Frequency response magnitude for Daubechies 6tap filters
Lowpass Highpass
c [k]m
H(z) 2
G(z) 2d [k]
m-2
H(z) 2c [k]
m-1
G(z) 2
d [k]m-1
c [k]m-2
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
Angular frequency (normalized by pi)
Fouriertransformm
agnitude
Haar frequency response
Lowpass Highpass
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-
-
-
-
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-
-
-
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-
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1 0 1 2 32
0
2
4
6
8
10
12
k
c_
m1[k]
.. Circular convolution
. Symmetric extension (with duplication)
Symmetric extension (without duplication)
Wavelet extrapolation
1 0 1 2 33
2.5
2
1.5
1
0.5
0
0.5
1
k
d_
m1[k]
.. Circular convolution
. Symmetric extension (with duplication)
Symmetric extension (without duplication)
Wavelet extrapolation
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-
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-
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-
-
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101
102
103
103
104
105
106
10
7
108
L
conditionnumber
o with preconditioning
+ without preconditioning
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30
0.5
1
1.5
2
2.5x 10
5
P
conditionnumber
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101
102
103
104
103
104
105
106
107
108
L
numberofoperations
o hierarchical algorithm
+ nonhierarchical algorithm
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-
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28 2 1>.:. AlIlaral UU!-(ll't.-Galerkiu Solutiou of \\"aw
El111atiou
and adaptive solution strategies. Tlw discussion of tlw p l ' (
~ c e d i n g sections suggl'sts sl'wralgeneral solution strategies
f(Jr solving m u l t i s c a l ( ~ equations.
1. Nou-h:ic'f'(J,1'(;hical (J,pp1'O(J,che.'i. In a
non-hierarchical approach. tlw goal is t.o COlllputI' tlw
lllllnerical solution. 1/,1/1(:1:). for a single vahw of 1T/..
wlwl' l l l ' l ~ c l e c lin order to determine '/lin (;r.). A
non-hierarchicaJ approach can })(' i l l l p k n w n t . ( ~ d
usingeither a single scale formulation or a lllultiscah
f(H'lnulation. so consideration lH'edsto })(' given t o whe ther th
e f()l'lnation of the nlllltiscak equations is . i u s t i f i ( ~
d . I f wehave (J, IJ7'i01"i knowledge of the behavior of the
solution. as f()r e X l u n p h ~ in t.he caseof s t l ' ( ~ S S
concentrations around a hole in a stressed Plastic plate. then tlw
forlllationof the multiseale equations will al low us to eliminate
sonH' of tlw degrpes of f r e ( ~ d ( J l l lwhkh (10 not lie
within the region of high gradient.13eylkin. Coifman and Rokhlin
[22] have devPloped fast algorithms f(n' tlw applicatiollof the
multiHcale wavelet-Galerkill diffenmtial operator (awl othpl'
oIH'rators) 1.0 al'-
