $x$ is related to $y$ and if $y$ is related to $z$, then $x$ is related to
  $z$ for any three points $x,y,z \in X$ --- then we call $\mathscr{R}$ and
  \textbf{\textit{equivalence relation}} on $X$. For each $x\in X$ the
  collection of all points which are related to it is written $\mathscr{R}(x)$
  and called the \textbf{\textit{equivalence class}} of $x$.

  \paragraph{Theorems}
  \begin{itemize}[leftmargin=0.45in]
    \item[\theorem{12.1}] $\mathscr{R}(x) = \mathscr{R}(y)$ whenever $(x,y) \in
      \mathscr{R}$.
  \end{itemize}

\end{document}


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\input{../preamble.tex}

\title{\normalsize Solving the quantum master equation of coupled harmonic oscillators with Lie algebra methods}

\begin{document}
\frame{\titlepage}
\section{Introduction}
\begin{frame}[t]{Hilbert space $\mathcal{H}$ and Liouville space $\mathfrak{L}$}
    \begin{columns}[t]
        \begin{column}{0.5\textwidth}
            Hilbert space $\mathcal{H}$
            \begin{itemize}
                \item Density operator $\hat{\rho}$
                \item Hamiltonian $\hat{\mathscr{H}}$
                \item von Neumann equation :
                    \begin{equation}
                        \pdv{t}\hat{\rho} = -i[\hat{\mathscr{H}},\hat{\rho}]
                        \label{eq:von Neumann equation in Hilbert space}
                    \end{equation}
            \end{itemize}
        \end{column}
        \begin{column}{0.5\textwidth}
            \alert{Liouville space} $\mathfrak{L}$
            \begin{itemize}
                \item Superket $\lket{\rho}$
                \item Liouvillian $\hhat{\mathscr{L}}$
                \item von Neumann equation :
                    \begin{equation}
                        \pdv{t}\lket{\rho} = \hhat{\mathscr{L}}\lket{\rho}
                    \end{equation}
            \end{itemize}
        \end{column}
    \end{columns}
    $\Longrightarrow$ Eq. \eqref{eq:von Neumann equation in Hilbert space} as the problem in linear algebra.
    \begin{equation*}
        \vb*{x} = A\vb*{b}
    \end{equation*}
\end{frame}

\begin{frame}{Quantum master equation}
    Von Neumann equation : $\partial_t \lket{\rho} =  -i[\hat{\mathscr{H}},\hat{\rho}]$
    \begin{description}\footnotesize
        \item[$+$] Total Hamiltonian :
                    $\hat{H}_T = \hat{H} + \hat{H}_E + \hat{H}_I$.
        \item[$+$] Markovian interaction.
        \item[$+$] Initial state is separable :
                    $\hat{\rho}_T(0) \approx \hat{\rho}(0) \otimes \hat{\rho}_E(0)$.
        \item[$+$] Born approximation :
            $\hat{\rho}_T(t) \approx \hat{\rho}(t) \otimes \hat{\rho}_E(0)$.
        \item[$+$] Weak coupled regime : $\hat{H}_I \ll \hat{H}, \hat{H}_E$.
        \item[$+$] Rotating wave approximation.
    \end{description}\normalsize
    \alert{Quantum master equation}~\footfullcite{Manzano2020} :
    {\small($\hat{\mathcal{L}}_i$ is Lindblad jump operator)}
    \begin{equation}
        \pdv{t}\hat{\rho}(t)
        = -i[\hat{\mathscr{H}},\hat{\rho}] + \frac{1}{2}\sum_i \qty(
                2\hat{\mathcal{L}}_i \hat{\rho}(t) \hat{\mathcal{L}}_i^\dag
                    - \{\hat{\mathcal{L}}_i^\dag\hat{\mathcal{L}}_i,\hat{\rho}(t)\})
    \end{equation}
\end{frame}

\begin{frame}{Lie algebra $\mathfrak{g}$ and Lie braket $[\cdot,\cdot]$}
    \alert{Lie algebra} $\mathfrak{g}$ is vector space with \alert{Lie bracket} $[\cdot,\cdot]$.
    \begin{itemize}
        \item Alternativity : $[X,Y] = -[Y,X]$.
        \item Linearity : $[aX+bY,Z] = a[X,Z] + b[Y,Z]$.
        \item Jacobi identity : $[[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0$.
    \end{itemize}
    ex) Commutator $[X,Y] \coloneqq XY - YX$.

    Lie algebra is not commutative. \\
    $\Longrightarrow$ Left translation and right transition :
    \begin{equation}
        \hhat{L}[\hat{O}]\lket{A} \coloneqq \hat{O}\hat{A}
        \qc
        \hhat{R}[\hat{O}]\lket{A} \coloneqq \hat{A}\hat{O}
    \end{equation}
\end{frame}

\begin{frame}{A linear chain of $N$ coupled harmonic oscillators}
    Hamiltonian :
    \begin{equation}
        \hat{\mathscr{H}}
        = \sum_{k=1}^N \varepsilon_k \hat{a}_k^\dag \hat{a}_k
            + \sum_{k=1}^{N-1} \kappa_k (\hat{a}_k^\dag\hat{a}_{k+1}
                + \hat{a}_k\hat{a}_{k+1}^\dag)
    \end{equation}
    Quantum master equation :
    \begin{equation}
        \pdv{t}\hat{\rho}(t)
        = -i[\hat{\mathscr{H}},\hat{\rho}]
            + \sum_{k=1}^N \gamma_k (2\hat{a}_k\hat{\rho}\hat{a}_k^\dag
                - \{\hat{a}_k^\dag\hat{a}_k,\hat{\rho}\})
        \label{eq:quantum master equation for harmonic oscillators}
    \end{equation}
    where \quad\qquad $\hat{a}_k, \hat{a}_k^\dag$ is bosonic mode operators,
    \begin{equation*}
        \varepsilon_k : \text{energy}, \
        \kappa_k : \text{coupling constant}, \
        \gamma_k : \text{loss rate}
    \end{equation*}
\end{frame}

