defect_budget.tex

Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations

defect_budget.tex

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \draw[black!13,line width=.25pt] (0.9,0.6) -- (0.9,3.9);
4 \draw[] (0.9,0.54) -- (0.9,0.66);
5 \node[font=\small,align=center,font=\scriptsize] at (0.9,0.34) {0};
6 \draw[black!13,line width=.25pt] (2.9,0.6) -- (2.9,3.9);
7 \draw[] (2.9,0.54) -- (2.9,0.66);
8 \node[font=\small,align=center,font=\scriptsize] at (2.9,0.34) {2};
9 \draw[black!13,line width=.25pt] (4.9,0.6) -- (4.9,3.9);
10 \draw[] (4.9,0.54) -- (4.9,0.66);
11 \node[font=\small,align=center,font=\scriptsize] at (4.9,0.34) {4};
12 \draw[black!13,line width=.25pt] (6.9,0.6) -- (6.9,3.9);
13 \draw[] (6.9,0.54) -- (6.9,0.66);
14 \node[font=\small,align=center,font=\scriptsize] at (6.9,0.34) {6};
15 \draw[black!13,line width=.25pt] (8.9,0.6) -- (8.9,3.9);
16 \draw[] (8.9,0.54) -- (8.9,0.66);
17 \node[font=\small,align=center,font=\scriptsize] at (8.9,0.34) {8};
18 \draw[black!13,line width=.25pt] (0.9,0.6) -- (8.9,0.6);
19 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.74,0.6) {0};
20 \draw[black!13,line width=.25pt] (0.9,1.425) -- (8.9,1.425);
21 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.74,1.425) {0.25};
22 \draw[black!13,line width=.25pt] (0.9,2.25) -- (8.9,2.25);
23 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.74,2.25) {0.5};
24 \draw[black!13,line width=.25pt] (0.9,3.075) -- (8.9,3.075);
25 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.74,3.075) {0.75};
26 \draw[black!13,line width=.25pt] (0.9,3.9) -- (8.9,3.9);
27 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.74,3.9) {1};
28 \draw[->,line width=.6pt] (0.9,0.6) -- (9.02,0.6);
29 \draw[->,line width=.6pt] (0.9,0.6) -- (0.9,4.02);
30 \node[font=\small,align=center,font=\scriptsize] at (4.9,0.05) {number of steps $n$};
31 \node[font=\small,align=center,anchor=west,font=\scriptsize] at (0.9,4.3) {induced $\ell^1$ operator norm};
32 \draw[line width=1.1pt] plot coordinates {(0.9,0.6) (1.9,1.0125) (2.9,1.038281) (3.9,0.98833) (4.9,0.935257) (5.9,0.893369) (6.9,0.862911) (7.9,0.841342) (8.9,0.826144)};
33 \draw[dashed,line width=.85pt] plot coordinates {(0.9,0.6) (1.9,1.0125) (2.9,1.425) (3.9,1.8375) (4.9,2.25) (5.9,2.6625) (6.9,3.075) (7.9,3.4875) (8.9,3.9)};
34 \path[fill=black] (0.9,0.6) circle (0.045);
35 \path[fill=black] (1.9,1.0125) circle (0.045);
36 \path[fill=black] (2.9,1.038281) circle (0.045);
37 \path[fill=black] (3.9,0.98833) circle (0.045);
38 \path[fill=black] (4.9,0.935257) circle (0.045);
39 \path[fill=black] (5.9,0.893369) circle (0.045);
40 \path[fill=black] (6.9,0.862911) circle (0.045);
41 \path[fill=black] (7.9,0.841342) circle (0.045);
42 \path[fill=black] (8.9,0.826144) circle (0.045);
43 \node[font=\small,align=center] at (11.3,3.75) {$\|\Delta\|_1=1/8$};
44 \node[font=\small,align=center,font=\small] at (11.3,2.8) {solid: exact defect\\dashed: $n/8$};
45 \node[font=\small,align=center,font=\small] at (11.3,1.55) {The bound is a guarantee,\\not a fitted trend.};
46 \end{tikzpicture}

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