%%%%%%%%%%%%%%%%%%%%%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}


\section{Introduction}

Many databases have large numbers of missing or \nullv{} values;  these can arise for a
variety of reasons, including missing source data,
missing columns during data integration, de-normalized databases, or outlier detection and
cleaning~\cite{kim2003}.  Such \nullv{} values can lead to incorrect or ill-defined query results~\cite{rubin1976}, and as such removing these values from data
before processing is often desirable.

% Solution
One common approach is to manually replace missing values using a statistical or predictive model based on other values in the table and record.
This process is called \emph{imputation}.
In this paper, we introduce new methods and theory to selectively apply
imputation to a subset of records, dynamically, during query execution.
We instantiate these ideas in a new system, \ProjectName{}\footnote{\ProjectUrl{}}.  \emph{The key insight behind \ProjectName{} is that imputation
only needs to be performed on the data relevant to a particular query and
that this subset is generally much smaller than the entire database.}
% Background
While traditional imputation methods work over the entire data set and replace all missing
values, running a sophisticated imputation algorithm over a large data set can be very
expensive: our experiments show that even a relatively simple decision tree algorithm takes \acsbaseresulthours{} to train and run
on a 600K row database.

A simpler approach might drop all rows with any missing values, which can not only introduce
bias into the results, but also result in discarding much of the data.
In contrast to existing systems~\cite{burgette2010multiple,akande2015empirical}, \ProjectName{} avoids imputing over the entire data set. Specifically, \ProjectName{}
carefully plans the placement of imputation or row-drop operations inside the query plan, resulting in a
significant speedup in query execution while reducing the number of dropped rows. Unlike previous work, the focus of our work is not on the imputation algorithms themselves (we can employ almost any such algorithm), but rather on placing imputation operations
optimally in query plans.   Specifically, our optimization algorithms generate the
Pareto-optimal trade-offs between imputation cost for result quality,
and allow the analyst to specify their desired plan from this frontier.

% \todobox{find appropriate location for this}
% {In a previous project, she
% worked with the American Community Survey data (671,153 rows), and performing imputation on the whole
% dataset took around 75 minutes\todo{Here, we can reference the number from our own experiment (done), or the number from Akande et al: ``performing imputation on just a 1.5 percent random sample took 45 minutes''}.}

% % Prior work
% There is relevant prior work in three broad areas: statistics, databases, and time-series forecasting.
% Prior work in statistics\todo{add more specific subarea qualifier} has focused on imputation quality~\cite{burgette2010multiple} and runtime cost~\cite{akande2015empirical}.
% Prior work in databases has established simple semantics for handling tuples with missing values~\cite{codd1973understanding,grant1977null}.
% Prior work in time-series forecasting has incorporated forecasting extensions into domain-specific databases and characterized their behavior~\cite{parisi2011embedding,parisi2013temporal,duan2007processing}.
% \ProjectName{} contributes the first optimizer to plan data imputation on a per query basis, allowing users to use standard SQL on real-world data sets with missing values
% and express preferences in quality of data imputation versus running time. The guiding design principle behind \ProjectName{} is that the user should never see missing data or have to modify their queries to account for it.

% Implications
Our approach enables an exploratory analysis workflow in which the analyst can issue
standard SQL queries over a data set, even if that data has missing values.
\cameraready{
Using dynamic imputation, these queries execute between \lowxalphazero{} and
\highxalphazero{} times faster than the traditional approach of first imputing all missing
values and then running queries. Empirical results obtained using dynamic imputation
on real-world datasets show errors within \lowsmapealphazero{} to \highsmapealphazero{} percent of
the traditional approach (see~\Cref{sec:experiments} for more details). Alternately,
configuring dynamic imputation to prioritize query runtime yields further speedups of
\lowxalphaone{} to \highxalphaone{} times, with errors of \lowsmapealphaone{} to
\highsmapealphaone{} percent.
}

\subsection{Contributions}
\ProjectName{} is designed to enable early data exploration, by allowing analysts to run
their queries without an explicit base-table imputation step. To do so, it leverages a number of contributions
to minimize imputation time while producing comparable results to traditional approaches.

These contributions include:
\begin{itemize}
\item \textbf{Relational algebra extended with imputation:}
  We extend the standard relational algebra with two new operators to represent imputation operations: \textit{Impute} ($\mu$) and \textit{Drop} ($\delta$).
  The \textit{Impute} operation fills in missing data values using any statistical imputation
  technique, such as chained-equation decision trees ~\cite{burgette2010multiple}.
  The \textit{Drop} operation simply drops tuples which contain \nullv{} values.
\item \textbf{Model of imputation quality and cost:}
  We extend the traditional cost model for query plans to incorporate a measure of the
  \textit{quality} of the imputations performed.
  We use the cost model to abstract over the imputation algorithm used.
  To add an imputation technique, it is sufficient to characterize it with two functions: one to describe its running time and one to describe the quality of its results.
\item \textbf{Query planning with imputation:}
  We present the first query planning algorithm to jointly optimize for running time and the quality of the imputed results.
  It does so by maintaining multiple sets of Pareto-optimal plans according to the cost model.
  By deferring selection of the final plan, we make it possible for the user to trade off running time and result quality.
%\item \textbf{Proof of optimality:}
% We prove that our cost-model and planning algorithm produce a sound and complete final Pareto frontier, making the final plan choice optimal under our search space and the user's preferences.
\end{itemize}

% Summary of related work

% Although this manual imputation solves the problem of missing data, in the age of big data it may be very expensive to run an imputation algorithm on an entire dataset.
% Additionally, it may not be necessary to completely clean the data to make it usable.
% Some users may be willing to run queries on dirty data, simply ignoring any missing values, as long as they do not have to pay the cost of imputation.
% Others may want to run queries on a subset of the data, and so do not need to impute
% every field in every record. Yet others may want to customize the
% imputation algorithm for the tradeoffs and demands of a particular domain.