\section{Induced Lie algebra}
\begin{frame}{Effective non-Hermite Hamiltonian $\hat{\mathscr{H}}_\mathrm{eff}$}
    \alert{Effective non-Hermite Hamiltonian} $\hat{\mathscr{H}}_\mathrm{eff}$ :
    \begin{equation}
        -i(\hat{\mathscr{H}}_\mathrm{eff}\hat{\rho}
            - \hat{\rho}\hat{\mathscr{H}}_\mathrm{eff}^\dag)
        \coloneqq -i[\hat{\mathscr{H}},\hat{\rho}]
            - \sum_{k=1}^N \gamma_k \{\hat{a}_k^\dag\hat{a}_k,\hat{\rho}\}_+
    \end{equation}
    Eigenvalue (energy) :
    \begin{equation}
        \varepsilon_k \longrightarrow \varepsilon_k - i\gamma_k
    \end{equation}
\end{frame}

\begin{frame}{Bosonic mode superoperators}
    Bosonic mode superoperators :
    \begin{align}
        \hhat{L}_k^-\lket{A} &= \hat{a}_k\hat{A},
            & \hhat{L}_k^+\lket{A} &= \hat{a}_k^\dag\hat{A}, \\
        \hhat{R}_k^-\lket{A} &= \hat{A}\hat{a}_k,
            & \hhat{R}_k^+\lket{A} &= \hat{A}\hat{a}_k^\dag.
    \end{align}
    As well as $[\hat{a}_j,\hat{a}_k^\dag]=\delta_{jk}$,
    \begin{equation}
        [\hhat{L}_i^-,\hhat{L}_j^+] = \delta_{ij},
        \quad
        [\hhat{R}_i^-,\hhat{R}_j^+] = \delta_{ij}.
    \end{equation}
    The set
    \begin{equation}
        \alt<2>{
            \{\hhat{\mathbbm{1}}_\mathfrak{L},\hhat{L}_i^\pm,\hhat{R}_i^\pm,\hhat{\mathscr{L}}\}
        }{
            \{\hhat{\mathbbm{1}}_\mathfrak{L},\hhat{L}_i^\pm,\hhat{R}_i^\pm,\hhat{L}_j^+\hhat{L}_k^-,\hhat{R}_j^-\hhat{R}_k^+,\hhat{L}_j^-\hhat{R}_k^-\}
        }
    \end{equation}
    spans Lie algebra $\mathfrak{g}$.
\end{frame}

\begin{frame}{Liouvillian $\hhat{\mathscr{L}}$ for harmonic oscillators}
    Liouvillian (right action) :
    \small
    \begin{align}
        \hhat{\mathscr{L}}
        &= \sum_{k=1}^N \qty[(i\varepsilon_k - \gamma_k)\hhat{R}_k^+\hhat{R}_k^-
                - (i\varepsilon_k + \gamma_k) \hhat{L}_k^+\hhat{L}_k^-
                    + 2\gamma_k\hhat{L}_k^-\hhat{R}_k^-] \nonumber \\
        &\qquad + \sum_{k=1}^N i\kappa_k \qty(\hhat{L}_k^+ \hhat{L}_{k+1}^-
                + \hhat{L}_{k+1}^+\hhat{L}_k^- - \hhat{R}_{k+1}^+\hhat{R}_k^-
                    - \hhat{R}_k^+\hhat{R}_{k+1}^-)
    \end{align}
    \normalsize
    \alert{Time evolution superoperator} :
    \begin{equation}
        \pdv{t} \hhat{\mathcal{U}}(t) = \hhat{\mathscr{L}}\hhat{\mathcal{U}}(t)
    \end{equation}
    where $\hhat{\mathcal{U}}(0) = \hhat{\mathbbm{1}}_\mathfrak{L}$.
\end{frame}

\begin{frame}{Approaches to solve master equation}
    Two approaches to solve master equation.
    \begin{enumerate}
        \item Eigendecomposition (time-independent)
        \item Wei-Norman expansion (time-dependent)
    \end{enumerate}
\end{frame}

\section{Eigendecomposition}
\begin{frame}{Regular representation in Lie algebra}
    Regular representation $\hhat{\mathcal{R}}(Z)$ :
    \begin{equation}
        [\hhat{Z},\hhat{X}_i] = \hhat{\mathcal{R}}_{ij}(\hhat{Z}) \hhat{X}_j.
    \end{equation}
    Interested in $\hhat{\mathcal{R}}(\hhat{\mathscr{L}})$,
    \begin{equation}
        \{\hhat{\mathbbm{1}}_\mathfrak{L},\hhat{L}_i^\pm,\hhat{R}_i^\pm,\hhat{\mathscr{L}}\}
            \longrightarrow \{\hhat{L}_i^\pm,\hhat{R}_i^\pm\}
    \end{equation}
    Eigenvalues of $\hhat{\mathcal{R}}(\hhat{\mathscr{L}})$ :
    \begin{equation}
        \{\pm\lambda_1,\ldots,\pm\lambda_N,\pm\lambda_1^*,\ldots,\pm\lambda_N^*\}
    \end{equation}
    where $\lambda=(\lambda_1,\ldots,\lambda_N)$ are the eigenvalues of $\hhat{\mathscr{H}}_\mathrm{eff}$.
\end{frame}

\begin{frame}{Collective creation and annihilation operators}
    Eigenvectors of $\hhat{\mathcal{R}}(\hhat{\mathscr{L}})$ :
    \begin{align}
        \hhat{P}_i^+ &= \sum_k c_{ik}(\hhat{L}_k^+ - \hhat{R}_k^-),
            & \hhat{P}_i^- &= \sum_k c_{ik}\hhat{L}_k^-, \\
        \hhat{Q}_i^+ &= \sum_k c_{ik}(\hhat{R}_k^+ - \hhat{L}_k^-),
            & \hhat{Q}_i^- &= \sum_k c^*_{ik}\hhat{R}_k^-.
    \end{align}
    $\Longrightarrow$ Collective creation and annihilation operators :
    \begin{gather}
        [\hhat{P}_i^-,\hhat{P}_j^+] = \delta_{ij},
        \quad
        [\hhat{Q}_i^-,\hhat{Q}_j^+] = \delta_{ij} \\
        \begin{aligned}
            [\hhat{\mathcal{L}},\hhat{P}_i^+] &= \lambda_i\hhat{P}_i^+
                & [\hhat{\mathcal{L}},\hhat{P}_i^-] &= -\lambda_i\hhat{P}_i^-  \\
            [\hhat{\mathcal{L}},\hhat{Q}_i^+] &= \lambda_i^*\hhat{Q}_i^+
                & [\hhat{\mathcal{L}},\hhat{Q}_i^-] &= -\lambda_i^*\hhat{Q}_i^-
        \end{aligned}
    \end{gather}
\end{frame}

\begin{frame}{Ground state and ladder state}
    Ground state in $\mathfrak{L}$ :
    \begin{gather}
        \hhat{P}_i^-\lket{0} = \hhat{Q}_i^-\lket{0} = 0_\mathcal{H} \\
        \hhat{P}_i^+ \lbra{0} = \hhat{Q}_i^+\lbra{0} = 0_\mathcal{H}
    \end{gather}
    Higher rungs of the ladder :
    \begin{align}
        \lket{\alpha,\beta} &= \frac{1}{\sqrt{\alpha!\beta!}}
            \hhat{\vb*{P}}^{+\alpha} \hhat{\vb*{Q}}^{+\beta}\lket{0} \\
        \lbra{\alpha,\beta} &= \frac{1}{\sqrt{\alpha!\beta!}}
            \lbra{0}\!\hhat{\vb*{P}}^{+\alpha} \hhat{\vb*{Q}}^{+\beta}
    \end{align}
    where $\alpha=(\alpha_1,\ldots,\alpha_N), \beta=(\beta_1,\ldots,\beta_N)$.
\end{frame}

\begin{frame}{Collective creation and annihilation operators}
    Baker–Campbell–Hausdorff formula :
    \begin{equation}
        e^{A}Be^{-A} = \sum_{n=0}^\infty\frac{1}{n!}
            \underbrace{[A,[A,\cdots,[A}_{n},B]\cdots]]
    \end{equation}

