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International Research Training Group IGDK 1754
A Level-Set Framework for Shape Optimisation
Daniel Kraft
November 30th, 2015
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 1
International Research Training Group IGDK 1754
Overview
1. The Level-Set MethodLevel Sets and the Speed MethodThe Hopf-Lax Formula
2. Gradient-Descent MethodsShape CalculusSteepest-Descent Directions
3. Numerical ResultsImage SegmentationPDE-Constrained Shape Optimisation
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 2
International Research Training Group IGDK 1754
The Level-Set Method
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 3
International Research Training Group IGDK 1754
The Level-Set Function
Geometries as level sets of φ : D ⊂ Rn → R:
Ω = Ω(φ) = x ∈ D | φ(x) < 0Γ = ∂Ω = x ∈ D | φ(x) = 0
Also irregular shapes are possible:
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 4
International Research Training Group IGDK 1754
The Level-Set Function
Geometries as level sets of φ : D ⊂ Rn → R:
Ω = Ω(φ) = x ∈ D | φ(x) < 0Γ = ∂Ω = x ∈ D | φ(x) = 0
Also irregular shapes are possible:
φ(x , y) =√x2 + y2 − 1 or φ(x , y) = x2 + y2 − 1
D Γ
Ω
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 4
International Research Training Group IGDK 1754
The Level-Set Function
Geometries as level sets of φ : D ⊂ Rn → R:
Ω = Ω(φ) = x ∈ D | φ(x) < 0Γ = ∂Ω = x ∈ D | φ(x) = 0
Also irregular shapes are possible:
φ(x , y) = max (|x | − 1, |y | − 1)
D Γ
Ω
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 4
International Research Training Group IGDK 1754
The Level-Set Function
Geometries as level sets of φ : D ⊂ Rn → R:
Ω = Ω(φ) = x ∈ D | φ(x) < 0Γ = ∂Ω = x ∈ D | φ(x) = 0
Also irregular shapes are possible:
φ(x , y) = min(√
(x − 2)2 + y2 − 1,√(x + 2)2 + y2 − 2
)− c
D D D
c = 0 c = 12 c = 1
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 4
International Research Training Group IGDK 1754
The Speed Method in Normal Direction
Speed eld F : D → R in normal direction:
a
b
c
Level-Set Equation
φt(x , t) + F (x) |∇φ(x , t)| = 0, φ0(x , 0) = φ0(x)
It has a unique viscosity solution, see Crandall, Ishii, Lions [3] and Giga [5].Original work by Osher, Sethian [6].
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 5
International Research Training Group IGDK 1754
The Speed Method in Normal Direction
Speed eld F : D → R in normal direction:
a
b
c
Level-Set Equation
φt(x , t) + F (x) |∇φ(x , t)| = 0, φ0(x , 0) = φ0(x)
It has a unique viscosity solution, see Crandall, Ishii, Lions [3] and Giga [5].Original work by Osher, Sethian [6].
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 5
International Research Training Group IGDK 1754
Changes in Topology Are Possible
-4 -2 0 2 4
-4
-2
0
2
4
0
0.2
0.4
0.6
0.8
1
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 6
International Research Training Group IGDK 1754
Changes in Topology Are Possible
-4 -2 0 2 4
-4
-2
0
2
4
-4
-2
0
2
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 6
International Research Training Group IGDK 1754
The Sign of F
Important is the sign of F :
Theorem
Let F = F+ − F− be the decomposition in positive and negative parts, and
φ±t (x , t) + F±(x)∣∣∇φ±(x , t)∣∣ = 0, φ±(x , 0) = ±φ0(x).
Then:
φ(x , t) =
φ+(x , t) F (x) > 0φ0(x) F (x) = 0
−φ−(x , t) F (x) < 0
F (x) = 0 means φ(x , ·) is constant: → Geometric constraints!
One can reduce all considerations to the case F ≥ 0.
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 7
International Research Training Group IGDK 1754
The Sign of F
Important is the sign of F :
Theorem
Let F = F+ − F− be the decomposition in positive and negative parts, and
φ±t (x , t) + F±(x)∣∣∇φ±(x , t)∣∣ = 0, φ±(x , 0) = ±φ0(x).
Then:
φ(x , t) =
φ+(x , t) F (x) > 0φ0(x) F (x) = 0
−φ−(x , t) F (x) < 0
F (x) = 0 means φ(x , ·) is constant: → Geometric constraints!
One can reduce all considerations to the case F ≥ 0.
