Civil Engineering Reference
In-Depth Information
0
12
.
95
14
3
.
93
28
3
.
13
−
2
2
12
.
31
16
1
.
76
30
1
.
33
−
2
4
11
.
99
18
0
.
519
32
5
.
79
−
3
6
11
.
64
20
0
.
273
34
1
.
82
−
3
8
10
.
55
22
0
.
175
36
6
.
21
−
4
10
7
.
47
24
0
.
130
38
1
.
51
−
4
12
4
.
76
26
0
.
086
40
3
.
35
−
5
Table 6 and Fig. 47.
Reduction in the energy norm of the error when the PCG
method is applied to a cantilever problem with 544 unknowns. The slow decrease
at the beginning and again in the middle is typical for the CG method
x
k
−
x
∗
log
→
k
Then
z
i
Az
j
=
λ
j
δ
ij
, and (3.12) again follows. Moreover,
(C
−
1
A)
z
j
=
λ
j
z
j
for
all
. The rest of the proof of (1) follows as in Lemma 3.5.
Since the numbers
λ
j
in (4.4) are actually the eigenvalues of
C
−
1
A
, the
assertion (2) follows by the arguments used in Theorem 3.7.
Preconditioning also helps to reduce the effect of the following difficulty. In
principle, in using gradient methods we want to choose the
direction of steepest
descent
. Which direction gives the steepest descent depends on the metric of the
space. For the sim
ple gr
adient method, we implicitly use the Euclidean metric.
But if
=
√
x
Cx
is
a b
etter approximation to the metric
x
C
:
x
A
than the
=
√
x
x
, then
C
is a good choice for preconditioning. By
Theorem 4.4, the
oscillation
of the quotient
Euclidean metric
x
x
Ax
x
Cx
(
4
.
5
)
is the main determining factor for the rate of convergence. We shall make use of
similar arguments in the following section.
Although the widely applicable methods described below can be used for the
solution of second order boundary-value problems, for large problems of fourth
order, we usually have to tailor the preconditioning to the problem. This is due
to the strong growth of order
h
−
4
of
κ
. There are three common approaches to
constructing special methods:
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