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- Committer: Simon Funke
- Date: 2012-06-05 16:27:08 UTC
- Revision ID: simon.funke@gmail.com-20120605162708-8p7zz6eblgfdgalt
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- files added:
- 50turbines
- files modified:
- libadjoint.tex
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libadjoint.tex
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}
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\frame{
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\frametitle{The functional gradient of Example problem}
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\begin{figure}[h!]
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\includegraphics[width=0.45\textwidth]{pics/turbine_optimisation_forward}
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\caption{The predicted velocity field. The velocity in the two turbine areas is clearly reduced.}
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\end{figure}
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\begin{figure}
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\includegraphics[width=0.45\textwidth]{pics/turbine_optimisation_gradient}
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\caption{The sensitivity solution of the power output. More (Less) drag in the dark red (blue) areas leads to more power.}
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\end{figure}
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}
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\frame
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{
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\frametitle{Adjoint vs. finite differences for gradient computation}
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\begin{block}{Adjoint \hspace{2cm} \hspace{2.5cm} Finite differences}
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\vspace{-0.5cm}
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\begin{columns}
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% TODO: Make a table out of it
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\column[t]{.5\textwidth}
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\begin{itemize}
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\item Computational expense \textbf{independent} of the number of controls.
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\item Does \textbf{not} suffer from truncation error.
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\item DOLFIN (Python) can \textbf{automatically} derive and solve the adjoint equation.
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\end{itemize}
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\column[t]{.5\textwidth}
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\begin{itemize}
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\item Computational expense increases linearly with the number of controls.
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\item Suffers from truncation error.
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\item Easy to implement.
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\end{itemize}
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\end{columns}
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\end{block}
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}
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\note{
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\begin{itemize}
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\item For 1: We saw that the computational expense using the adjoint is independent of the choice of $m$. The finite difference approach requires a functional evaluation for each control, hence its expense increases linearly with the number of controls.
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\item For 2: Even for the optimal choice of h in [f(x+h) - f(x)] / h only about 1/2 to 2/3 of the digits are correct.
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\item For 3: Finite difference has the advantage of its ease implementation. However, there are powerful tools available to facilitate the development of an adjoint model.
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\end{itemize}
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}
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%\frame
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%{
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%\frametitle{Adjoint vs. finite differences for gradient computation}
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%\begin{block}{Adjoint \hspace{2cm} \hspace{2.5cm} Finite differences}
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% \vspace{-0.5cm}
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% \begin{columns}
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% % TODO: Make a table out of it
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% \column[t]{.5\textwidth}
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% \begin{itemize}
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% \item Computational expense \textbf{independent} of the number of controls.
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% \item Does \textbf{not} suffer from truncation error.
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% \item DOLFIN (Python) can \textbf{automatically} derive and solve the adjoint equation.
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% \end{itemize}
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% \column[t]{.5\textwidth}
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% \begin{itemize}
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% \item Computational expense increases linearly with the number of controls.
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% \item Suffers from truncation error.
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% \item Easy to implement.
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% \end{itemize}
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% \end{columns}
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%\end{block}
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%}
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%\note{
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%\begin{itemize}
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% \item For 1: We saw that the computational expense using the adjoint is independent of the choice of $m$. The finite difference approach requires a functional evaluation for each control, hence its expense increases linearly with the number of controls.
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% \item For 2: Even for the optimal choice of h in [f(x+h) - f(x)] / h only about 1/2 to 2/3 of the digits are correct.
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% \item For 3: Finite difference has the advantage of its ease implementation. However, there are powerful tools available to facilitate the development of an adjoint model.
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%\end{itemize}
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%%}
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\section{Applications}
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\frame{
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\end{itemize}
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}
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%\frame
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%{
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%\frametitle{Cheap functional evaluation for linear problems}
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%
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%Consider a linear problem (i.e. $F$ and $J$ are linear in $u$) of the form:
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%\begin{align*}
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% J(u) & = (j, u), & j \in \mathbb{R}^n, \\
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% F(u, m) & = Au - b(m) = 0, & A \in \mathbb{R}^{n \times n}, b\in \mathbb{R}^n.
