Fast Solvers for Cahn-Hilliard Inpainting€¦ · Jessica Bosch David Kay Martin Stoll Andrew J....
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MAX PLANCK INSTITUTE
FOR DYNAMICS OF COMPLEX
TECHNICAL SYSTEMS
MAGDEBURG
Preconditioning Conference 2013June, 19-21, 2013
Oxford, UK
Fast Solvers for Cahn-Hilliard Inpainting
Jessica Bosch David KayMartin Stoll Andrew J. Wathen
Max Planck Institute for Dynamics of Complex Technical Systems,Research group Computational Methods in Systems and Control Theory
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
c©2012 Thomas Rolle
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
c©2012 Thomas Rolle
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
1 Phase Separation
2 Cahn-Hilliard System
3 Inpainting Model
4 Preconditioning
5 Numerical Results
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Phase SeparationTwo-Phase Structure
Ω ⊂ Rd , d ∈ 2,3u = u(x , t): concentration
u ∈ [0,1]
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Phase SeparationTwo-Phase Structure
Ω ⊂ Rd , d ∈ 2,3
u = u(x , t): concentration
u ∈ [0,1]
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Phase SeparationTwo-Phase Structure
Ω ⊂ Rd , d ∈ 2,3u = u(x , t): concentration
u ∈ [0,1]
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Phase SeparationTwo-Phase Structure
Ω ⊂ Rd , d ∈ 2,3u = u(x , t): concentration
u ∈ [0,1]
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Phase SeparationEnergy Functional
E(u) =
∫Ω
γε
2|∇u|2 +
1εψ(u) dx
Smooth potential
ψ(u) = u2(u − 1)2
Non-smooth potential
ψ(u) =
12u(1 − u), u ∈ [0,1]
∞, otherwise
= ψ0(u) + I[0,1](u)
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Phase SeparationEnergy Functional
E(u) =
∫Ω
γε
2|∇u|2 +
1εψ(u) dx
Smooth potential
ψ(u) = u2(u − 1)2
Non-smooth potential
ψ(u) =
12u(1 − u), u ∈ [0,1]
∞, otherwise
= ψ0(u) + I[0,1](u)
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Phase SeparationEnergy Functional
E(u) =
∫Ω
γε
2|∇u|2 +
1εψ(u) dx
Smooth potential
ψ(u) = u2(u − 1)2
Non-smooth potential
ψ(u) =
12u(1 − u), u ∈ [0,1]
∞, otherwise
= ψ0(u) + I[0,1](u)
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Cahn-Hilliard SystemMoreau-Yosida Regularization
E(u) =
∫Ω
γε
2|∇u|2 +
1ε
(ψ0(u) + I[0,1](u)) dx
↓
ϑν(uν) B12ν
(|max (0,uν − 1)|2 + |min (0,uν)|2)
↓
E1(uν) =
∫Ω
γε
2|∇uν|2 +
1εψ0(uν) + ϑν(uν) dx
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Cahn-Hilliard SystemMoreau-Yosida Regularization
E(u) =
∫Ω
γε
2|∇u|2 +
1ε
(ψ0(u) + I[0,1](u)) dx
↓
ϑν(uν) B12ν
(|max (0,uν − 1)|2 + |min (0,uν)|2)
↓
E1(uν) =
∫Ω
γε
2|∇uν|2 +
1εψ0(uν) + ϑν(uν) dx
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Cahn-Hilliard SystemMoreau-Yosida Regularization
E(u) =
∫Ω
γε
2|∇u|2 +
1ε
(ψ0(u) + I[0,1](u)) dx
↓
ϑν(uν) B12ν
(|max (0,uν − 1)|2 + |min (0,uν)|2)
↓
E1(uν) =
∫Ω
γε
2|∇uν|2 +
1εψ0(uν) + ϑν(uν) dx
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Cahn-Hilliard SystemRegularized Cahn-Hilliard System
∂tu(t) = −gradH−1E(u(t))
Regularized system
∂tuν = −∆(γε∆uν −1εψ′0(uν) − θν(uν))
∂uν∂n
=∂∆uν∂n
= 0 on ∂Ω
[Hintermuller/Hinze/Tber ’11]
θν(uν) B1ν
(max (0,uν − 1) + min (0,uν))
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Cahn-Hilliard SystemRegularized Cahn-Hilliard System
∂tu(t) = −gradH−1E(u(t))
Regularized system
∂tuν = −∆(γε∆uν −1εψ′0(uν) − θν(uν))
∂uν∂n
=∂∆uν∂n
= 0 on ∂Ω
[Hintermuller/Hinze/Tber ’11]
θν(uν) B1ν
(max (0,uν − 1) + min (0,uν))
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Cahn-Hilliard SystemPhase Separation in 2D
n = 0 n = 5 n = 50 n = 500
Taken from [Bosch/Stoll/Benner ’12].
