10
1 A Bilinear Conjugate-Gradient Inversion Algorithm
The derivative of Φ (R) is computed in a straight-forward (though slightly tedious)
manner to be:
dΦ (R) (J + αv)
dα
= =
v
(x)∗
·
∂Φ (R) (J)
∂J (x) + v
(y)∗
·
∂Φ (R) (J)
∂J (y) + v
(z)∗
·
∂Φ (R) (J)
∂J (z)
+α
v
(x)∗
·
∂Φ (R) (v)
∂J (x) + v
(y)∗
·
∂Φ (R) (v)
∂J (y) + v
(z)∗
·
∂Φ (R) (v)
∂J (z)
.
(1.17)
The vector dot-product denotes a sum over the cell indices, klm, as before, and the
components of the gradient vectors are given in (1.11).
Similarly,
dΦ (x)
dα
(ρ + αu, J + αv)
= =
A
∗
0x · B 0x + α
A
∗
0x · B 1x + A
∗
1x · B 0x
+α
2
A
∗
0x · B 2x + A
∗
1x · B 1x
+ α
3 A
∗
1x · B 2x
dΦ (y)
dα
(ρ + αu, J + αv)
= =
A
∗
0y · B 0y + α
A
∗
0y · B 1y + A
∗
1y · B 0y
+α
2
A
∗
0y · B 2y + A
∗
1y · B 1y
+ α
3 A
∗
1y · B 2y
dΦ (z)
dα
(ρ + αu, J + αv)
= =
A
∗
0z · B 0z + α
A
∗
0z · B 1z + A
∗
1z · B 0z
+α
2
A
∗
0z · B 2z + A
∗
1z · B 1z
+ α
3 A
∗
1z · B 2z
(1.18)
where
A
∗
0x = J
(x)∗
· Q
(x)T (u) + v
(x)∗
· Q
(x)T (ρ) + v
(x)∗
· G
(xx)H
+v
(y)∗
· G
(xy)H
+ v
(z)∗
· G
(xz)H
A
∗
1x = 2v
(x)∗
· Q
(x)T (u)
B 0x = Q
(x) (ρ) · J
(x)
+ G
(xx)
· J
(x)
+G
(xy)
· J
(y)
+ G
(xz)
· J
(z)
− E
(0x)
B 1x = Q
(x) (u) · J
(x)
+ Q
(x) (ρ) · v
(x)
+ G
(xx)
· v
(x)
+G
(xy)
· v
(y)
+ G
(xz)
· v
(z)