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 7
International Research Training Group IGDK 1754
Mayer's Problem
Paths suited to F ≥ 0:
St(x) =ξ ∈W 1,∞([0, t]) | ξ(0) = x , |ξ′(τ)| ≤ F (ξ(τ)) for all τ ∈ [0, t]
Reachable set:
Rt(x) = ξ(t) | ξ ∈ St(x)
Mayer's Problem
V (x , t) = infξ∈St (x)
φ0(ξ(t)) = infy∈Rt (x)
φ0(y)
Hamilton-Jacobi-Bellman: The level-set equation!
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 8
International Research Training Group IGDK 1754
Mayer's Problem
Paths suited to F ≥ 0:
St(x) =ξ ∈W 1,∞([0, t]) | ξ(0) = x , |ξ′(τ)| ≤ F (ξ(τ)) for all τ ∈ [0, t]
Reachable set:
Rt(x) = ξ(t) | ξ ∈ St(x)
Mayer's Problem
V (x , t) = infξ∈St (x)
φ0(ξ(t)) = infy∈Rt (x)
φ0(y)
Hamilton-Jacobi-Bellman: The level-set equation!
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 8
International Research Training Group IGDK 1754
The Hopf-Lax Formula
Optimal-control theory implies a Hopf-Lax Formula:
Let d solve the Eikonal equation for the speed F :
F (x) |∇d(x)| = 1
d is the F -induced distance.
Theorem (Hopf-Lax Formula)
Let F ≥ 0 be Lipschitz continuous and have compact support in D. For all x
with F (x) > 0 and φ0(x) > 0, the solution of the level-set equation is given as:
φ(x , t) = inf φ0(y) | d(x , y) ≤ t
See also Falcone, Giorgi, Loreti [4] and Capuzzo-Dolcetta [2].
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 9
International Research Training Group IGDK 1754
The Hopf-Lax Formula
Optimal-control theory implies a Hopf-Lax Formula:
Let d solve the Eikonal equation for the speed F :
F (x) |∇d(x)| = 1
d is the F -induced distance.
Theorem (Hopf-Lax Formula)
Let F ≥ 0 be Lipschitz continuous and have compact support in D. For all x
with F (x) > 0 and φ0(x) > 0, the solution of the level-set equation is given as:
φ(x , t) = inf φ0(y) | d(x , y) ≤ t
See also Falcone, Giorgi, Loreti [4] and Capuzzo-Dolcetta [2].
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 9
International Research Training Group IGDK 1754
Representation of the Level-Set Domain
Distance to Initial Geometry
d0(x) = infy∈Ω0
d(x , y)
→ When does the advancing front hit x?
Ecient computation possible with Fast Marching (Sethian [7]).
Theorem (Representation Formula)
Let F ≥ 0. The time evolution of Ω0 is given by:
Ωt = x ∈ D | d0(x) < tΓt = x ∈ D | d0(x) = t
Generalisation is possible for arbitrary signs of F :→ Composite Fast Marching
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 10
International Research Training Group IGDK 1754
Representation of the Level-Set Domain
Distance to Initial Geometry
d0(x) = infy∈Ω0
d(x , y)
→ When does the advancing front hit x?
Ecient computation possible with Fast Marching (Sethian [7]).
Theorem (Representation Formula)
Let F ≥ 0. The time evolution of Ω0 is given by:
Ωt = x ∈ D | d0(x) < tΓt = x ∈ D | d0(x) = t
Generalisation is possible for arbitrary signs of F :→ Composite Fast Marching
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 10
International Research Training Group IGDK 1754
Demonstration
-4 -2 0 2 4
-4
-2
0
2
4
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 11
International Research Training Group IGDK 1754
Non-Fattening
Classical result for Γt = ∂Ωt : Non-fattening in a topological sense, consideringthe space-time. (See Barles, Soner, Souganidis [1].)
Based on our formula, one can easily deduce:
Theorem (Topological Non-Fattening)
Let F ≥ 0 and Ω0 = Ω0 ∪ Γ0. Then Ωt = Ωt ∪ Γt for all t ≥ 0.If F ≤ 0 and (Ω0 ∪ Γ0) = Ω0, then (Ωt ∪ Γt) = Ωt for all t ≥ 0.
Theorem (Measure-Theoretic Non-Fattening)
Let |Γ0| = 0. Then |Γt | = 0 for all t ≥ 0.
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 12
International Research Training Group IGDK 1754
Non-Fattening
Classical result for Γt = ∂Ωt : Non-fattening in a topological sense, consideringthe space-time. (See Barles, Soner, Souganidis [1].)
