Extrap: initial commit on map of methods

Research/Extrapolation methods/notes.tex

@@ -6,6 +6,7 @@
 \usepackage{graphicx}
 \usepackage{hyperref}
 \usepackage{subcaption}
+\usepackage{tikz}
 
 \newcommand{\dxdy}[2]{\frac{d #1}{d #2}}
 \newcommand{\dxdyk}[3]{\frac{d^{#3} #1}{d {#2}^{#3}}}
@@ -604,4 +605,45 @@ This is trivial to show.
 If $Q_k$ is derived from another method but shares the column space of $F_{n,k-1}$ then there is still mathematical equivalence between methods 2 and 3.
 If one minimizes with respect to a different norm then the methods correspond to other methods discussed here.
 
+\section{Map of equivalences}
+
+\begin{figure}
+  \centering
+  \begin{tikzpicture}
+    \matrix (m) [column sep=2cm, row sep=1em]{
+      \node[align=left] (multisecant) {$\begin{bmatrix} \vec{1}^\top \\ F_{n,k} \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$,\\ $X_{n,k} \vec{u} = \hat{\vec{x}}$}; & 
+      \node (overdetermined) {$\begin{bmatrix} \vec{1}^\top \\ B^\top F_{n,k} \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$}; &
+      \node (preMPE) {}; &
+      \node (Anderson) {Anderson mixing}; \\
+      & & & \node (MPE) {MPE}; \\
+      \node[align=left] (preBroyden) {$\begin{bmatrix} \vec{1}^\top & \vec{1}^\top \\ F_{n,k} & B \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$,\\ $\begin{bmatrix} X_{n,k} & C \end{bmatrix} \vec{u} = \hat{\vec{x}}$};
+      & & & \node (GMRES) {GMRES}; \\
+      & \node (prepreRRE) {}; & \node (preRRE) {}; & \node (RRE) {RRE}; \\
+      & & & \node (GCR) {GCR}; \\
+      & \node (prepreMMPE) {}; & \node (preMMPE) {}; & \node (MMPE) {MMPE}; \\
+      \node (Broyden) {Generalized Broyden};
+      & \node (preTEA) {}; & \node (preBiCG) {}; & \node (BiCG) {BiCG}; \\
+      & & & \node (TEA) {TEA}; \\
+      };
+    \path[->,very thick]
+      (multisecant) edge node[above] {$k \neq d$} (overdetermined) edge node[right] {$k < d$} (preBroyden)
+      (preBroyden) edge node[right,align=left] {$(C \Delta)^\top (X_{n,k} \Delta) = 0$, \\ $B = \hat{J}_{n-1,k} C$} (Broyden)
+      (overdetermined) edge node[above] {$B=F_{n,k-1}$} (preMPE)
+      (preMPE) edge node[above] {+ relaxation} (Anderson)
+      (prepreRRE) edge node[above] {$B=F_{n,k} \Delta$} (preRRE)
+      (preRRE) edge[red] (RRE)
+      (prepreMMPE) edge node[above] {$B=\begin{bmatrix} \vec{q}_1 & \dots & \vec{q}_k \end{bmatrix}$} (preMMPE)
+      (preMMPE) edge[red] (MMPE)
+      (preTEA) edge node[above] {$B=\begin{bmatrix} \vec{q} & A^\top \vec{q} & \dots \end{bmatrix}$} (preBiCG)
+      (preBiCG) edge[blue] (BiCG)
+      (BiCG) edge[red] (TEA);
+    \draw[-, very thick] (overdetermined) edge (preTEA);
+    \draw[->,very thick,red] (preMPE) |- (MPE);
+    \draw[->,very thick,blue] (preMPE) |- (GMRES);
+    \draw[->,very thick,blue] (preRRE) |- (GCR);
+  \end{tikzpicture}
+  \caption{Interconnectivity of extrapolation, acceleration and quasi-Newton methods.
+  Red arrows indicated $\fxi{n} = \vec{x}_{n+1}-\vec{x}_n$ while blue arrows indicate $\fxi{n} = (A-I) \vec{x}_n + \vec{b}$ and $\fxi{n+1} = A \fxi{n}$.}
+\end{figure}
+
 \end{document}
\ No newline at end of file