Slide 36
Slide 36 text
Continuous formula
First order optimality condition
Y
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[
ˆ„(x, t)
ˆt
+ Èv(x, t), Ò
x
„(x, t)Í ≠ H
ú(x, t, v(x, t)) = ‘
x
„(x, t), x œ , t > 0,
ˆfl(x, t)
ˆt
+ Ò
x
· (fl(x, t)v(x, t)) + ‘
x
fl(x, t) = 0, x œ , t > 0,
v(x, t) = Ò
pH(x, t, Ò
x
„(x, t)), x œ , t > 0,
„(x, 0) = g(x), fl(x, T) = c, x œ ,
(5)
In PDHG method, the updates for „ and fl are related to the first (linear wrt „) and
second line (linear wrt fl), respectively.
We use implicit time discretization
We can have a larger CFL number ( t
x
max
p
ÎÒ
pH(x, t, p)Î), e.g., in example 2,
CFL number is 0.036 for explicit scheme, but it can be about 10.8 in our method.
Meng, Hao, Liu, Yang, Li, Osher PDHG HJ Aug 10, 2023 4 / 8
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