Optimization and Computational Fluid Dynamics - Department of ...

5 Efficient Deterministic Approaches for Aerodynamic Shape Optimization 123
Table 5.2 Forward differentiated evaluation trace
it is
v−1 = x1
=1.5
˙v−1 =˙x1
=1.0
v0 = x2
=0.5
˙v0 =˙x2
=0.0
v1 = v−1/v0 =1.5/0.5 =3.0000
˙v1 =˙v−1/v0 − v−1 ˙v0/v0/v0
=(˙v−1 − v1 ˙v0)/v0 =(1.0−3.0 × 0.0)/0.5 =2.0000
v2 =sin(v1) =sin(3.0) = 0.1411
˙v2 =cos(v1)˙v1 =(−0.99) × 2.0 = −1.9800
v3 = v1 + v2 =3.0+0.1411 = 3.1411
˙v3 =˙v1 +˙v2 =2.0− 1.98 = 0.0200
y = v3
=3.1411
˙y = v3 ˙
=0.0200
v1 = ∂y
∂v1
= ∂y ∂v3
∂v3 ∂v1
∂(v1 + v2)
= v3
∂v1
∂v1
= v3
∂v1
∂v2
+ v3
∂v1
∂(sin v1)
= v3(1 + )
∂v1
This can also be evaluated by the iterative equations
v1 = v3
v2 = v3
(5.29)
(5.30)
= v3 + v3 cos(v1) . (5.31)
v1 = v1 + v2 cos(v1) .
Thus applying the chain rule to every line in the evaluation trace of Table
5.1, we obtain the reverse differentiated code in Table 5.3. Note that the
adjoint statements are lined up vertically underneath the original statements
that spawned them.
Note also that line 7 of Table 5.3 belongs to the expansion of Eq. (5.29),
lines 8 and 9 to the expansion of Eq. (5.30) and line 10 to the expansion of
Eq. (5.31).
As with the forward propagation method, the floating-point operation
count of the added lines is a small multiple of that for the underlying code
to evaluate y. But this time the complete gradient has been computed.