2.1 Modelling Tools
31
π
∗
3 (a; γ ) = γ [a artanh a + ln
1 − a 2 ].
Observe that π
∗
1 , π
∗
2 and π
∗
3 approach finite values for a → 1. For γ > 0 all π
∗
(a; γ )
are smooth, however for γ → 0 all π
∗
(a; γ ) converge to the non-smooth indicator
function I {|a|−1} of |a| ≤ 1, i.e.
π
∗
(a; γ → 0) = I {|a|−1} :=
⎧
⎨
⎩
0
|a| ≤ 1
for
∞
|a| > 1
(2.28)
−1
−0.5
0
0.5
1
−1
0.5
0
0.5
1
˙
α
π 1 ( ˙
α; γ)
1
1 + γ
| ˙
α|
1+γ
γ
−1 −0.5
0
0.5
1
−1
−0.5
0
0.5
1
˙
α
a 1 ( ˙
α; γ)
γ
−1
−0.5
0
0.5
1
−1
0.5
0
0.5
1
˙
α
π 2 ( ˙
α; γ)
˙
α 2 + γ 2 − γ
γ
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
˙
α
a 2 ( ˙
α; γ)
γ
−1
−0.5
0
0.5
1
−1
0.5
0
0.5
1
˙
α
π 3 ( ˙
α; γ)
γ ln cosh
˙
α
γ
γ
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
˙
α
a 3 ( ˙
α; γ)
γ
Fig. 2.2 Alternative families of dissipation potentials π( ˙
α; γ ) together with resulting subdifferentials a = a( ˙
α; γ ) for regularization parameter γ ∈ {0, 0.1, 0.2, 0.3}. For γ → 0 all π( ˙
α; γ )
converge to the non-smooth π( ˙
α) = | ˙
α|