I want to generate a plot similar to this one, which allows for different colors when given different parameters : (This is the Moreau-Yosida regularization of the absolute value) Moreau-Yosida of regularization

the Moreau-Yosida regularization is given by :

$$f_{\lambda}(x):= \inf_{u\in \mathbb{R}}\left\lbrace f(u)+ \dfrac{1}{2 \lambda} |x-u|^2 \right\rbrace $$

I want to show these functions in the same plot:

  • Absolute value function
  • the Moreau-Yosida regularization of the $0-norme$ with different values of $\lambda$ to get this kind of color progression (aesthetically, I think a luminosity progression with a single color would look better than the rainbow colors) ( $|x|_0 = 0$ if $x=0$ and $|x|_0 = 1$ otherwise)
  • the Moreau-Yosida regularization of the Absolute value function with different values of $\lambda$ and the same criteria as above.

The following code is my first attempt to set up the visualization I want to get:

   1/(2 \[Lambda]) * (Abs[x]^2 - Max[Abs[x]^2 - 2 \[Lambda], 0]), 
   1/(2 b) * (Abs[x]^2 - Max[Abs[x] - b, 0]^2)}, {x, -2, 
   2}] , {\[Lambda], 1/1000, 1}, {b, 1/1000, 1}]
  • $\begingroup$ Thank you @dionys for editing, I hope my English get better someday. $\endgroup$ – Aymane Fihadi Oct 18 '14 at 20:10


Thank you for comment from ybeltukov: Exclusions->None:

fun[b_, x_] := 1/(2 b)*(Abs[x]^2 - Max[Abs[x] - b, 0]^2)
Legended[ParametricPlot[{u, fun[a, u]}, {u, -2, 2}, {a, 0, 1}, 
  ColorFunction -> {ColorData["Rainbow"][#4] &}, Exclusions -> None, 
  ImageSize -> 500], BarLegend["Rainbow"]] 

enter image description here

  • 1
    $\begingroup$ Exclusions -> None removes artifacts. It is also 2.5 times faster. $\endgroup$ – ybeltukov Oct 18 '14 at 13:01
  • $\begingroup$ @ybeltukov thank you I have edited $\endgroup$ – ubpdqn Oct 18 '14 at 13:08
  • $\begingroup$ Thank you very much @ubpdqn. Can you pleas explain the part {ColorData["Rainbow"][#4] &} of the code? ( what is the function of the argument #4 and the & in the end) ;) And how can we do the Color gradient with one color?, say from gray 10 % to gray 100% (black) $\endgroup$ – Aymane Fihadi Oct 18 '14 at 14:03
  • 1
    $\begingroup$ @AymaneFihadi #4 just the slot for b in parameter space (x,b). I suggest looking at documentation of ColorFunction and ColorData and playing. $\endgroup$ – ubpdqn Oct 18 '14 at 21:46

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