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Inpainting ModelIdea
Original image f withinpainting domain D.
Inpainted image.
ω(x) =
0, if x ∈ Dω0, if x ∈ Ω \ D
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Inpainting ModelIdea
Original image f withinpainting domain D.
Inpainted image.
ω(x) =
0, if x ∈ Dω0, if x ∈ Ω \ D
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Inpainting ModelModified Cahn-Hilliard Equation
Regularized modified Cahn-Hilliard system
∂tuν = −∆(γε∆uν −1εψ′0(uν) − θν(uν))+ω(x)(f − uν)
∂uν∂n
=∂∆uν∂n
= 0 on ∂Ω
Smooth variant: [Bertozzi/Esedoglu/Gillette ’07]
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Inpainting ModelTime Discretization
Two energies
H−1 : E1(uν) =∫
Ω
γε2 |∇uν|2 + 1
εψ0(uν) + ϑν(uν) dx
L2 : E2(uν) = 12
∫Ωω(f − uν)2 dx
Convexity splitting [Elliott/Stuart ’93, Eyre ’97]
u(n)ν − u(n−1)
ν
τ= −∆H−1(E11(u(n)
ν ) − E12(u(n−1)ν ))
−∆L2(E21(u(n)ν ) − E22(u(n−1)
ν ))
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Inpainting ModelTime Discretization
Two energies
H−1 : E1(uν) =∫
Ω
γε2 |∇uν|2 + 1
εψ0(uν) + ϑν(uν) dx
L2 : E2(uν) = 12
∫Ωω(f − uν)2 dx
Convexity splitting [Elliott/Stuart ’93, Eyre ’97]
u(n)ν − u(n−1)
ν
τ= −∆H−1(E11(u(n)
ν ) − E12(u(n−1)ν ))
−∆L2(E21(u(n)ν ) − E22(u(n−1)
ν ))
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningLinear System
We want to solve Ax = b where
A =
(A BC −D
)with A and D symmetric and positive definite and B and Csymmetric positive semi-definite.
Note: In the smooth case we have B = C.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningLinear System
We want to solve Ax = b where
A =
(A BC −D
)with A and D symmetric and positive definite and B and Csymmetric positive semi-definite.
Note: In the smooth case we have B = C.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningSmooth Case – Coefficient Matrix
The coefficient matrix becomes
A =
(M γεKγεK −γε[(1
τ + C2)M + C1K ]
)where M = MT > 0, K = KT
≥ 0 and C1 >1ε , C2 > ω0.
A is symmetric and indefinite:
A =
(I 0
γεKM−1 I
) (M 00 S
) (I γεM−1K0 I
).
S is the Schur complement which is symmetric negativedefinite.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningSmooth Case – Coefficient Matrix
The coefficient matrix becomes
A =
(M γεKγεK −γε[(1
τ + C2)M + C1K ]
)where M = MT > 0, K = KT
≥ 0 and C1 >1ε , C2 > ω0.
A is symmetric and indefinite:
A =
(I 0
γεKM−1 I
) (M 00 S
) (I γεM−1K0 I
).
S is the Schur complement which is symmetric negativedefinite.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningSmooth Case – Preconditioner
We consider the block-triangular preconditioner
P =
(M 0γεK −S
)where S is a Schur complement preconditioner.
The preconditioned matrix becomes
A = P−1A =
(I γεM−1K0 −S−1S
)which has in the idealized case S = S only two distincteigenvalues.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningSmooth Case – Preconditioner
We consider the block-triangular preconditioner
P =
(M 0γεK −S
)where S is a Schur complement preconditioner.
The preconditioned matrix becomes
A = P−1A =
(I γεM−1K0 −S−1S
)which has in the idealized case S = S only two distincteigenvalues.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningSmooth Case – Schur Complement Approximation
The Schur complement
S = −γε[(1τ
+ C2)M + C1K ] − γ2ε2KM−1K
is approximated by
S = −
√γε(
1τ
+ C2)M + γεK
︸ ︷︷ ︸AMG
M−1
√γε(
1τ
+ C2)M + γεK
︸ ︷︷ ︸AMG
.
Note: S ∧= S if C1 = 2
√γε(1
τ + C2).
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningSmooth Case – Schur Complement Approximation
The Schur complement
S = −γε[(1τ
+ C2)M + C1K ] − γ2ε2KM−1K
is approximated by
S = −
√γε(
1τ
+ C2)M + γεK
︸ ︷︷ ︸AMG
M−1
√γε(
1τ
+ C2)M + γεK
︸ ︷︷ ︸AMG
.