    Ladder state :
    \begin{equation}
        e^{\mathscr{L}t} \lket{\alpha,\beta}
            = e^{(\alpha\cdot\lambda + \beta\cdot\lambda^*)t}
                \lket{\alpha,\beta}
    \end{equation}

    Density superket $\lket{\rho}$ evoluted as
    \begin{equation}
        \lket{\rho(t)} = \sum_{\alpha,\beta} e^{(\alpha\cdot\lambda + \beta\cdot\lambda^*)t}
                        \lket{\alpha,\beta}\lbraket{\alpha,\beta}{\rho_0}.
    \end{equation}
\end{frame}

\section{Wei-Norman expansion}
\begin{frame}{Collective creation and annihilation operators}
    Expansion of $\hhat{\mathscr{L}}$ by basis :
    \begin{equation}
        \hhat{\mathscr{L}}(t) = \sum_{k=1}^m c_k(t) \hhat{X}_k.
    \end{equation}

    Time evolution superoperator :
    \begin{equation}
        \hhat{\mathcal{U}}(t) = \prod_k^n \hhat{\mathcal{U}}_k(t)
                              = \prod_k^n \exp g_k(t)\hhat{X}_k
    \end{equation}
    where
    \begin{equation}
        \pdv{t}\hhat{\mathcal{U}}_k(t) =  g_k(t)\hhat{X}_k \hhat{\mathcal{U}}_k(t)
        \qc
        \hhat{\mathcal{U}}_k(0) = \hhat{\mathbbm{1}}_\mathfrak{L}.
    \end{equation}
\end{frame}

\begin{frame}{Collective creation and annihilation operators}
    Liouvillian is split as :
    \begin{equation}
        \hhat{\mathscr{L}} = \underbrace{\hhat{\mathscr{L}}_S}_\text{solvable}
                                + \underbrace{\hhat{\mathscr{L}}_R}_\text{remaining}
    \end{equation}

    Time evolution superoperator :
    \begin{equation}
        \pdv{t}\hhat{\mathcal{U}}_S(t) =  \hhat{\mathscr{L}}_S \hhat{\mathcal{U}}_S(t)
        \qc
        \pdv{t}\hhat{\mathcal{U}}_R(t)
            =  (\hhat{\mathcal{U}}_S^{-1}\hhat{\mathscr{L}}_R\hhat{\mathcal{U}}_S)
                \hhat{\mathcal{U}}_R(t)
    \end{equation}
    However, $\hhat{\mathcal{U}}_R(t)$ is not always solvable.
\end{frame}
\end{document}


\chapter{Conclusion}
\label{ch_conclusion}

% \markright{Conclusion}

We have presented recent methods for restoration of spherical data with noise following a Poisson distribution. A denoising method was proposed, which used a variance stabilization method and multiscale transforms on the sphere.
Experiments have shown it is very efficient for the denoising of  astrophysical data set such as Fermi data.  Two spherical multiscale transforms, the wavelet and the curvelets, were used.
Then, we have described an extension of the denoising method in order to take into account missing data, and we have shown that this inpainting method
could be a useful tool to estimate the diffuse emission.
Then, we have introduced a new denoising method the sphere which takes into account a background model. The simulated data have shown that it is relatively robust to
errors in the model, and can therefore be used for diffuse background modeling and source detection.
Finally, an extension to multichannel denoising and deconvolution has been proposed, which proved very efficient on simulated data.


\section{Appendix}

\subsection{Model documentation}

\subsubsection{Overview}
The French power model FLORE is a dispatch and investment model of the French power mix. It is a partial equilibrium model of the wholesale electricity market, which determines optimal investment and hourly generation.

FLORE minimizes total cost with respect to investment and production under a set of constraints. The model is linear, deterministic, and solved in hourly time step for each year, from 2014 to 2050.

Generation is modelled as twelve distinct technologies: three VRE with zero marginal costs - onshore wind, offshore wind and solar, three fossil-based thermal technologies - coal, combined cycle gas turbine (CCGT), open cycle gas turbine (OCGT), three nuclear technologies (historical nuclear, retrofitted nuclear and new nuclear), and three hydro systems - run-of-river, conventional dams (lakes) and pumped storage.
Hourly VRE generation is limited by specific generation profiles.
Dispatchable power plants produce whenever the price is above their variable costs, and unless they are limited by their ramping constraints.
Hydro storage and dispatch is optimized by the model, under water reservoir constraints and pumping losses.
At the end its lifetime, each capacity is decommissioned.

A particularity of the model is to represent explicitly the cost of extending the lifetime of nuclear power plants. Nuclear plants normally close after 40 years, but their lifetime can be extended to 60 years, if upgrade costs are paid for.

Demand is exogenous and assumed to be perfectly price inelastic at all times.

FLORE is currently calibrated for France. Exports are given exogenously.

\subsubsection{Total System Costs}

The model minimizes total system costs $C$ with respect to several constraints and decision variables. Total system costs are the sum of fixed costs (representing the sum of capital and fixed O\&M costs), and variable costs, over all hours $h$, weeks $w$, years $y$ and generation technologies $tec$.

Fixed costs are the product of the capacity installed $CAPA_{tec,y}$ for each technology in year $y$, by its annualized capacity cost $c_{tec}^{inv}$ plus fixed O\&M costs $c_{tec}^{qfix}$.
Variable generation costs are the product of $GENE_{tec,w,y,h}$, the generation from technology $tec$ in year $y$, week $w$ and hour $h$, by variable generation costs $c_{tec}^{var}$.
\begin{equation}
C =\sum_{tec,y} CAPA_{tec,y} \cdot ( c_{tec}^{inv} + c_{tec}^{qfix} ) + \sum_{tec,y,w,h} GENE_{tec,y,w,h} \cdot c_{tec}^{var}
\end{equation}
Throughout this model description, capital letters denote choice variable for the model, while lowercase letters denote exogenous data.

\subsubsection{Supply and demand}

Power balance between supply and demand is the central constraint of the model. Demand is the sum of national load $d$, pumping for pumped-storage hydro $PUMP$ and exports flow $flows$.
Supply is the sum of generation over all technologies.
$$\forall h,y \quad \sum_{tec} GENE(tec,h,y) \geq \sum_{tec} LOAD(h,y) + PUMP(h,y) + flow(h,y)$$

In this framework, demand is perfectly price-inelastic. Cost minimization is thus equivalent to welfare-maximization.