Based on our formula, one can easily deduce:
Theorem (Topological Non-Fattening)
Let F ≥ 0 and Ω0 = Ω0 ∪ Γ0. Then Ωt = Ωt ∪ Γt for all t ≥ 0.If F ≤ 0 and (Ω0 ∪ Γ0) = Ω0, then (Ωt ∪ Γt) = Ωt for all t ≥ 0.
Theorem (Measure-Theoretic Non-Fattening)
Let |Γ0| = 0. Then |Γt | = 0 for all t ≥ 0.
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 12
International Research Training Group IGDK 1754
Non-Fattening
Classical result for Γt = ∂Ωt : Non-fattening in a topological sense, consideringthe space-time. (See Barles, Soner, Souganidis [1].)
Based on our formula, one can easily deduce:
Theorem (Topological Non-Fattening)
Let F ≥ 0 and Ω0 = Ω0 ∪ Γ0. Then Ωt = Ωt ∪ Γt for all t ≥ 0.If F ≤ 0 and (Ω0 ∪ Γ0) = Ω0, then (Ωt ∪ Γt) = Ωt for all t ≥ 0.
Theorem (Measure-Theoretic Non-Fattening)
Let |Γ0| = 0. Then |Γt | = 0 for all t ≥ 0.
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 12
International Research Training Group IGDK 1754
Gradient-Descent Methods
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 13
International Research Training Group IGDK 1754
Shape Calculus
Theorem
Let f ∈ L1loc(D) and assume F ≥ 0. Then:
J(t) =
∫Ωt
f dx =
∫Ω0
f dx +
∫ t
0
∫Γs
Ff dσ ds
This yields the shape derivative of J:
dJ (Ωt ;F ) = J ′(t) =
∫Γt
Ff dσ
(In a weak sense, since J is absolutely continuous.)
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 14
International Research Training Group IGDK 1754
Shape Calculus
Theorem
Let f ∈ L1loc(D) and assume F ≥ 0. Then:
J(t) =
∫Ωt
f dx =
∫Ω0
f dx +
∫ t
0
∫Γs
Ff dσ ds
This yields the shape derivative of J:
dJ (Ωt ;F ) = J ′(t) =
∫Γt
Ff dσ
(In a weak sense, since J is absolutely continuous.)
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 14
International Research Training Group IGDK 1754
Total Shape Dierential
For a shape-dependent integrand:
J(Ω) =
∫Ω
f (x , Ω) dx ,
where for all xed x ∈ D:
f (x , Ωt) = f (x , Ω0) +
∫ t
0
f ′(x , Ωs) ds.
Theorem (Total Shape Dierential)
Denote J(t) = J(Ωt). Then J is absolutely continuous and
dJ (Ωt ;F ) = J ′(t) =
∫Γt
Ff dσ +
∫Ωt
f ′ dx .
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 15
International Research Training Group IGDK 1754
Total Shape Dierential
For a shape-dependent integrand:
J(Ω) =
∫Ω
f (x , Ω) dx ,
where for all xed x ∈ D:
f (x , Ωt) = f (x , Ω0) +
∫ t
0
f ′(x , Ωs) ds.
Theorem (Total Shape Dierential)
Denote J(t) = J(Ωt). Then J is absolutely continuous and
dJ (Ωt ;F ) = J ′(t) =
∫Γt
Ff dσ +
∫Ωt
f ′ dx .
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 15
International Research Training Group IGDK 1754
A Chain Rule
Scalar, shape-dependent quantity:
J(Ω) =
∫Ω
f (x ,G (Ω)) dx
(Multiple G 's are also possible.)
Theorem (Chain Rule)
Let f ∈ C 1, then t 7→ J(t) = J(Ωt) is absolutely continuous and
dJ (Ωt ;F ) = J ′(t) =
∫Γt
Ff dσ +
∫Ωt
∂f
∂GG ′ dx .
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 16
International Research Training Group IGDK 1754
A Chain Rule
Scalar, shape-dependent quantity:
J(Ω) =
∫Ω
f (x ,G (Ω)) dx
(Multiple G 's are also possible.)
Theorem (Chain Rule)
Let f ∈ C 1, then t 7→ J(t) = J(Ωt) is absolutely continuous and
dJ (Ωt ;F ) = J ′(t) =
∫Γt
Ff dσ +
∫Ωt
∂f
∂GG ′ dx .