% In this paper, we present \ProjectName{}, a database system which is designed to interact with a dirty dataset as though it were clean.
% To achieve this goal, we perform imputation on the fly, during query execution.
% Performing imputation at query time allows our system to impute only the data necessary to run the query, and it allows users to flexibly trade imputation quality for computation time.

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:


\section{Introduction}

In connectomics, neuroscientists annotate neurons and their connectivity within
3D volumes to gain insight into the functional structure of the brain. Rapid
progress in automatic sample preparation and electron microscopy (EM)
acquisition techniques has made it possible to image large volumes of brain
tissue at nanometer resolution. With a voxel size of
$4\times4\times40~\text{nm}^3$, a cubic millimeter volume is one petabyte of
data. With so much data, manual annotation is not feasible, and automatic
annotation methods are needed~\cite{jain2010,Liu2014,GALA2014,kaynig2015large}.

Automatic annotation by segmentation and classification of brain tissue is
challenging~\cite{isbi_challenge} and all available methods make errors, so
the results must be \emph{proofread} by humans. This crucial
task serves two purposes: 1) to correct errors in the segmentation, and 2) to
increase the body of labeled data from which to train better automatic
segmentation methods. Recent proofreading tools provide intuitive user
interfaces to browse segmentation data in 2D and 3D and to identify and manually
correct errors~\cite{markus_proofreading,raveler,mojo2,haehn_dojo_2014}. Many
kinds of errors exist, such as inaccurate boundaries, but the most common are
\emph{split errors}, where a single segment is labeled as two, and \emph{merge
errors}, where two segments are labeled as one
(Fig.~\ref{fig:merge_and_slit_errors}). With user interaction, split errors can
be joined, and the missing boundary in a merge error can be defined with
manually-seeded watersheds~\cite{haehn_dojo_2014}. However, the visual
inspection to find errors takes the majority of the time, even with
semi-automatic correction tools~\cite{proofreading_bottleneck}.

\begin{figure}[t]
\begin{center}
  \includegraphics[width=\linewidth]{merge_and_split_errors.pdf}
\end{center}
\vspace{-4mm}
   \caption{The most common proofreading corrections are fixing split errors (red arrows) and merge errors (yellow arrow). A fixed segmentation matches the cell borders.}
\label{fig:merge_and_slit_errors}
\end{figure}

Our goal is to automatically detect potential split and merge errors to reduce visual
inspection time. Further, to reduce correction time, we propose
corrections to the user to accept or reject. We call this process \textit{guided
proofreading}.

We train a classifier for split error detection with a convolutional neural network
(CNN). This takes as input patches of membrane segmentation probabilities, cell
segmentation masks, and boundary masks, and outputs a split-probability score. As we
must process large data, this classifier only operates on cell boundaries, which
reduces computation over methods that analyze every pixel. For merge errors, we
invert and reuse the split classification network, and ask it to rate a
set of generated boundaries that hypothesize a split.

Possible erroneous regions are sorted by their score, and a candidate correction is generated for each
region. Then, a user works through this list of regions and corrections. In a
forced choice setting, the user either selects a correction or skips it to
advance to the next region. In an automatic setting, errors with a high probability are automatically corrected first, given an appropriate
probability threshold, after which the user would take over. Finally, to test
the limits of performance, we create an oracle which only accepts corrections
that improve the segmentation, based on knowledge of the ground truth. This is
guided proofreading with a perfect user.

We evaluate these methods on multiple connectomics datasets. For the forced
choice setting, we perform a quantitative user study with 20 novice users who
have no previous experience of proofreading EM data. We ask participants to
proofread a small segmentation volume in a fixed time frame. In a
between-subjects design, we compare guided proofreading to the semi-automatic
\textit{focused proofreading} approach by Plaza~\cite{focused_proofreading}. In
addition, we compare against the manual interactive proofreading tool
\textit{Dojo} by Haehn~\etal~\cite{haehn_dojo_2014}. We also asked four domain
experts to use guided proofreading and focused proofreading for comparison.

This paper makes the following contributions.
%
First, we present a CNN-based boundary classifier for split errors, plus a merge
error classifier that inverts the split error classifier. This is used to
propose merge error corrections, removing the need to manually draw the missing
edge. These classifiers perform well without much training data, which is
expensive to collect for connectomics data.
%
Second, we developed a guided proofreading approach to correcting segmentation
volumes, and an assessment scenario comparing forced-choice interaction with
automatic and oracle proofreading.
%
Third, we present the results of a quantitative user study assessing
guided proofreading. Our method is able to reduce segmentation
error faster than state-of-the-art semi-automatic tools for both novice and
expert users.

Guided proofreading is applicable to all existing automatic segmentation methods that
produce a label map. As such, we believe that our approach is a promising
direction to proofread segmentations more efficiently and better tackle large
volumes of connectomics imagery.