Generation is constrained by capacities installed and the load factor (also called availability), for all technologies except wind, PV and run-of-river hydro, which present a specific generation profile.
$$GENE(tec\_noprof,y,w,h) \leq CAPA(tec\_noprof,y)*loadfactor(tec\_noprof)$$
For wind and PV, a generation profile is an additional constraint:
$$GENE(tec\_res,y,w,h) \leq CAPA(tec\_res,y)*res\_profiles(h,tec\_res)$$
For run-of-river hydro, generation is fixed exogenously for each week:
$$GENE("river",y,w,h) = river\_flow(w)$$

Investment is exogenous for historical nuclear and the three hydro technologies. The capacity of historical nuclear technology is decreasing based on plant-level data, with each plant capacity being phased-out when it reaches 40 years old. For hydro, we assume constant capacities for run-of-river and lakes. Pumped-hydro potential is expected to grow by 3.2GW by 2050, starting from 5 GW in 2014, with a discharge time of 20 hours, as in ADEME (Visions ADEME 2030).
$$CAPA_{tec\_ex,y} = capa_exo_{y,tec\_ex} $$

The capacity of all other technologies is determined endogenously by the model. Capacity is installed when optimal, and then decommissioned when it reaches its lifetime. The initial capacity, already installed in 2014, is supposed to be decommissioned linearly from 2014 onwards in a period of 40 years.
\begin{align*}
CAPA_{tec\_end,y+1} &= CAPA_{tec\_end,y} + INVE_{tec\_end,y} - DECO\_{tec\_end,y} \\
DECO_{tec\_end,y}  &= INVE_{tec\_end,y-lifetime(tec\_end)} + (tec\_data_{tec\_end,'initcap'}/40)\$(ord(y) <= 40)
\end{align*}

\subsubsection{Power System Inflexibilities}

The model includes ramp-up and ramp-down constraints for nuclear and coal power plants, to represent the fact that generation cannot vary too quickly from one hour to the next. These constraints are modelled as a maximum variation rate:
The ramp up constraint is:
$$\forall tec\_ramp,y,w,h \quad GENE(tec\_ramp,y,w,h+1) \leq GENE(tec\_ramp;y;w;h) \dot (1+ramp\_rate)$$
The ramp-down constraint is:
$$\forall tec\_ramp,y,w,h \quad GENE(tec\_ramp,y,w,h+1) \geq GENE(tec\_ramp;y;w;h) \dot (1-ramp\_rate)$$
We take the value of $0.08$ for coal and $0.05$ for nuclear - for both ramp-up and ramp-down, as in \citet{ADEME2015}.

\subsubsection{Model limitation}

This model was designed to represent the French power system from 2014 to 2050, optimizing investment and dispatch over many assumptions on nuclear costs. As such, it had to make several simplifying assumptions.

This model does not provide a detailed representation of infra-hourly events. In particular, ancillary services are not represented.

We do not model power plants at the plant level, but we rather use representative technologies. Thus, one limitation is the absence of constraints related to unit commitment, such as start-up cost and delay, minimum load, and part-load efficiency.
However, this default is in part compensated for nuclear. We use real data on age limit (or retirement), so the model retrofits plants whose capacity is in line with plant-level data. On the contrary, new nuclear will be installed continuously, while in reality it composed of discrete power plants with a significant size (1.3 GW for the EPR technology).
For renewable, the representative technology assumption is less of an issue as these technologies are modular, and very small in size. The continuous approach is here a good assumption for costs. However, this approach does not enable one to distinguish the different potentials within a country, both for solar and wind.

%%%%%%%%%%%%%%%%

\clearpage

\subsubsection{Model Calibration}

\label{app:calibration}

The model is calibrated using capital cost, fixed O\&M and variable O\&M costs. All payments are annualized with a 5\% interest rate.

%The corresponding LCOE with usual load factor are given in the Figure \ref{fig:calibration_LCOE}.
%\begin{figure}[!ht]
%	\centering
%	\includegraphics[width=12cm]{figures/calibration_LCOE.png}
%	\caption{LCOE computed from input data of the model}
%	\label{fig:calibration_LCOE}
%\end{figure}

\begin{table}[!ht]
	%\colorTable
	\centering
	\caption{Main model parameters}
	\begin{tabular}{llr}
		\toprule
		parameter&value&unit\\
		\midrule
		Rate of pure time preference &0&\%\\
		Financial interest rate&5&\%\\
		\bottomrule
	\end{tabular}
\end{table}

\begin{table}[!ht]
	\centering
	\caption{Investment costs (euros/kW)}
	\label{tab:Investment _costs}
	\begin{adjustbox}{width=1\textwidth}
	\small
	\begin{tabular}{llllllllllll}
		\toprule
		 Year & onshore & PV & Coal & CCGT & TAC & river & lake & pump  \\
		\midrule
		2014 & 1650 & 1560 & 1500 & 800 & 400 & 3000 & 2000 & 2000  \\
		2020 & 1572 & 1194 & 1500 & 800 & 400 & 3000 & 2000 & 2000 \\
		2030 & 1443 & 869 & 1500 & 800 & 400 & 3000 & 2000 & 2000  \\
		2040 & 1313 & 735 & 1500 & 800 & 400 & 3000 & 2000 & 2000 \\
		2050 & 1184 & 600 & 1500 & 800 & 400 & 3000 & 2000 & 2000 \\
		\bottomrule
	\end{tabular}
\end{adjustbox}
\end{table}

\begin{table}[!ht]
	\centering
	\caption{Fixed O\&M costs (euro/MWh)}
	\label{tab:OM_costs}
		\begin{adjustbox}{width=1\textwidth}
	\small
	\begin{tabular}{llllllllllll}
		\toprule
		Year & onshore & PV & Coal & CCGT & TAC & river & lake & pump & nuc\_hist & nuc\_renov & nuc\_new \\
		\midrule
		2014 & 35 & 25 & 30 & 20 & 15 & 60 & 60 & 20 & 188 & 188 & 100 \\
		2020 & 35 & 25 & 30 & 20 & 15 & 60 & 60 & 20 & 188 & 188 & 100 \\
		2030 & 35 & 25 & 30 & 20 & 15 & 60 & 60 & 20 & 188 & 188 & 100 \\
		2040 & 35 & 25 & 30 & 20 & 15 & 60 & 60 & 20 & 188 & 188 & 100 \\
		2050 & 35 & 25 & 30 & 20 & 15 & 60 & 60 & 20 & 188 & 188 & 100 \\
		\bottomrule
	\end{tabular}
\end{adjustbox}
\end{table}