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%\end{align*}
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%The associated adjoint equation is $A^*\lambda = j$.
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%
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%Then:
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%\begin{equation*}
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% J(u) = (A^*\lambda, u) = (\lambda, Au ) = (\lambda, b(m) ).
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%\end{equation*}
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%
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%$\to$ With the adjoint, the functional evaluation for \textbf{any} $m$ requires only an inner product! %TOO: Check if inner product is the right word ere
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%}
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\frame{
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\frametitle{The turbine optimisation problem}
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\begin{figure}[h!]
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\includegraphics[width=0.4\textwidth]{50turbines/initial_turbines.png}
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\includegraphics[width=0.4\textwidth]{50turbines/final_turbines.png}\\
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\includegraphics[width=0.4\textwidth]{50turbines/functional_plot.png}
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\caption{Initial and optimised turbine positions and the power increase.}
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\end{figure}
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}
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\frame
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%}
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\section{Three ways to adjoint a model}
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\frame{
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\frametitle{Outline}
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\tableofcontents[currentsection]
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\begin{center}
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\includegraphics[width=0.8\textwidth]{pics/adjoint_types_continuous}
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\end{center}
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\onslide<2->
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\begin{block}{The continuous adjoint is ...}
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... a system of linear equations that can be discretised and solved in a similar manner to the forward model.
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\begin{itemize}
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\item \color{darkgreen} No problems with non-differentiable numerical schemes in the forward discretisation.
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\item \color{red} Specific to one problem.
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\end{itemize}
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\end{block}
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%\onslide<2->
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%\begin{block}{The continuous adjoint is ...}
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%... a system of linear equations that can be discretised and solved in a similar manner to the forward model.
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%\begin{itemize}
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%\item \color{darkgreen} No problems with non-differentiable numerical schemes in the forward discretisation.
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%\item \color{red} Specific to one problem.
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%\end{itemize}
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%\end{block}
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}
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%\subsection{Automatic differentiation}
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\begin{center}
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\includegraphics[width=0.8\textwidth]{pics/adjoint_types_AD}
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\end{center}
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\onslide<2->
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\begin{block}{The adjoint of the source code...}
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... is usually obtained by applying an AD tool.
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\begin{itemize}
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\item \color{darkgreen} Can sometimes be obtained automatically from the source code
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\item \color{red} Does not work for some models
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\end{itemize}
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\end{block}
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%\onslide<2->
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%\begin{block}{The adjoint of the source code...}
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%... is usually obtained by applying an AD tool.
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%\begin{itemize}
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%\item \color{darkgreen} Can sometimes be obtained automatically from the source code
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%\item \color{red} Does not work for some models
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%\end{itemize}
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%\end{block}
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}
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\note{
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The model is considered as a sequence of primitive instructions, each of which can be differentiated.
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\begin{center}
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\includegraphics[width=0.8\textwidth]{pics/adjoint_types_libadjoint}
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\end{center}
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\onslide<2->
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\begin{block}{Adjoint of the discretised equations}
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\begin{itemize}
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\item \color{darkgreen} Consistent gradient
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%\item \color{darkgreen} Circumvents many problems with the AD approach.
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\item \color{red}{Complicated to develop by hand}
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\item \color{darkgreen}{\texttt{Libadjoint} facilitates the development}
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\end{itemize}
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\end{block}
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%\onslide<2->
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%\begin{block}{Adjoint of the discretised equations}
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%\begin{itemize}
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%\item \color{darkgreen} Consistent gradient
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%%\item \color{darkgreen} Circumvents many problems with the AD approach.
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%\item \color{red}{Complicated to develop by hand}
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%\item \color{darkgreen}{\texttt{Libadjoint} facilitates the development}
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%\end{itemize}
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%\end{block}
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}
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\note{
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similar to the "continuous adjoint" approach, but uses the discretised equations instead of the continuous equations.
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\item DOLFIN
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\end{itemize}
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\vspace{1cm}
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\onslide<2->
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\begin{center}
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\huge{Thank you}
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\end{center}
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