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Fig. 12.3
Wavelet
compression for level
L
=
7
(
top
) and number of non-zero
entries (
bottom
)
12.2.3 Multilevel Preconditioning
One of the advantages of multi-scale discretizations is their ability to precondition
large linear systems due to the norm equivalences. With (
12.3
)for
s
=
0wehave
N
L
for every
u
∈
V
L
with coefficient vector
u
∈ R
that
2
2
.
u
,
M
u
=
u
L
2
(G)
∼|
u
|
Therefore, the condition number of
M
is bounded, independent of the level
L
,i.e.
κ(
M
)<c
,
. Denote by
D
the diagonal matrix with entries 2
α
∀
L
∈ N
for an index
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