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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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