Frequency domain seismic forward modelling: A tool for waveform ...

can write directly from the Figure 2.5:
D(n; 0) n 2 =2+2(n=2) 2 =2+2 n n 1
=n 2
2 2 +1 4 +1 = 7 4 n2 : (2.13)
In a similar manner the following equations can be derived:
D(n; 2) 19 4 n2 (2.14)
D(n; 3) 25 4 n2 (2.15)
D(n; 4) 31 4 n2 (2.16)
and equation 2.12 can be expanded in the following form using 2.16:
S(n; 4) 31 31
4 n2 +4
4 (n=2)2 +4S(n=4; 4) =
31
4 n2 (1+1)+16S(n=4; 4) =
:::= 31
X
4 n2 log 2 (n)
i=1
1 = 31
4 n2 log 2 (n): (2.17)
Substituting this into the equations (2.9) to (2.11) and using (2.13) to (2.15) the
following expressions can be obtained:
S(n; 3) 31 4 n2 log 2
(n)+O(n 2 ) (2.18)
S(n; 2) 31 4 n2 log 2
(n)+O(n 2 ) (2.19)
S(n; 0) 31 4 n2 log 2
(n)+O(n 2 ) (2.20)
which gives us a total memory requirementof 31
2 n2 log 2
(n) for the matrices L ~
and U ~
together. George and Liu (1981) have shown that the theoretical minimal memory
requirements to perform the LU decomposition on an n by n grid is of the same
order of magnitude, so that nested dissection can therefore be assumed to be an
\optimal" grid ordering to within, at least, an order of magnitude. George and Liu
(1981) also showed that nested dissection gives an optimal number of operations
((n; 0))
(n; 0) 829
84 n3 ; (2.21)
47