kuposter.tex

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\begin{document}

\begin{poster}{columns=4,Faculteit=Ingenieurswetenschappen} %Bioingenieurswetenschappen}
{ % the formatting is done here, as the desired font size will depend on the title length
\spaceskip=1\fontdimen2\font plus 1\fontdimen3\font minus \fontdimen4\font % Make larger spaced in the title 
  \Huge\bfseries Parareal computation of SDEs with time-scale separation
}
{ % Names (and subtitle) go(es) here
  Fr\'ed\'eric Legoll\textsuperscript{1},Tony Leli\`evre\textsuperscript{1}, \underline{Keith Myerscough}\textsuperscript{2} and Giovanni Samaey\textsuperscript{2}\\
  {\footnotesize \textsuperscript{1}Ecole des Ponts ParisTech, \textsuperscript{2}KU Leuven}}
{ % Personal images (photo/QR code to website
  \includegraphics[width=\footerheight]{specific/qrcode}%
  \includegraphics[width=\footerheight]{specific/photo.jpg}
}
{ % Personal details
  Keith Myerscough\\{\texttt keith.myerscough@kuleuven.be}\\people.cs.kuleuven.be/\~{}keith.myerscough/
}
{% Text to state association
In association with:%
}
{% Logo's of associated institutes/universities/companies/funding etc.
  \includegraphics[height=.8\footerheight]{specific/logo_EcolePonts} % 
}
{ % Department name
  Department of Computer Science%
}
{ % Department logo
  \includegraphics[height=\footerheight]{images/logo_CS}% 
}

%
\begin{posterbox}%
[name=micro-macro modelling, column=0, row=0, boxColorOne=KULeuvenFaculteit!15!white, borderColor=KULeuvenFaculteit]%
%[name=micro-macro modelling, column=0, row=0, boxColorOne=KULeuvenFaculteit!15!white, textborder=open, borderColor=KULeuvenFaculteit, headerColorOne=KULeuvenFaculteit, headerborder=open, headerBorderColor=KULeuvenFaculteit,  headerFontColor=white]%
%{Micro-macro modelling}%
{In a nutshell}%
%\begin{noindentitemize}
%\item The microscopic system is governed by known equations of motion, which we consider `exact'
%\item From the microscopic state we can extract macroscopic quantities
%\item We assume we have some (possibly inaccurate) model for the evolution of the macroscopic variables
%\item Observables of interest are defined as functions of the macroscopic state
%\end{noindentitemize}
\begin{description}
\item[Goal] Simulate slow-fast SDEs over long time, quickly
\item[Model] Slow-fast system of SDEs, and a macroscopic model taken from the ``fast'' limit
\item[Method] Parallel-in-time algorithm that iteratively improves the macroscopic result
\item[Result] Reduction in wall clock time
\item[Bonus] Lower variance than full microscopic model
\end{description}
\end{posterbox}

\begin{posterbox}[name=microscopic, column=0,below=micro-macro modelling]{Microscopic model}
\begin{noindentitemize}
%\item We study as the microscopic system a slow-fast system of coupled SDEs
\item Slow-fast system of coupled SDEs
\begin{align}
\dd X &= (-X_t^3 + X_t + Y_t^2) \dd t + \sqrt{\frac{2}{\beta}} \dd B_t^{(x)}  \label{eq:xParisdyn}\\
\dd Y &= \frac{1}{\epsilon}(X_t - Y_t) \dd t + \sqrt{\frac{2}{\beta\epsilon}} \dd B_t^{(y)}. \label{eq:yParisdyn}
\end{align}
%
%\item Modeled as an ensemble $\mathcal{X}_t = \lbrace X_t^p,Y_t^p,W_t^p \rbrace_{p=1}^P$ of particles with weight $W^p$
\item Modeled as an ensemble $\mathcal{X}_t$ of particles with positions ($X_t^p$, $Y_t^p$) and weight $W^p$
\item Time integrator: a Lie-Trotter splitting, updating $X_t$ first, then $Y_t$
\item Validation: \emph{deterministic} solution given by the Fokker-Planck equation, akin to the macroscopic model
\end{noindentitemize}
\end{posterbox}

\begin{posterbox}%
[name=parareal micro-macro, column=1, above=bottom, boxColorOne=KULeuvenFaculteit!15!white, borderColor=KULeuvenFaculteit]%
{The parareal algorithm}
%\begin{noindentitemize}
%\item The parareal algorith iteratively improves the macroscopic propagator, by computing the discrepancies between the macroscopic and the microscopic models \emph{in parallel}
%\item By employing the microscopic propagator only in parallel, there is a reduction in \emph{wall clock time} if there are fewer iterations needed than time steps
%\item The error is dominated by variance (see figures $\rightarrow$)%the variance due to the stochastic nature of the microscopic propagator
%\item This variance is greatly reduced by using a particle propagator with {\color{KULeuvenSecundair}correlated noise} for the macroscopic model in computing the discrepancies between the models
%\end{noindentitemize}
\begin{description}
\item[Iteratively improves] the macroscopic propagator by computing the discrepancies between the {\color{KULeuvenBlauw}macroscopic} and the {\color{KULeuvenFaculteit}microscopic} models \emph{in parallel}
\item[Parallel] use of the microscopic propagator gives a reduction in \emph{wall clock time} if there are fewer iterations needed than time steps
\item[Variance] of the stochastic microscopic propagator dominates the error (see figures $\rightarrow$)%the variance due to the stochastic nature of the microscopic propagator
\item[Reduction in variance] by using a particle propagator with {\color{KULeuvenSecundair}correlated noise} for the macroscopic model in computing the discrepancies between the models
\end{description}
\input{specific/sketch}
\end{posterbox}

