Fast Solvers for Cahn-Hilliard Inpainting€¦ · Jessica Bosch David Kay Martin Stoll Andrew J. Wathen Max Planck Institute for Dynamics of Complex Technical Systems, Research group

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



c©2012 Thomas Rolle



1 Phase Separation

2 Cahn-Hilliard System

3 Inpainting Model

4 Preconditioning

5 Numerical Results



Phase SeparationTwo-Phase Structure

Ω ⊂ Rd , d ∈ 2,3u = u(x , t): concentration

u ∈ [0,1]




Ω ⊂ Rd , d ∈ 2,3

u = u(x , t): concentration

u ∈ [0,1]





u ∈ [0,1]





u ∈ [0,1]



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)




E(u) =

∫Ω

γε

2|∇u|2 +

1εψ(u) dx

Smooth potential

ψ(u) = u2(u − 1)2


ψ(u) =

12u(1 − u), u ∈ [0,1]

∞, otherwise

= ψ0(u) + I[0,1](u)




E(u) =

∫Ω

γε

2|∇u|2 +

1εψ(u) dx

Smooth potential

ψ(u) = u2(u − 1)2


ψ(u) =

12u(1 − u), u ∈ [0,1]

∞, otherwise

= ψ0(u) + I[0,1](u)



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




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 +





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 +




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))



Cahn-Hilliard SystemRegularized Cahn-Hilliard System


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))



Cahn-Hilliard SystemPhase Separation in 2D

n = 0 n = 5 n = 50 n = 500

Taken from [Bosch/Stoll/Benner ’12].



Inpainting ModelIdea

Original image f withinpainting domain D.

Inpainted image.

ω(x) =

0, if x ∈ Dω0, if x ∈ Ω \ D



Inpainting ModelIdea

Original image f withinpainting domain D.

Inpainted image.

ω(x) =

0, if x ∈ Dω0, if x ∈ Ω \ D



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]



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 ))



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 ))



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.



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.



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.



PreconditioningSmooth Case – Coefficient Matrix

The coefficient matrix becomes

A =

(M γεKγεK −γε[(1

τ + C2)M + C1K ]


≥ 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.



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.



PreconditioningSmooth Case – Preconditioner


P =

(M 0γεK −S



A = P−1A =

(I γεM−1K0 −S−1S




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).





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).




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].




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].



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.



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 ]


≥ 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.



PreconditioningNon-Smooth Case – Preconditioner


P =

(M 0K −S



A = P−1A =

(I M−1(γεK + 1

νGA MGA )

0 −S−1S




PreconditioningNon-Smooth Case – Preconditioner


P =

(M 0K −S



A = P−1A =

(I M−1(γεK + 1

νGA MGA )

0 −S−1S




PreconditioningNon-Smooth Case – Schur Complement Approximation


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

.




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.




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.




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.



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



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



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


Numerical ResultsZebra

n = 0 n = 57 n = 758



Numerical ResultsQR Code

n = 0 n = 16715



Numerical ResultsInpainting in 3D

n = 0 n = 82 n = 160



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.




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.





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.





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.





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.





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.







FEM vs. FFT Cahn-Hilliard Equations

FEM vs. FFTSmooth Case

n = 0

n = 880



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



Cahn-Hilliard Equations


∂tu = −∆(γε∆u −1ε

)

∂u∂n

=∂∆u∂n

= 0 on ∂Ω





Smooth potential ψ(u) = u2(u − 1)2

∂tu = −∆(γε∆u −1εψ′(u))

∂u∂n

=∂∆u∂n

= 0 on ∂Ω

[Elliott ’89]





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]


Documents

Fast Solvers for Cahn-Hilliard Inpainting€¦ · Jessica Bosch David Kay Martin Stoll Andrew J. Wathen Max Planck Institute for Dynamics of Complex Technical Systems, Research group