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 16
International Research Training Group IGDK 1754
Gradients in H1
Generic form of the shape derivative:
dJ (Ω;F ) =
∫Γ
f (. . .)F dσ
→ linear functional in H1(D), operating on F
dJ (Ω; ·) lives on the boundary Γ (Hadamard-Zolésio structure theorem)
H1 Shape Gradient
Riesz representative F ∈ H1(D) of dJ (Ω; ·) as gradient:
∀G ∈ H1(D) : 〈F ,G 〉
β
=
∫D
(FG +
β
〈∇F ,∇G 〉) dx = dJ (Ω;G )
→ the solution denes a speed eld on the domain D
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 17
International Research Training Group IGDK 1754
Gradients in H1
Generic form of the shape derivative:
dJ (Ω;F ) =
∫Γ
f (. . .)F dσ
→ linear functional in H1(D), operating on F
dJ (Ω; ·) lives on the boundary Γ (Hadamard-Zolésio structure theorem)
H1 Shape Gradient
Riesz representative F ∈ H1(D) of dJ (Ω; ·) as gradient:
∀G ∈ H1(D) : 〈F ,G 〉
β
=
∫D
(FG +
β
〈∇F ,∇G 〉) dx = dJ (Ω;G )
→ the solution denes a speed eld on the domain D
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 17
International Research Training Group IGDK 1754
Gradients in H1
Generic form of the shape derivative:
dJ (Ω;F ) =
∫Γ
f (. . .)F dσ
→ linear functional in H1(D), operating on F
dJ (Ω; ·) lives on the boundary Γ (Hadamard-Zolésio structure theorem)
H1 Shape Gradient
Riesz representative F ∈ H1(D) of dJ (Ω; ·) as gradient:
∀G ∈ H1(D) : 〈F ,G 〉β =
∫D
(FG + β 〈∇F ,∇G 〉) dx = dJ (Ω;G )
→ the solution denes a speed eld on the domain D
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 17
International Research Training Group IGDK 1754
Gradient-Descent Method
One step in the gradient descent:
1. evaluate shape-dependent quantities for Ω
2. calculate functional dJ (Ω; ·)3. nd Riesz representative: gradient F
4. evolve Ω with a line search in direction −F
Repeat until no more changes are made.
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 18
International Research Training Group IGDK 1754
Numerical Results
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 19
International Research Training Group IGDK 1754
Image Segmentation
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 20
International Research Training Group IGDK 1754
Image Segmentation
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 20
International Research Training Group IGDK 1754
Cost and Shape Derivative
Consider D ⊂ Rn, e. g., D = [0, 1]2.Let u : D → R be a grey-scale image.
Denition (Cost Function)
J(Ω) =
∫Ω
(u − u)2 dx − 2γ · σ · |Ω| for ∅ 6= Ω ⊂ D
Shape Derivative
dJ (Ω;F ) =
∫Γ
((u − u)2
(1−
γ
σ
)− γσ
)F dσ
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 21
International Research Training Group IGDK 1754
Cost and Shape Derivative
Consider D ⊂ Rn, e. g., D = [0, 1]2.Let u : D → R be a grey-scale image.
Denition (Cost Function)
J(Ω) =
∫Ω
(u − u)2 dx − 2γ · σ · |Ω| for ∅ 6= Ω ⊂ D
Shape Derivative
dJ (Ω;F ) =
∫Γ
((u − u)2
(1−
γ
σ
)− γσ
)F dσ
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 21
International Research Training Group IGDK 1754
Cost and Shape Derivative
Consider D ⊂ Rn, e. g., D = [0, 1]2.Let u : D → R be a grey-scale image.
Denition (Cost Function)
J(Ω) =
∫Ω
(u − u)2 dx − 2γ · σ · |Ω| for ∅ 6= Ω ⊂ D
Shape Derivative
dJ (Ω;F ) =
∫Γ
((u − u)2
(1−
γ
σ
)− γσ
)F dσ
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 21
International Research Training Group IGDK 1754
Eect of β
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 22
International Research Training Group IGDK 1754
Eect of β
β = 1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 22
International Research Training Group IGDK 1754
Eect of β
β = 10−2
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
-1
0
1
2
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 22
International Research Training Group IGDK 1754
Descent Run
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 23
International Research Training Group IGDK 1754
Descent Run
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 23
International Research Training Group IGDK 1754
Descent Run
0 20 40 60 80 1001e-7
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
Steps
Cost
Gradient Norm
Step Length
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 23
International Research Training Group IGDK 1754
PDE-Constrained Shape Optimisation
Let D ⊂ R2 be compact, B ⊂ D, f ∈ L2(D) and ud ∈ L2(B).