\begin{table}
	\centering
	\caption{\coo\ price}
	\label{tab:CO2_price}
	\begin{tabular}{llll}
		\toprule
		Year & \coo\ price & & \\
		\midrule
		& Low & High \\
		2014 & 21 & 42 \\
		2020 & 28 & 56 \\
		2030 & 50 & 100 \\
		2040 & 75 & 150 \\
		2050 & 100 & 200 \\
		\bottomrule
	\end{tabular}
\end{table}

\begin{table}[!ht]
	\centering
	\caption{Fuel prices}
	\label{tab:Fuel_prices}
	\begin{tabular}{llll}
		\toprule
		euro/MWh & Coal & Gas & Uranium \\
		\midrule
		2014 & 10,3 & 25,3 & 7,30 \\
		\bottomrule
	\end{tabular}
\end{table}

\begin{table}
	\centering
	\caption{Efficiencies}
	\label{tab:Efficiencies}
	\begin{tabular}{lll}
		\toprule
		Coal & CCGT & TAC \\
		\midrule
		0,43 & 0,61 & 0,410 \\
		\bottomrule
	\end{tabular}
\end{table}

%%%%%%%%%%%%%%%%%%
\clearpage
\subsection{Additional material}

\subsubsection{Plausible costs of retrofitted plants}

\begin{table}[!htp]
	\centering
	\caption{Historical values of French nuclear plants in the literature, in the best case}
	\label{tab:historical_costs}
	\begin{tabular}{llll}
		\toprule
		Item & Cost & Unit & Source \\
		\midrule
		Historical OPEX (excluding fuel) & 13.3 & euro/MWh & Boccard, 2014 \\
		Fuel & 5.7 & euro/MWh & Cour des Comptes, 2014, p. 13 \\
		Waste & 3.6 &  & Cour des Comptes, 2014, p. 24 \\
		Decommissioning & 1 &  & Cour des Comptes, 2014, p. 24 \\
		Capital rate & 5 & \% &  \\
		Lifetime extension & 20 & years & \\
		availability & 83.5 & \% & \\
		\bottomrule
	\end{tabular}
\end{table}

\begin{table}[!htp]
	\centering
	\caption{Retrofit costs according to EDF}
	\label{tab:costs_grand_carenage}
	\begin{tabular}{lll}
		\toprule
		Additional CAPEX & 74730 & million euros \\
		\midrule
		Additional OPEX & 25160 & million euros \\
		Capacity concerned & 53200 & MW \\
		\bottomrule
	\end{tabular}
\end{table}

\begin{table}[!htp]
	\centering
	\caption{Sensitivity tests}
	\label{tab:sensitivity_tests}
	\begin{tabular}{lll}
		\toprule
		CAPEX & 30 & \% \\
		\midrule
		Historical OPEX & 27.5 & euro/MWh \\
		Capital rate & 10 & \% \\
		Lifetime extension & 10 & years \\
		Availability & 70 & \% \\
		Waste & x2 &  \\
		Decommissioning & x2 &  \\
		Insurance & 8.5 & euro/MWh \\
		\bottomrule
	\end{tabular}
\end{table}

\clearpage

\subsubsection{Strategies examined}
\label{sec:strategies_examined}

\begin{figure}[!htp]
	\centering
	\includegraphics[height=7cm]{figures/strategies.png}
	\caption{Nuclear capacity of existing plants, including retrofit, in the 27 strategies}
	\label{fig:strategies}
\end{figure}

\clearpage

The first period includes all reactors that reach 40 years before 2021, which represents 14 reactors. The second period includes all the reactors reaching 40 years between 2021 and 2025, which represent 23 additional reactors. The final period goes up to 2050 and represents the remaining 21 reactors.

\begin{table}[!hb]
	\centering
	\caption{Share of retrofitted nuclear plants in each period}
	\label{tab:shareNukeRetrofit}
	\small
	\begin{tabular}{llll}
		\toprule
		Strategy & Period1 & Period2 & Period3 \\
		\midrule
		S1 & 0\% & 0\% & 0\% \\
		S2 & 0\% & 0\% & 50\% \\
		S3 & 0\% & 0\% & 100\% \\
		S4 & 0\% & 50\% & 0\% \\
		S5 & 0\% & 50\% & 50\% \\
		S6 & 0\% & 50\% & 100\% \\
		S7 & 0\% & 100\% & 0\% \\
		S8 & 0\% & 100\% & 50\% \\
		S9 & 0\% & 100\% & 100\% \\
		S10 & 50\% & 0\% & 0\% \\
		S11 & 50\% & 0\% & 50\% \\
		S12 & 50\% & 0\% & 100\% \\
		S13 & 50\% & 50\% & 0\% \\
		S14 & 50\% & 50\% & 50\% \\
		S15 & 50\% & 50\% & 100\% \\
		S16 & 50\% & 100\% & 0\% \\
		S17 & 50\% & 100\% & 50\% \\
		S18 & 50\% & 100\% & 100\% \\
		S19 & 100\% & 0\% & 0\% \\
		S20 & 100\% & 0\% & 50\% \\
		S21 & 100\% & 0\% & 100\% \\
		S22 & 100\% & 50\% & 0\% \\
		S23 & 100\% & 50\% & 50\% \\
		S24 & 100\% & 50\% & 100\% \\
		S25 & 100\% & 100\% & 0\% \\
		S26 & 100\% & 100\% & 50\% \\
		S27 & 100\% & 100\% & 100\% \\
		\bottomrule
	\end{tabular}
\end{table}

\clearpage

\subsubsection{Optimum power mixes}
\label{app:optimumMixes}

\begin{figure}[!htp]
	\centering
	\subfloat[Optimum for retrofitted nuclear below or equal to 40 \emwh] {
		\includegraphics[height=6.5cm]{figures/powerMix_R40N100MedD.png}
		\label{fig:nukeShare40}
	}
	\subfloat[Optimum for retrofitted nuclear around 70 \emwh] {
		\includegraphics[height=6.5cm]{figures/powerMix_R70N100MedD.png}
		\label{fig:nukeShare70}
	}
\end{figure}

\begin{figure}[!htp]
	\centering
		\subfloat[Optimum for retrofitted nuclear around 80 \emwh] {
		\includegraphics[height=6.5cm]{figures/powerMix_R80N100MedD.png}
		\label{fig:nukeShare80}
	}
	\subfloat[Optimum for retrofitted nuclear around 90 \emwh] {
		\includegraphics[height=6.5cm]{figures/powerMix_R90N100MedD.png}
		\label{fig:nukeShare90}
	}
	\caption{Optimum power mix for various nuclear retrofitting costs}
\end{figure}

\begin{figure}[!ht]
	\centering
	\includegraphics[height=8cm]{figures/powerMixSensitivity.png}
	\caption{Optimal nuclear shares and sensitivity analysis}
	\label{fig:powerMixSensitivity}
\end{figure}

