multilevel approach for signal restoration problems with toeplitz matrices malena español, tufts...
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![Page 1: Multilevel approach for signal restoration problems with Toeplitz matrices Malena Español, Tufts University Misha Kilmer, Tufts University](https://reader035.vdocuments.mx/reader035/viewer/2022062517/56649eb55503460f94bbdb80/html5/thumbnails/1.jpg)
Multilevel approach for signal restoration problems with Toeplitz matrices
Malena Español, Tufts UniversityMisha Kilmer, Tufts University
![Page 2: Multilevel approach for signal restoration problems with Toeplitz matrices Malena Español, Tufts University Misha Kilmer, Tufts University](https://reader035.vdocuments.mx/reader035/viewer/2022062517/56649eb55503460f94bbdb80/html5/thumbnails/2.jpg)
10th Copper Mountain Conference on Iterative Methods
2
Outline Background Multilevel Method Algorithm Implementation Numerical Example Conclusion and Future Work
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10th Copper Mountain Conference on Iterative Methods
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Ill-posed Problem
A problem is ill-posed if its solution is not unique, or its solution does not depend continuously
on the data
)()( ,
kindfirst ofequation integral Fredholm :Example
sgdttft)K(s
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10th Copper Mountain Conference on Iterative Methods
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Discrete Ill-Posed problem
holds condition Picard Discrete
yoscillator more become ectorssingular v
noise (white)unknown is
gap without aluessingular v Decaying
:Properties
e
matrix dconditione-ill large, a is where
,
model theand ,given , Find
nm
truetrue
true
RA
ebbAx
bAx
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10th Copper Mountain Conference on Iterative Methods
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Need for regularization
)(
1
)(
1ii
)(
1i
†
)(
1i
†
)(
)(
:are solutions squareleast Then the
. of SVD theLet
Arank
i
Arank
itrue
i
T
i
trueT
Arank
i i
trueT
true
Arank
i i
trueT
truetrue
errorxeubu
ebuebAx
bubAx
AVUA
ii
i
i
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10th Copper Mountain Conference on Iterative Methods
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Regularization
small is if 0
large is if 1
where
solution dRegularize
1i
i
ii
n
i i
T
ireg
bux i
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10th Copper Mountain Conference on Iterative Methods
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Regularization methods
CGLS.or LSQR, eg.
methods, Krylov of iterationsk :Methods Iterative
min
:tionRegulariza Tikhonov
: (TSVD) SVD Truncated
2
2
22
2
1i22
2
1i
LxbAx
bux
bux
x
n
i i
T
i
iTik
k
i i
T
TSVD
i
i
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10th Copper Mountain Conference on Iterative Methods
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Two-Level Method
cii
ic
i
iiii
iiii
iii
iiii
iii
iiii
iii
xxx
rxA
xAbr
xPxx
rxA
PARA
rRr
xAbr
bxA
Solve""
Solve""
Solve""
1
111
1
1
Pre-Smoothing
Post-Smoothing
Coarse-Grid Correction
Restriction
Prolongation
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10th Copper Mountain Conference on Iterative Methods
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Multilevel Method
If End
Solve
,
Solve
Else
Solve
gridcoarsest If
function
1
111
11
cii
ic
i
iiii
iiii
iii
iiiiiii
iiii
iii
iii
iii
xxx
rxA
xAbr
xPxx
),bMGM(Ax
PARArRr
xAbr
bxA
bxA
),bMGM(Ax
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10th Copper Mountain Conference on Iterative Methods
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Restriction and Prolongation Operators
11000000
00110000
00001100
00000011
11000000
00110000
00001100
00000011
2
2TW
Haar wavelet transform
W2W1
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10th Copper Mountain Conference on Iterative Methods
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Wavelet Domain
2
1
2
1
43
21
ˆ
ˆ
ˆ
ˆˆˆ
ˆˆ
.ˆ and ˆ ,ˆ where
ˆˆˆ