Note: S ∧= S if C1 = 2
√γε(1
τ + C2).
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningSmooth Case – Schur Complement Approximation
Lemma
λ(S−1S) ∈
12,1 +
C1
2√γε(1
τ + C2)
Proof.Using the Rayleigh quotient, define a =
√γε(C2 + 1
τ )M12 v and
b = γεM−12 Kv, we can write
vT Sv
vT Sv=
1 + C1
2√γε(C2+ 1
τ )
2aT baT a+bT b
1 + 2aT baT a+bT b
.
The Lemma results from 2aT baT a+bT b ∈ [0,1].
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningSmooth Case – Schur Complement Approximation
Lemma
λ(S−1S) ∈
12,1 +
C1
2√γε(1
τ + C2)
Proof.Using the Rayleigh quotient, define a =
√γε(C2 + 1
τ )M12 v and
b = γεM−12 Kv, we can write
vT Sv
vT Sv=
1 + C1
2√γε(C2+ 1
τ )
2aT baT a+bT b
1 + 2aT baT a+bT b
.
The Lemma results from 2aT baT a+bT b ∈ [0,1].
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningNon-Smooth Case – Coefficient Matrix
In every Newton step k , the coefficient matrix becomes
A =
(M γεK + 1
νGA MGAK −[(1
τ + C2)M + C1K ]
)
where M = MT > 0, K = KT≥ 0 and C1 > 0, C2 > ω0 and
GA = GA (u(k−1)) = diag(
0, if 0 ≤ u(k−1)(xi) ≤ 11, otherwise
).
A is non-symmetric.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningNon-Smooth Case – Coefficient Matrix
In every Newton step k , the coefficient matrix becomes
A =
(M γεK + 1
νGA MGAK −[(1
τ + C2)M + C1K ]
)where M = MT > 0, K = KT
≥ 0 and C1 > 0, C2 > ω0 and
GA = GA (u(k−1)) = diag(
0, if 0 ≤ u(k−1)(xi) ≤ 11, otherwise
).
A is non-symmetric.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningNon-Smooth Case – Preconditioner
We consider the block-triangular preconditioner
P =
(M 0K −S
)where S is a Schur complement preconditioner.
The preconditioned matrix becomes
A = P−1A =
(I M−1(γεK + 1
νGA MGA )
0 −S−1S
)which has in the idealized case S = S only two distincteigenvalues.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningNon-Smooth Case – Preconditioner
We consider the block-triangular preconditioner
P =
(M 0K −S
)where S is a Schur complement preconditioner.
The preconditioned matrix becomes
A = P−1A =
(I M−1(γεK + 1
νGA MGA )
0 −S−1S
)which has in the idealized case S = S only two distincteigenvalues.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningNon-Smooth Case – Schur Complement Approximation
The Schur complement
S = −[(1τ
+ C2)M + C1K ] − KM−1(γεK +1ν
GA MGA )
is approximated by
S = −
√
1τ
+ C2M + K
︸ ︷︷ ︸AMG
M−1
√
1τ
+ C2M + (γεK +1ν
GA MGA )
︸ ︷︷ ︸AMG
.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningNon-Smooth Case – Schur Complement Approximation
0 500 100010
−1
100
101
Index
Eig
enva
lues
ν=10−1
ν=10−3
ν=10−5
ν=10−7
0 2000 400010
−2
10−1
100
101
102
IndexE
igen
valu
es
N=289N=1089N=4225
ε = 0.8, C1 = 3ε , C2 = 3 · 105.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningNon-Smooth Case – Schur Complement Approximation
0 500 1000
10−0.8
10−0.4
100
Index
Eig
enva
lues
ν=10−1
ν=10−3
ν=10−5
ν=10−7
0 2000 400010
−2
10−1
100
101
IndexE
igen
valu
es
N=289N=1089N=4225
ε = 0.8, C1 = 3ε , C2 = 3 · 107.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
PreconditioningNon-Smooth Case – Schur Complement Approximation
0.5 0.75 1−0.2
0
0.2
0.4
Eigenvalue Real Part
Eig
enva
lue
Imag
inar
y P
art
ν=10−1
ν=10−3
ν=10−5
ν=10−7
0.2 0.4 0.6 0.8 1 1.2−0.2
0
0.2
Eigenvalue Real PartE
igen
valu
e Im
agin
ary
Par
t
N=289N=1089N=4225
ε = 0.01, C1 = 3ε , C2 = 3 · 105.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsIteration Numbers – Smooth
0 100 200 300 400
10
15
20
Time step
Num
ber
of B
iCG
iter
atio
ns
N=16641N=66049N=263169N=1050625
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsIteration Numbers – Non-Smooth
0 200 40020
30
40
50
60
Time step
Ave
rage
num
ber
of B
iCG
iter
atio
nspe
r N
ewto
n st
ep
N=16641N=66049N=263169N=1050625
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsNon-Smooth vs. Smooth
n = 0 n = 134 n = 2024
n = 0 n = 158 n = 3276
Figure: Non-smooth (above) and smooth (below).Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 23/27
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsZebra
n = 0 n = 57 n = 758
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsQR Code
n = 0 n = 16715
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsInpainting in 3D
n = 0 n = 82 n = 160
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsResults and Outlook
ResultsNon-smooth Cahn-Hilliard inpainting model.