%%%%%%%%%%%%

\clearpage

\subsubsection{Analysis with PRIM}

%\paragraph{Choice of threshold}

\begin{figure}[!htp]
	\centering
	\includegraphics[width=6cm]{figures/quantile_regret.pdf}
	\caption{Ordered distribution of regret for the three candidate strategies S9, S24 and S18}
	\label{fig:quantile_regret}
\end{figure}

%\paragraph{Efficiency frontier}

\begin{figure}[!htp]
	\centering
	\subfloat[With the cluster from S9] {
		\includegraphics[width=6cm]{figures/low_regret_frontier_S9.pdf}
	}
\subfloat[With the cluster from S18 and S24] {
		\includegraphics[width=6cm]{figures/low_regret_frontier_S18.pdf}
}
	\caption{Trade-offs and efficiency fontier using the PRIM-generated cluster from strategy S9 and strategy S18 and S24}
	\label{fig_app:low_regret_frontier}
\end{figure}

%\paragraph{Regrets of the strategies S9, S18 and S27}
\begin{figure}[!ht]
	\centering
	\subfloat[Regret of the full-retrofit strategy S27] {
		\makebox[5.5cm][c]{\includegraphics[height=5cm]{figures/vulnerabilities_S27.pdf}}
		\label{fig_app:vulnerabilities_S27}
	}

	\subfloat[Regret of the full-retrofit strategy S9] {
		\makebox[5.5cm][c]{\includegraphics[height=5cm]{figures/vulnerabilities_S9.pdf}}
	} \qquad
	\subfloat[Regret of the full-retrofit strategy S18]{
		\makebox[6cm][c]{\includegraphics[height=5cm]{figures/vulnerabilities_S18.pdf}}
		\label{fig_app:vulnerabilities_S9_S18_S27}
	}
	\caption{Regret of the full-retrofit strategy S9, S18 and S27}
\end{figure}
%\paragraph{Optimal trajectory depending on implicit probabilities}
\begin{figure}[!h]
	\centering
	\subfloat[With the cluster from strategy S9] {
		\includegraphics[width=6cm]{figures/odds_S9.pdf}
	}
	\subfloat[With the cluster from strategy S18 and S24] {
		\includegraphics[width=6cm]{figures/odds_S18.pdf}
	}
	\caption{Optimal strategy depending on the implicit odds of the decision maker}
	\label{fig_app:odds}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
\clearpage
\subsubsection{Comparison with official scenarios}

\begin{figure}[!htp]
	\centering
	\subfloat[Nuclear share (\%)] {
		\includegraphics[height=6cm]{figures/dnte_nuc_shares.pdf}
	}
\subfloat[Demand levels (TWh/year)] {
		\includegraphics[height=6cm]{figures/dnte_demands.pdf}
	}
	\caption{Characteristics of the four official scenarios in the French national debate \\Source: \citet{DNTE_gt2} and authors' calculations}
	\label{fig:DNTE_scenarios}
\end{figure}


\documentclass[11pt]{amsart}
\usepackage{geometry}
\geometry{letterpaper}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{siunitx}
\usepackage{tikz}
\usepackage{lscape}
\usetikzlibrary{dsp,fit,positioning}

\input{../util/airfoils.tex}
\input{../util/wing.tex}
\input{../util/coordinate_systems.tex}
\input{../util/control.tex}

\newcommand{\qhat}{\hat{q}}
\newcommand{\cbar}{\bar{c}}
\newcommand{\qbar}{\bar{q}}
\newcommand{\cmd}{\mathrm{cmd}}
\newcommand{\ff}{\mathrm{ff}}
\newcommand{\eff}{\mathrm{eff}}
\newcommand{\app}{\mathrm{app}}
\newcommand{\wind}{\mathrm{wind}}
\newcommand{\kite}{\mathrm{kite}}
\newcommand{\nom}{\mathrm{nom}}
\newcommand{\aero}{\mathrm{aero}}
\newcommand{\geom}{\mathrm{geom}}

\begin{document}

\section{Servo model}
Expressing everything in terms of shaft angle $\theta$, which is $\theta_m / g$.
\begin{eqnarray}
\dot{\theta}_{\mathrm{ref}} &=& \omega_{\mathrm{ref}}
(\theta_{\mathrm{cmd}} - \theta_{\mathrm{ref}}) \\
v &=& k_p (\theta_{\mathrm{ref}} - \theta) +
k_d (\dot{\theta}_{\mathrm{ref}} - \dot{\theta}) \\
i &=& (v - k_{\tau} g \dot{\theta}) / Z \\
I_{\mathrm{servo}}\, g \ddot{\theta} &=& k_{\tau} i - b \dot{\theta} / g +
\tau_{\mathrm{ext}} / g \\
\tau_{\mathrm{ext}} &=& \frac{1}{2} \rho v^2 A c C_{m_{\delta_i}} \theta
\end{eqnarray}
$\omega_{\mathrm{ref}} = 100$ rad/s, $k_p = 1000$ V/rad, $k_d = 100$
V/rad-s, $k_{\tau} = 0.239$ N-m/A, $g = 160$,
$I_{\mathrm{servo}}= 2.3 \times 10^{-4}$ kg-m$^2$.
$Z = \sqrt{R^2 + (g \dot{\theta} L)^2}$.
$R = 0.886$.  The friction constant $b$ is between 70 and 170
N-m/(rad/s) of shaft velocity.  For the ailerons, $C_{m_{\delta_a}} =
0.0026$.

\begin{equation}
\frac{\Theta_{\mathrm{ref}}(s)}{\Theta_{\mathrm{cmd}}(s)} =
\frac{\omega_{\mathrm{ref}}}{s + \omega_{\mathrm{ref}}}
\end{equation}

\begin{equation}
V(s) = (k_p + k_d s) (\Theta_{\mathrm{ref}}(s) - \Theta(s))
\end{equation}

\begin{equation}
I_{\mathrm{servo}}\, g s^2 \Theta(s) =
k_{\tau} (k_p + k_d s) (\Theta_{\mathrm{ref}}(s) - \Theta(s)) / Z -
k_{\tau}^2 g s \Theta(s) / Z
\end{equation}

\begin{equation}
\frac{\Theta(s)}{\Theta_{\mathrm{cmd}}(s)} =
\frac{\omega_{\mathrm{ref}}}{s + \omega_{\mathrm{ref}}}
\cdot
\frac{k_{\tau} (k_p + k_d s)}
{g Z I_{\mathrm{servo}} s^2 + (k_{\tau} k_d + k_{\tau}^2 g) s +
(k_{\tau} k_p - \frac{Z \bar{q} A c}{g} C_{m_{\delta_i}})}
\end{equation}

