Algorithmic Differentiation in Python with Application Examples

Minimal Surface problem with PYADOLC
Example where the Reverse Mode Excels
(discretized) objective function:
u : [0, 1] 2 → R , u ∈ C 1
s
Z 1 Z 1
u ↦→
1 +
0 0
≈
m−1 X m−1 X
O ij (u)
„ « ∂u 2
+
∂x
„ « ∂u 2
∂y
i=0 j=0
"
Õ ij (u) := h 2 1 + (u #
i+1,j+1 − u i,j ) 2 + (u i,j+1 − u i+1,j ) 2
4
Nonlinear Program with box constraints:
u ∗ ∈ R m×m = argmin u Õ(u)
therefore ∇ u Õ(u) ∈ R m×m , e.g. m = 50
yields a gradient with 2500 elements ⇒ use
reverse mode
Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 22 / 27