Efficient, robust preconditioners.Fast convergence rates.Nearly mesh independent iteration numbers.Better results with the non-smooth model.Application to 3D problems.
OutlookGrey/color inpainting.Vector-valued Cahn-Hilliard systems.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsResults and Outlook
ResultsNon-smooth Cahn-Hilliard inpainting model.Efficient, robust preconditioners.
Fast convergence rates.Nearly mesh independent iteration numbers.Better results with the non-smooth model.Application to 3D problems.
OutlookGrey/color inpainting.Vector-valued Cahn-Hilliard systems.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsResults and Outlook
ResultsNon-smooth Cahn-Hilliard inpainting model.Efficient, robust preconditioners.Fast convergence rates.
Nearly mesh independent iteration numbers.Better results with the non-smooth model.Application to 3D problems.
OutlookGrey/color inpainting.Vector-valued Cahn-Hilliard systems.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsResults and Outlook
ResultsNon-smooth Cahn-Hilliard inpainting model.Efficient, robust preconditioners.Fast convergence rates.Nearly mesh independent iteration numbers.
Better results with the non-smooth model.Application to 3D problems.
OutlookGrey/color inpainting.Vector-valued Cahn-Hilliard systems.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsResults and Outlook
ResultsNon-smooth Cahn-Hilliard inpainting model.Efficient, robust preconditioners.Fast convergence rates.Nearly mesh independent iteration numbers.Better results with the non-smooth model.
Application to 3D problems.
OutlookGrey/color inpainting.Vector-valued Cahn-Hilliard systems.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsResults and Outlook
ResultsNon-smooth Cahn-Hilliard inpainting model.Efficient, robust preconditioners.Fast convergence rates.Nearly mesh independent iteration numbers.Better results with the non-smooth model.Application to 3D problems.
OutlookGrey/color inpainting.Vector-valued Cahn-Hilliard systems.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsResults and Outlook
ResultsNon-smooth Cahn-Hilliard inpainting model.Efficient, robust preconditioners.Fast convergence rates.Nearly mesh independent iteration numbers.Better results with the non-smooth model.Application to 3D problems.
OutlookGrey/color inpainting.
Vector-valued Cahn-Hilliard systems.
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Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsResults and Outlook
ResultsNon-smooth Cahn-Hilliard inpainting model.Efficient, robust preconditioners.Fast convergence rates.Nearly mesh independent iteration numbers.Better results with the non-smooth model.Application to 3D problems.
OutlookGrey/color inpainting.Vector-valued Cahn-Hilliard systems.
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FEM vs. FFT Cahn-Hilliard Equations
FEM vs. FFTSmooth Case
n = 0
n = 880
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FEM vs. FFT Cahn-Hilliard Equations
FEM vs. FFTNon-Smooth Case
0 100 2000
200
400
600
Time step
Ave
rage
num
ber
of B
iCG
iter
atio
nspe
r N
ewto
n st
ep
ν=10−3
ν=10−4
ν=10−5
ν=10−6
ν=10−7
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FEM vs. FFT Cahn-Hilliard Equations
Cahn-Hilliard Equations
∂tu(t) = −gradH−1E(u(t))
∂tu = −∆(γε∆u −1ε
)
∂u∂n
=∂∆u∂n
= 0 on ∂Ω
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FEM vs. FFT Cahn-Hilliard Equations
Cahn-Hilliard Equations
∂tu(t) = −gradH−1E(u(t))
Smooth potential ψ(u) = u2(u − 1)2
∂tu = −∆(γε∆u −1εψ′(u))
∂u∂n
=∂∆u∂n
= 0 on ∂Ω
[Elliott ’89]
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FEM vs. FFT Cahn-Hilliard Equations
Cahn-Hilliard Equations
∂tu(t) = −gradH−1E(u(t))
Non-smooth potential ψ(u) = ψ0(u) + I[0,1](u)
∂tu = −∆(γε∆u −1ε
(ψ′0(u) + µ))
µ ∈ ∂β[0,1](u)
0 ≤ u ≤ 1∂u∂n
=∂∆u∂n
= 0 on ∂Ω
[Blowey/Elliott ’91/’92]
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