a generalized mbo diffusion generated motion for …a generalized mbo diffusion generated motion for...
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A generalized MBO diffusion generated motionfor constrained harmonic maps
Dong WangDepartment of Mathematics, University of Utah
Joint work with Braxton Osting (U. Utah)
Workshop on Modeling and Simulation of Interface-related ProblemsIMS, NUS, Singapore, May. 3, 2018
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Motion by mean curvature
Mean curvature flow arises in a variety of physical applications:I Related to surface tensionI A model for the formation of grain boundaries in crystal growth
Some ideas for numerical computation:I We could parameterize the surface and compute
H = −12∇ · n
I If the surface is implicitly defined by the equation F (x , y , z) = 0, thenmean curvature can be computed
H = −12∇ ·(∇F|∇F |
)Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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MBO diffusion generated motion
In 1989, Merriman, Bence, and Osher (MBO) developed an iterative methodfor evolving an interface by mean curvature.Repeat until convergence:Step 1. Solve the Cauchy problem for the diffusion equation (heat equation)
ut = ∆u
u(x , t = 0) = χD,
with initial condition given by the indicator function χD of a domain D untiltime τ to obtain the solution u(x , τ).Step 2. Obtain a domain Dnew by thresholding:
Dnew =
x ∈ Rd : u(x , τ) ≥ 1
2
.
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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How to understand the MBO method?
Intuitively, from pictures, one can easily see:I diffusion quickly blunts sharp points on the boundary andI diffusion has little effect on the flatter parts of the boundary.
Formally, consider a point P ∈ ∂D. In localpolar coordinates, the diffusion equation isgiven by
∂u∂t
=1r∂u∂r
+∂2u∂r 2 +
1r 2
∂2u∂θ2 .
Considering local symmetry, we have
∂u∂t
=1r∂u∂r
+∂2u∂r 2
= H∂u∂r
+∂2u∂r 2 .
The 12 level set will move in the normal
direction with velocity given by the meancurvature, H.
Initial
t = 0.0025
t = 0.005
t = 0.01
t = 0.02
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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A variational point of view: Modica+Mortola
Define the energy
Jε(u) =
∫Ω
12|∇u|2 +
1ε2 W (u) du
where W (u) = 14
(u2 − u
)2 is a double well potential.Theorem (Modica+ Mortola, 1977) A minimizing sequence (uε)converges (along a subsequence) to χD in L1 for some D ⊂ Ω.Furthermore,
εJε(uε)→2√
23Hd−1(∂D) as ε→ 0.
Gradient flow. The L2 gradient flow of Jε gives the Allen-Cahnequation:
ut = ∆u − 1ε2 W ′(u) in Ω.
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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A variational point of view:
Operator/energy splitting. Repeat the following two steps untilconvergence:
I Step 1. Solve the diffusion equation until time τ with initialcondition u(x , t = 0) = χD
∂tu = ∆u
I Step 2*. Solve the (pointwise defined!) equation until time τ :
φt = −W ′(φ)/ε2, φ(x ,0) = u(x , τ), in Ω.
I Step 2. Rescaling t = ε−2t , we have as ε→ 0, ε−2τ →∞. So,Step 2* is equivalent to thresholding:
φ(x ,∞) =
1 if φ(x ,0) > 1/20 if φ(x ,0) < 1/2
.
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Analysis, extensions, applications, connections,and computation
I Proof of convergence of the MBO method to mean curvature flow [Evans1993,Barles and Georgelin 1995, Chambolle and Novaga 2006, Laux and Swartz2017, Swartz and Yip 2017, Laux and Yip 2018].
I Multi-phase problems with arbitrary surface tensions [Esedoglu and Otto 2015,Laux and Otto 2016]
I Numerical algorithms [Ruuth 1996, Ruuth 1998, Jiang et al. 2017]I Area or volume preserving interface motion [Ruuth and Weston 2003]I Image processing [Esedoglu et al. 2006, Merkurjev et al. 2013, Wang et al. 2017,
Jacobs 2017]I Problems of anisotropic interface motion [Merriman et al. 2000, Ruuth et al.