\begin{posterbox}[name=macroscopic, column=0,below=microscopic, above=bottom]{Macroscopic model}
%\begin{posterbox}[name=macroscopic, column=1,row=0, above=parareal micro-macro]{Macroscopic model}
\begin{noindentitemize}
%\item \emph{assume} $\epsilon \rightarrow 0$, then $X_t$ ``feels'' only the conditional distribution of $Y_t$, a Gaussian centred at $X_t$ with variance $\frac{1}{\beta}$. 
\item Only slow variable, \emph{assume} the fast $Y_t$ is equilibrated and use only the expected value of the term $Y_t^2$
\begin{align}
\dd Z &= (-Z_t^3 + Z_t + Z_t^2 + \frac{1}{\beta}) \dd t + \sqrt{\frac{2}{\beta}} \dd B^{(x)}, \label{eq:zParisdyn}
\end{align}
or in potential form
\begin{align}
\dd Z &= -\partial_z V_{\eff} (Z_t) \dd t + \sqrt{\frac{2}{\beta}} \dd B^{(x)}. \label{eq:zParisdyn_pot}
\end{align}
%
\item The associated Fokker-Planck equation reads
\begin{equation}
\partial_t \rho(z) = \partial_z \left( \rho(z) \partial_z V_{\eff} \right) + \frac{1}{\beta} \partial_{zz} \rho(z).
\end{equation}
\item The macroscopic state is represented by integral quantities over a regular grid
%\begin{equation}
%\rho_i = \int_{x_i}^{x_{i+1}} \rho(z) \dd z,
%\end{equation}
%where the $x_i$ form an evenly spaced grid
\end{noindentitemize}
\end{posterbox}

%\begin{posterbox}[name=coupling,column=0,below=notset, above=bottom]{Coupling}
\begin{posterbox}[name=macroscopic, column=1,row=0, above=parareal micro-macro]{Coupling}
\begin{description}
\item[Restriction]($\mathcal{R}$, from micro to macro) sum the weights of all particles in each bin
%\begin{equation}
%%\rho_i = \sum_{\mbox{all particles $p$ in bin $i$}} W^p
%\rho_i = \sum_{\mbox{\footnotesize all particles $p$ in bin $i$}} W^p
%\end{equation}
\item[Matching]($\mathcal{M}$, from macro to micro) reweight particles from a known microstate $(\bar{X}^p, \bar{Y}^p, \bar{W}^p)$
%\begin{equation}
%%W^p = \bar{W}^p \frac{\rho_i}{\bar{\rho}_i} \mbox{ for } {\lbrace p: x_i \leq X_t^p < x_{i+1} \rbrace} \mbox{ for } i=0,1,\ldots,N-1
%W^p = \bar{W}^p \frac{\rho_i}{\bar{\rho}_i}\quad \mbox{\footnotesize for all particles $p$ in bin $i$}
%\end{equation}
\item[Resampling]($\mathcal{M}^*$, optional) retrieve an ensemble with all particles equal in weight
\end{description}
\end{posterbox}

\begin{posterbox}[name=convergenceK, column=2, below=notset, above=bottom, borderColor=KULeuvenFaculteit]{Convergence in iterations}%the number of iterations}
\begin{tikzpicture}
\node (inK) {
\includegraphics[width=\textwidth]{specific/inK}
};
\node [hlnode, anchor=south west] (inKlabel) at ($(inK.south west)!.1!(inK.north east)$) {Iteration converges in a few steps, need only K/N of serial wall clock time};
\end{tikzpicture}
\end{posterbox}

\begin{posterbox}[name=convergenceP, column=3, below=notset, above=bottom, borderColor=KULeuvenFaculteit]{Convergence in number of particles}%the number of particles}
\begin{tikzpicture}
\node (invalue) {
\includegraphics[width=\textwidth]{specific/invalue}
};
\end{tikzpicture}
\end{posterbox}

\begin{posterbox}[name=diagram, column=2, row=0, span=2, textborder=none, headerColorOne=white, headerborder=none]{}
Parameters used below: $\beta=5, \epsilon=0.1, K=N=20, \Delta t = 0.025, \delta t=1.0\times 10^{-4}$
\end{posterbox}

%\begin{posterbox}[name=parameters, column=2, below=notset, above=convergenceK, span=2, textborder=none, headerColorOne=white, headerborder=none]{}
%Parameters used below: $\beta=5, \epsilon=1,N=20,\Delta t = 0.025, \delta t=1.0\times 10^{-4}$
%\end{posterbox}
%
\end{poster}

\end{document}

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