becomes domain In wavelet
b
b
x
x
AA
AA
bWbxWxAWWA
bxA
bAx
TTT
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10th Copper Mountain Conference on Iterative Methods
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Coarse-scale equation
x
1x̂111
22111
2
1
2
1
43
21
ˆˆˆ
ˆˆˆˆˆ
ˆ
ˆ
ˆ
ˆˆˆ
ˆˆ
bxA
xAbxA
b
bx
x
AA
AA
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10th Copper Mountain Conference on Iterative Methods
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.ˆˆˆ
Then,
ˆˆ
ˆˆˆ
ˆ
ˆˆ
ˆ
ˆˆ
ˆˆˆˆˆˆˆˆˆ
111
)ˆ(
1i
2211
221122111
1
bxA
xAueubu
xAebxAbxA
Arank
i i
T
i
T
i
trueT
true
iii
Coarse-scale equation
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10th Copper Mountain Conference on Iterative Methods
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p
p
p
xxLbxA
bAx
bxA
1
2
2111ˆ
111
111
ˆˆˆˆmin
3)or 2 ,ˆ,ˆLSQR(ˆ
solve we
ˆˆˆ
ofsolution dregularize aget To
1
Coarse-Grid Correction
Pre-smoothing
Pre-smoothing
Coarse-Grid Correction
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10th Copper Mountain Conference on Iterative Methods
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1
3
1
2
12
4
2 ˆˆ
ˆ
ˆ
ˆˆ
ˆ
ˆx
A
A
b
bx
A
A
Post-Smoothing
p
pprep
newx
xWxWxLrxA
A)ˆˆ(ˆ
ˆ
ˆmin 2211
2
2
2
4
2
ˆ2
newr
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10th Copper Mountain Conference on Iterative Methods
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Multilevel Method
If End
ˆ
),,,(Newtonˆ
ˆ
ˆ
;
If End
,3),(
gridfinest If
Else
)(Newtonor solvedirect
gridcoarsest If
function
122
21
2
111
1111
111
11
iinew
i
inew
inew
Tii
inew
iiinew
iiinew
iii
iTii
Ti
iiii
iii
iii
iii
xWxx
xLWrWWAx
xAbr
xWxx
),bMGM(Ax
WAWArWb
xAbr
bALSQRx
non
,bAx
),bMGM(Ax
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10th Copper Mountain Conference on Iterative Methods
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Toeplitz Matrices
),,,,,(: 1101)1(
0321
)3(012
)2(101
)1(210
mm
mmm
m
m
m
ttttttvectorToeplitz
tttt
tttt
tttt
tttt
A
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10th Copper Mountain Conference on Iterative Methods
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Toeplitz structure inheritance
A
43
21
ˆˆ
ˆˆˆ
AA
AAA
.ˆeach ofvector -Toeplitz theknow weMoreover,
Toeplitz. are ˆ and ˆ,ˆ,ˆ then Toeplitz, is If
:Theorem
4321
iA
AAAAA
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10th Copper Mountain Conference on Iterative Methods
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Pre-smoothing
productvector -matrixFast structure Toeplitz
LSQR. of iterations 3or 2 applyingby
ˆˆˆ
solve We
111
bxA
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10th Copper Mountain Conference on Iterative Methods
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Coarse-Grid Correction
.ˆon depend and where,ˆ
ˆmin
Method sNewton'by ˆˆˆˆmin
or
solvedirect aby ˆˆˆ
solve We
11
2
21
11
1
2
2111ˆ
111
1
xrDxDL
rd
DL
A
xLbxA
bxA
d
p
p
p
x
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10th Copper Mountain Conference on Iterative Methods
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Post-Smoothing
.ˆon depend ~ and ~, where,~
~ˆ
ˆ
min
Method sNewton'by
)ˆˆ(ˆˆ
ˆmin
solve We
221
2
2
2
1
2
4
2
2211
2
2
2
4
2
ˆ2
xrrDrcD
rd
DLW
A
A
xWxWxLrxA
A
d
p
pprep
newx
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10th Copper Mountain Conference on Iterative Methods
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Numerical Example
1.1 operator, derivative
)01.0,0( with
solution edgy :
matrix symmetric Toeplitz, Gaussian, :
pL
bNeeAxb
x
A
true
true
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10th Copper Mountain Conference on Iterative Methods
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Numerical Example
truex b
MGMxLSQRx
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10th Copper Mountain Conference on Iterative Methods
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Conclusions and Future work
General Cases – Non Structured Matrices
Parameter Selection Adaptive p-norm Extension to 2D, 3D