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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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 1/27
Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
c©2012 Thomas Rolle
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 2/27
Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
c©2012 Thomas Rolle
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 2/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 3/27
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]
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 4/27
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]
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 4/27
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]
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 4/27
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]
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 4/27
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)
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 5/27
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)
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 5/27
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)
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 5/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 6/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 6/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 6/27
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))
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 7/27
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))
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 7/27
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].
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 8/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 9/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 9/27
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]
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 10/27
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 ))
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 11/27
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 ))
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 11/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 12/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 12/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 13/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 13/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 14/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 14/27
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).
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 15/27
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).
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 15/27
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].
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 16/27
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].
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 16/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 17/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 17/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 18/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 18/27
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
.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 19/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 20/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 20/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 20/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 21/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 22/27
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
Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsZebra
n = 0 n = 57 n = 758
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 24/27
Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsQR Code
n = 0 n = 16715
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 25/27
Phase Separation Cahn-Hilliard System Inpainting Model Preconditioning Numerical Results
Numerical ResultsInpainting in 3D
n = 0 n = 82 n = 160
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 26/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 27/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 27/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 27/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 27/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 27/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 27/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 27/27
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.
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 27/27
FEM vs. FFT Cahn-Hilliard Equations
FEM vs. FFTSmooth Case
n = 0
n = 880
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 28/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 29/27
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
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 30/27
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]
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 30/27
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]
Max Planck Institute Magdeburg J. Bosch, Fast Solvers for Cahn-Hilliard Inpainting 30/27