\end{document}


\documentclass{article}
\usepackage{amsfonts}
\usepackage[operators]{cryptocode}
\usepackage[utf8]{inputenc}

\title{Introduction of the committed encryption}
\author{Weikeng Chen\\
\vspace{0.12em}
for CS294-144 Blockchain course project}
\date{May 2018}

\begin{document}

\maketitle

\section{Motivation}

We want to encrypt one message to several individual parties. With public-key encryption under a group $\mathbb{G}$ and the group generator $g\in\mathbb{G}$, we assume there are three parties, and their public/private key pairs are $(g^{x_1},x_1)$,  $(g^{x_2},x_2)$, and $(g^{x_3},x_3)$, respectively, we can encrypt a message $m$ as follows:
\[
\begin{array}{l}
r_1\sample[0,...,|\mathbb{G}|-1],~r_2\sample[0,...,|\mathbb{G}|-1],~r_3\sample[0,...,|\mathbb{G}|-1],\\
k\sample\{0,1\}^\lambda,\\
c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1})\oplus k,\\
c_2=g^{r_2}\parallel\mathsf{KDF}(g^{x_2r_2})\oplus k,\\
c_3=g^{r_3}\parallel\mathsf{KDF}(g^{x_3r_3})\oplus k,\\
c_4=\mathsf{AuthEnc}.\mathsf{Enc}_k(\mathsf{msg}).
\end{array}
\]
where $\mathsf{KDF}(\cdot)$ is a key derivation function and $\mathsf{AuthEnc}$ refers to an authenticated encryption scheme.

For honest encryption, all three parties can decrypt the message using their secret key. However, if the encryption is not honest, different parties may see different messages, we formalize as follows:
\[
\begin{array}{l}
r_1\sample[0,...,|\mathbb{G}|-1],~r_2\sample[0,...,|\mathbb{G}|-1],~r_3\sample[0,...,|\mathbb{G}|-1],\\
k\sample\{0,1\}^\lambda,\\
c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1})\oplus k,\\
c_2=g^{r_2}\parallel\mathsf{KDF}(g^{x_2r_2})\oplus k~\boxed{\oplus\Delta_2},\\
c_3=g^{r_3}\parallel\mathsf{KDF}(g^{x_3r_3})\oplus k~\boxed{\oplus\Delta_3},\\
c_4=\mathsf{AuthEnc}.\mathsf{Enc}_k(\mathsf{msg}).
\end{array}
\]
in which the party 2 obtains $\mathsf{k}\oplus\Delta_2$ and the party 3 obtains $\mathsf{k}\oplus\Delta_3$ instead of $\mathsf{k}$. Note that the party 2's key $\mathsf{k}\oplus\Delta_2$ may decrypt $c_4$ into non-$\perp$ result, as the authenticity of authenticated encryption does not necessarily protect against malicious encryption.

We want to provide a multi-recipient encryption that satisfies the following properties: (1) semantic security; (2) party 1 can detect whether $c_2$ and $c_3$ is generated honestly. Therefore, a malicious encryption cam be detected by the party 1 without knowing the secret keys of the party 2 and the party 3.

\section{Our construction, informally}
We encrypt the message as follows with a pseudorandom function:
\[
\begin{array}{l}
s\sample\{0,1\}^\lambda,~k=\mathsf{PRF}_s(1),\\
r_1\sample[0,...,|\mathbb{G}|-1],\\
\qquad c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1})\oplus s,\\
r_2=\mathsf{PRF}_s(2)\mod{|\mathbb{G}|},\\
\qquad c_2=g^{r_2}\parallel\mathsf{KDF}(g^{x_2r_2})\oplus k,\\
r_3=\mathsf{PRF}_s(3)\mod{|\mathbb{G}|},\\
\qquad c_3=g^{r_3}\parallel\mathsf{KDF}(g^{x_3r_3})\oplus k,\\
c_4=\mathsf{AuthEnc}.\mathsf{Enc}_k(\mathsf{msg}).
\end{array}
\]
assumed the Hash Diffie-Hellman (HDH) assumption holds on the group $\mathbb{G}$.

\smallskip
\noindent\textbf{Semantic security.} We can prove the semantic security of this construction with the random oracle heuristic.

By hybrid arguments, we have the following proof. We assume the distinguisher has all the public keys, but does not have any private keys of the three parties.

\smallskip
\noindent $\mathbf{H}_0$: the honest generation.

\smallskip
\noindent $\mathbf{H}_1$: in this hybrid, we change the $s$ in $c_1$ into $0^\lambda$:
\[
\begin{array}{l}
s\sample\{0,1\}^\lambda,~k=\mathsf{PRF}_s(1),\\
r_1\sample[0,...,|\mathbb{G}|-1],\\
\qquad \boxed{c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1}),}\\
r_2=\mathsf{PRF}_s(2)\mod{|\mathbb{G}|},\\
\qquad c_2=g^{r_2}\parallel\mathsf{KDF}(g^{x_2r_2})\oplus k,\\
r_3=\mathsf{PRF}_s(3)\mod{|\mathbb{G}|},\\
\qquad c_3=g^{r_3}\parallel\mathsf{KDF}(g^{x_3r_3})\oplus k,\\
c_4=\mathsf{AuthEnc}.\mathsf{Enc}_k(\mathsf{msg}).
\end{array}
\]

This hybrid is computationally indistinguishable with $\mathbf{H}_0$ because $\mathsf{KDF}(g^{x_1r_1})$ is computationally indistinguishable to a random $\lambda$-bit string assuming HDH.

\smallskip
\noindent $\mathbf{H}_2$: in this hybrid, we change $k$, $r_2$, and $r_3$ to randomly sampled values and discard $s$:

\[
\begin{array}{l}
\boxed{k\sample\{0,1\}^\lambda,}\\
r_1\sample[0,...,|\mathbb{G}|-1],\\
\qquad c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1}),\\
\boxed{r_2\sample[0,...,|\mathbb{G}|-1],}\\
\qquad c_2=g^{r_2}\parallel\mathsf{KDF}(g^{x_2r_2})\oplus k,\\
\boxed{r_3\sample[0,...,|\mathbb{G}|-1],}\\
\qquad c_3=g^{r_3}\parallel\mathsf{KDF}(g^{x_3r_3})\oplus k,\\
c_4=\mathsf{AuthEnc}.\mathsf{Enc}_k(\mathsf{msg}).
\end{array}
\]

This hybrid is computationally indistinguishable with $\mathbf{H}_1$ because of $\mathsf{PRF}$ with hidden seed $s$.