Denition (Cost Function)
Find Ω with B ⊂ Ω ⊂ D that minimises
J(Ω) =1
2‖u − ud‖2L2(B) + α |Γ | .
State Equation
u ∈ H1(Ω) solves the state equation:−∆u + u = f in Ω
∂u∂ν = 0 on Γ
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 24
International Research Training Group IGDK 1754
PDE-Constrained Shape Optimisation
Let D ⊂ R2 be compact, B ⊂ D, f ∈ L2(D) and ud ∈ L2(B).
Denition (Cost Function)
Find Ω with B ⊂ Ω ⊂ D that minimises
J(Ω) =1
2‖u − ud‖2L2(B) + α |Γ | .
State Equation
u ∈ H1(Ω) solves the state equation:−∆u + u = f in Ω
∂u∂ν = 0 on Γ
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 24
International Research Training Group IGDK 1754
The Shape Derivative
Equations in Weak Form
Find u and p such that (for each v ∈ H1(D)):∫Ω
(〈∇u,∇v〉+ uv) dx =
∫Ω
fv dx∫Ω
(〈∇p,∇v〉+ pv) dx =
∫B
(u − ud)v dx
Shape Derivative
dJ (Ω;F ) =
∫Γ
(fp − 〈∇u,∇p〉 − up + ακ)F dσ
We require: F = 0 on B
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 25
International Research Training Group IGDK 1754
The Shape Derivative
Equations in Weak Form
Find u and p such that (for each v ∈ H1(D)):∫Ω
(〈∇u,∇v〉+ uv) dx =
∫Ω
fv dx∫Ω
(〈∇p,∇v〉+ pv) dx =
∫B
(u − ud)v dx
Shape Derivative
dJ (Ω;F ) =
∫Γ
(fp − 〈∇u,∇p〉 − up + ακ)F dσ
We require: F = 0 on B
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 25
International Research Training Group IGDK 1754
Descent Run
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 26
International Research Training Group IGDK 1754
Descent Run
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 26
International Research Training Group IGDK 1754
Descent Run
0 100 200 300 400 500 600 7001e-5
1e-4
1e-3
1e-2
1e-1
1e+0
1e+1
1e+2
Steps
Cost
Gradient Norm
Step Length
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 26
International Research Training Group IGDK 1754
Conclusion
I Level sets allow a exible description of shapes.
I A Hopf-Lax formula can be employed for the time evolution.
I This yields a special shape calculus.
I One can formulate gradient methods for shape optimisation.
Thanks for your attention!
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 27
International Research Training Group IGDK 1754
Conclusion
I Level sets allow a exible description of shapes.
I A Hopf-Lax formula can be employed for the time evolution.
I This yields a special shape calculus.
I One can formulate gradient methods for shape optimisation.
Thanks for your attention!
Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 27
International Research Training Group IGDK 1754
References I
I G. Barles, H. M. Soner, and P. E. Souganidis.Front Propagation and Phase Field Theory.SIAM Journal on Control and Optimization, 31(2):439469, March 1993.
I Italo Capuzzo-Dolcetta.A Generalized Hopf-Lax Formula: Analytical and Approximations Aspects.In Fabio Ancona, editor, Geometric Control and Nonsmooth Analysis,volume 76 of Series on Advances in Mathematics for Applied Sciences, pages136150. World Scientic, 2008.
I Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions.User's Guide to Viscosity Solutions of Second Order Partial DierentialEquations.Bulletin of the American Mathematical Society, 27(1):167, July 1992.
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International Research Training Group IGDK 1754
References II
I M. Falcone, T. Giorgi, and P. Loreti.Level Sets of Viscosity Solutions: Some Applications to Fronts andRendez-Vous Problems.SIAM Journal on Applied Mathematics, 54(5):13351354, 1994.
I Yoshikazu Giga.Surface Evolution Equations: A Level Set Approach, volume 99 ofMonographs in Mathematics.Birkhäuser, 2006.
I Stanley J. Osher and James A. Sethian.Fronts Propagating with Curvature-Dependent Speed: Algorithms Based onHamilton-Jacobi Formulations.Journal of Computational Physics, 79:1249, 1988.
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International Research Training Group IGDK 1754
References III
I James A. Sethian.A Fast Marching Level Set Method for Monotonically Advancing Fronts.Proceedings of the National Academy of Sciences, 93(4):15911595, 1996.
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