2001, Bonnetier et al. 2010, Elsey et al. 2016]I Diffusion generated motion using signed distance function [Esedoglu et al. 2009]I High order geometric motion [Esedoglu et al. 2008]I Harmonic map of heat flow [E and Wang 2000, Wang et al. 2001, Ruuth et al.
2002]I Nonlocal threshold dynamics method [Caffarelli and Souganidis 2010]I Wetting problem on solid surfaces [Xu et al. 2017]I Graph partitioning and data clustering [Van Gennip et al. 2013]I Auction dynamics [Jacobs et al. 2017]I Centroidal Voronoi Tessellation [Du et al. 1999]I Quad meshing [Viertel and Osting 2017]
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Generalized energies
I Let Ω ⊂ Rd be a bounded domain with smooth boundary.I Let T ⊂ Rk be "target set" and f : Rk → R+ be a smooth function such that
T = f−1(0).I T is the set of global minimizers of f . Roughly, we want f (x) ≈ dist2(x ; T ).
Consider a generalized variational problem,
infu:Ω→T
E(u) where E(u) =12
∫Ω|∇u|2 dx
Relax the energy to obtain:
minu∈H1(Ω,Rk )
Eε(u) where Eε(u) =
∫Ω
12|∇u|2 +
1ε2
f (u(x)) dx .
Examples.
k T f(x) comment1 ±1 1
4 (x2 − 1)2 Allen-Cahn2 S1 1
4 (|x |2 − 1)2 Ginzburg-Landauk Sk−1 1
4 (|x |2 − 1)2
n2 O(n) 14‖x
t x − In‖2F Orthogonal matrix valued fields
k coordinate axes, Σk14∑
i 6=j x2i x2
j Dirichlet partitionsRP2 Landau-de Gennes model
...Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Orthogonal matrix valued fields
Let On ⊂ Mn = Rn×n be the group of orthogonal matrices.
infA:Ω→On
E(A), where E(A) :=12
∫Ω
‖∇A‖2F dx .
Relaxation:
minA∈H1(Ω,Mn)
Eε(A), where Eε(A) :=
∫Ω
12‖∇A‖2
F +1
4ε2 ‖AtA− In‖2
F dx .
The penalty term can be written:
14ε2 ‖A
tA− In‖2F =
1ε2
n∑i=1
W (σi (A)) , where W (x) =14
(x2 − 1
)2.
Gradient Flow. The gradient flow of Eε is
∂tA = −∇AE2,ε(A) = ∆A− ε−2A(AtA− In).
Special cases.I For n = 1, we recover Allen-Cahn equation.I For n = 2, if the initial condition is taken to be in SO(2) ∼= S1, we
recover the complex Ginzburg-Landau equation.Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Diffusion generated motion for On valued fields.
Eε(A) :=
∫Ω
12‖∇A‖2
F +1
4ε2 ‖AtA− In‖2
F dx .
k T f(x) commentn2 O(n) 1
4‖xtx − In‖2
F Orthogonal matrix valued fields
Lemma. The nearest-point projection map, ΠT : Rn×n → T , for T = On isgiven by
ΠT A = A(AtA)−12 = UV t ,
where A has the singular value decomposition, A = UΣV t .Diffusion generated method. For i = 1, 2, . . . ,
I Step 1. Solve the diffusion equation until time τ with initial value givenby Ai (x):
∂tu = ∆u, u(x , t = 0) = Ai (x).
I Step 2. Point-wise, apply the nearest-point projection map:
Ai+1(x) = ΠT u(x , τ).
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Computational Example I: flat torus, n = 2
closed line defect vol. const. closed line defect
parallel lines defect parallel lines defectO(n) = SO(n) ∪ SO−(n), SO(2) ∼= S1
x is yellow ⇐⇒ det(A(x)) = 1 ⇐⇒ A(x) ∈ SO(n)x is blue ⇐⇒ det(A(x)) = −1 ⇐⇒ A(x) ∈ SO−(n).