\smallskip
\noindent $\mathbf{H}_3$: in this hybrid, we hide the key $k$:

\[
\begin{array}{l}
r_1\sample[0,...,|\mathbb{G}|-1],\\
\qquad c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1}),\\
r_2\sample[0,...,|\mathbb{G}|-1],\\
\qquad \boxed{c_2=g^{r_2}\parallel\mathsf{KDF}(g^{x_2r_2}),}\\
r_3\sample[0,...,|\mathbb{G}|-1],\\
\qquad \boxed{c_3=g^{r_3}\parallel\mathsf{KDF}(g^{x_3r_3}),}\\
k\sample\{0,1\}^\lambda,\\
c_4=\mathsf{AuthEnc}.\mathsf{Enc}_k(\mathsf{msg}).
\end{array}
\]

This hybrid is computationally indistinguishable with $\mathbf{H}_2$ due to the same reason for $\mathbf{H}_1\stackrel{\mathsf{c}}{\approx}\mathbf{H}_0$.

\smallskip
\noindent $\mathbf{H}_4$: in this hybrid, we change the message into $1^l$, where $l$ is the message size:

\[
\begin{array}{l}
r_1\sample[0,...,|\mathbb{G}|-1],\\
\qquad c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1}),\\
r_2\sample[0,...,|\mathbb{G}|-1],\\
\qquad c_2=g^{r_2}\parallel\mathsf{KDF}(g^{x_2r_2}),\\
r_3\sample[0,...,|\mathbb{G}|-1],\\
\qquad c_3=g^{r_3}\parallel\mathsf{KDF}(g^{x_3r_3}),\\
k\sample\{0,1\}^\lambda,\\
c_4=\mathsf{AuthEnc}.\mathsf{Enc}_k(\boxed{1^l}).
\end{array}
\]

This hybrid is computationally indistinguishable with $\mathbf{H}_3$ due to the message indistinguishability of the authenticated encryption.

According to the hybrid arguments, $\mathbf{H}_0\stackrel{\mathsf{c}}{\approx}\mathbf{H}_4$, but $\mathbf{H}_4$ can be simulated by a PPT simulator without the message. This shows that our construction achieves semantic security under HDH assumption.

\smallskip
\noindent\textbf{Accountability (in our definition).} As mentioned in the motivation section, we want the party 1 to be able to detect malicious encryption, in which different parties will see different results.

We first introduce the problem. In a nutshell, we want to see if $\Delta_1$, $\Delta_2$, ..., and $\Delta_4$ equal to zero pads and whether $k=k'$ in the following ciphertext:

\[
\begin{array}{l}
c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1})\oplus s,\\
c_2=g^{r_2}~\boxed{\oplus\Delta_1}\parallel\mathsf{KDF}(g^{x_2r_2})\oplus k ~\boxed{\oplus\Delta_2,}\\
c_3=g^{r_3}~\boxed{\oplus\Delta_3}\parallel\mathsf{KDF}(g^{x_3r_3})\oplus k ~\boxed{\oplus\Delta_4,}\\
c_4=\mathsf{AuthEnc}.\mathsf{Enc}_{k'}(\mathsf{msg}).
\end{array}
\]
where $k=\mathsf{PRF}_s(1)$, $r_2=\mathsf{PRF}_s(2)$, and $r_3=\mathsf{PRF}_s(3)$.

We now introduce the algorithm to detect the malicious encryption.
\[
\begin{array}{l}
\mathsf{Detect}(c_1,c_2,c_3,c_4,\mathbb{G},g,g^{x_1},g^{x_2},g^{x_3},x_1)
\end{array}
\]
which runs as follows:
\begin{enumerate}
    \item Computes $g^{x_1r_1}$ from $g^{r_1}$ and $x_1$.

    \item Recovers $s$ by XOR the second part of $c_1$ with $\mathsf{KDF}(g^{x_1r_1})$.

    \item Computes the following pseudorandom values:
    \begin{itemize}
        \item $k=\mathsf{PRF}_s(1)$.
        \item $r_2=\mathsf{PRF}_s(2)$.
        \item $r_3=\mathsf{PRF}_s(3)$.
    \end{itemize}

    \item Checks whether the first part of $c_2$ equals to $g^{r_2}$ and the first part of $c_3$ equals to $g^{r_3}$. If not, outputs $\mathsf{CHEAT}$ and terminates. Otherwise, we know $\Delta_1$ and $\Delta_3$ are both zero pads, and the ciphertext has the following format:
    \[
    \begin{array}{l}
    c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1})\oplus s,\\
    c_2=\boxed{g^{r_2}}\parallel\mathsf{KDF}(g^{x_2r_2})\oplus k \oplus\Delta_2,\\
    c_3=\boxed{g^{r_3}}\parallel\mathsf{KDF}(g^{x_3r_3})\oplus k \oplus\Delta_4,\\
    c_4=\mathsf{AuthEnc}.\mathsf{Enc}_{k'}(\mathsf{msg}).
    \end{array}
    \]

    \item Computes $g^{x_2r_2}$ from $r_2$ and $g^{x_2}$ and $g^{x_3r_3}$ from $r_3$ and $g^{x_3}$. Note that we only use the public keys of the party 2 and the party 3.

    \item Recovers $\Delta_2$ by XOR the second part of $c_2$ with $\mathsf{KDF}(g^{x_2r_2})\oplus k$ and $\Delta_4$ by XOR the second part of $c_4$ with $\mathsf{KDF}(g^{x_3r_3})\oplus k$.

    \item Checks if $\Delta_2$ and $\Delta_4$ are both zero pads. If not, outputs $\mathsf{CHEAT}$ and terminates. Otherwise, we know the ciphertext has the following format:
    \[
    \begin{array}{l}
    c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1})\oplus s,\\
    c_2=g^{r_2}\parallel\boxed{\mathsf{KDF}(g^{x_2r_2})\oplus k,}\\
    c_3=g^{r_3}\parallel\boxed{\mathsf{KDF}(g^{x_3r_3})\oplus k,}\\
    c_4=\mathsf{AuthEnc}.\mathsf{Enc}_{k'}(\mathsf{msg}).
    \end{array}
    \]

    \item Uses $k$ to decrypt $c_4$. If the decrypted result is $\perp$, the ciphertext is not valid, the algorithm outputs $\mathsf{CHEAT}$ and terminates. Otherwise, outputs $\mathsf{ACCEPT}$, as the ciphertext now has the following format the same as the one from honest encryption:

    \[
    \begin{array}{l}
    c_1=g^{r_1}\parallel\mathsf{KDF}(g^{x_1r_1})\oplus s,\\
    c_2=g^{r_2}\parallel\mathsf{KDF}(g^{x_2r_2})\oplus k,\\
    c_3=g^{r_3}\parallel\mathsf{KDF}(g^{x_3r_3})\oplus k,\\
    c_4=\mathsf{AuthEnc}.\boxed{\mathsf{Enc}_{k}}(\mathsf{msg}).
    \end{array}
    \]
\end{enumerate}
According to the discussion above, if the detection algorithm outputs $\mathsf{ACCEPT}$, three parties will see the same decrypted result.

\end{document}