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Computational Example II: sphere, n = 3
shrinking on sphere vol. const. on sphere
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Computational Example III: peanut, n = 3
x(t , θ) = (3t − t3),
y(t , θ) =12
√(1 + x2)(4− x2) cos(θ),
z(t , θ) =12
√(1 + x2)(4− x2) sin(θ)
peanut with closed geodesic
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Lyapunov functional for MBO iterates
Motivated by [Esedoglu + Otto, 2015], we define the functionalEτ : H1(Ω,Mn)→ R, given by
Eτ (A) :=1τ
∫Ω
n − 〈A,e∆τA〉F dx
Here, e∆τA denotes the solution to the heat equation at time τ withinitial condition at time t = 0 given by A = A(x).
Denoting the spectral norm by ‖A‖2 = σmax(A), the convex hull of Onis
Kn = conv On = A ∈ Mn : ‖A‖2 ≤ 1.
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Lyapunov functional for MBO iterates
Lemma. The functional Eτ has the following elementary properties.(i) For A ∈ L2(Ω,On), Eτ (A) = E(A) + O(τ).(ii) Eτ (A) is concave.(iii) We have
minA∈L2(Ω,On)
Eτ (A) = minA∈L2(Ω,Kn)
Eτ (A).
(iv) Eτ (A) is Fréchet differentiable with derivativeLτA : L∞(Ω,Mn)→ R at A in the direction B given by
LτA(B) = −2τ
∫Ω
〈e∆τA,B〉F dx .
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Stability
The sequential linear programming approach to minimizing Eτ (A)subject to A ∈ L∞(Ω,Kn) is to consider a sequence of functionsAs∞s=0 which satisfies
As+1 = arg minA∈L∞(Ω,Kn)
LτAs(A), A0 ∈ L∞(Ω,On) given.
Lemma. If eAsτ = UΣV t , the solution to the linear optimizationproblem,
minA∈L∞(Ω,Kn)
LτAs(A).
is attained by the function A? = UV t ∈ L∞(Ω,On).
Thus, As ∈ L∞(Ω,On) for all s ≥ 0 and these are precisely theiterations generated by the generalized MBO diffusion generatedmotion!
Theorem (Stability). [Osting + W., 2017] The functional Eτ isnon-increasing on the iterates As∞s=1, i.e., Eτ (As+1) ≤ Eτ (As).
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Convergence
We consider a discrete grid Ω = xi|Ω|i=1 ⊂ Ω and a standard finitedifference approximation of the Laplacian, ∆, on Ω. For A : Ω→ On,define the discrete functional
Eτ (A) =1τ
∑xi∈Ω
1− 〈Ai , (e∆τA)i〉F
and its linearization by
LτA(B) = −2τ
∑xi∈Ω
〈Bi , (e∆τA)i〉F .
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Convergence
Theorem (Convergence for n = 1.) [Osting + W., 2017]Let n = 1. Non-stationary iterations of the generalized MBO diffusiongenerated motion strictly decrease the value of Eτ and since thestate space is finite, ±1|Ω|, the algorithm converges in a finitenumber of iterations. Furthermore, for m := e−‖∆‖τ , each iterationreduces the value of J by at least 2m, so the total number of iterationsis less than Eτ (A0)/2m.
Theorem (Convergence for n ≥ 2.) [Osting + W., 2017]Let n ≥ 2. The non-stationary iterations of the generalized MBOdiffusion generated motion strictly decrease the value of Eτ . For agiven initial condition A0 : Ω→ On, there exists a partitionΩ = Ω+ q Ω− and an S ∈ N such that for s ≥ S,
det As(xi ) =
+1 xi ∈ Ω+
−1 xi ∈ Ω−.
Lemma. dist (SO(n), SO−(n)) = 2.Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Volume constrained problem
Definitions:I Dn: A diagonal matrix with diagonal entries 1 everywhere except
in the n−th position, where it is −1.I
T +(A) =
UV t , if det A > 0,
UDnV t , if det A < 0.I
T−(A) =
UDnV t , if det A > 0,
UV t , if det A < 0.I
∆E(x) =⟨T + (A(τ, x))− T− (A(τ, x)) ,A(τ, x)
⟩F ,
I
Ωλ+ = x : ∆E(x) ≥ λ, Ωλ
− = Ω \ Ωλ+.
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Volume constrained problem
Similar as that in [Ruuth+Weston, 2003], we treat the volume of Ωλ+ as afunction of λ and identify the value λ such that f (λ) = V .For the matrix case, the following lemma leads us to the optimal choice:Lemma. Assume λ0 satisfies f (λ0) = V . Then,
B? =
T +(A(τ, x)) if ∆E(x) ≥ λ0
T−(A(τ, x)) if ∆E(x) < λ0.
attains the minimum in
minB∈L∞(Ω,On)
LτA(B)
s.t. vol(x : B(x) ∈ SO(n)) = V .
Theorem (Stability). [Osting+W., 2017] For any τ > 0, the functional Eτ , isnon-increasing on the volume-preserving iterates As∞s=1, i.e.,Eτ (As+1) ≤ Eτ (As).
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Dirichlet partitions
Let U ⊂ Rd be either an open bounded domain with Lipschitz boundary.Dirichlet k-partition: A collection of k disjoint open sets, U1,U2, . . . ,Uk ⊆ U,attains
infU`⊂U
U`∩Um=∅
k∑`=1
λ1(U`) where λ1(U) := minu∈H1
0 (U)
‖u‖L2(U)=1
E(u).
E(u) =∫
U |∇u|2 dx is the Dirichlet energy and ‖u‖L2(U) := (∫
U u2(x)dx)12 .
I λ1(U) is the first Dirichlet eigenvalue of the Laplacian, −∆.I Monotonicity of eigenvalues =⇒ U = ∪k
`=1U`.
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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A mapping formulation of Dirichlet partitions [Cafferelli and Lin 2007]
Consider vector valued functions u = (u1, u2, . . . , uk ), that takes values in the singularspace, Σk , given by the coordinate axes,
Σk :=
x ∈ Rk :k∑
i 6=j
x2i x2
j = 0
.
The Dirichlet partition problem for U is equivalent to the mapping problem
min
E(u) : u = (u1, . . . , uk ) ∈ H1
0 (U; Σk ),
∫U
u2` (x) dx = 1 ∀ ` ∈ [k ]
,
where E(u) :=∑k`=1∫
U |∇u`|2 dx is the Dirichlet energy of u andH1
0 (U; Σk ) = u ∈ H10 (U,Rk ) : u ∈ Σk a.e..
Refer to minimizers u as ground states, and WLOG take u > 0 and quasi-continuous.
u is a ground state
⇐⇒
U =∐`
U` with U` = u−1` ((0,∞)) for ` ∈ [k ] is a Dirichlet partition.
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Diffusion generated method for computing Dirichlet partitions
Eε(u) :=
∫Ω
12‖∇u‖2
F +1
4ε2
∑i 6=j
u2i (x)u2
j (x) dx.
k T f(x) commentk coordinate axes, Σk
14∑
i 6=j x2i x2
j Dirichlet partitions
I Relaxed problem:min
u∈H1(Ω;Rk )Eε(u) s.t. ‖uj‖L2(Ω)
= 1.
The nearest-point projection map, ΠT : Rk → T , for T = Σk is given by
(ΠT x)i =
xi xi = maxj xj
0 otherwise
Diffusion generated method. For i = 1, 2, . . . ,I Step 1. Solve the diffusion equation until time τ
∂tφ = ∆φ, φ(x, t = 0) = ui (x).
I Step 2. Point-wise, apply the nearest-point projection map:
ui+1(x) = ΠTφ(x, τ).
I Step 3. Nomarlize:
ui+1(x) =ui+1(x)
‖ui+1‖L2(Ω)
.
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Results for 2D flat tori, k = 3− 9,11,12,15,16, and 20
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Results for 3D flat tori, k = 2
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Results for 3D flat tori, k = 4, tessellation by rhombic do-decahedra
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Results for 3D flat tori, k = 8, Weaire-Phelan structure
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Results for 3D flat tori, k = 12, Kelvin’s structure com-posed of truncated octahedra
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Results for 4D flat tori, k = 8, 24-cell honeycomb
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore
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Discussion and future directions for generalized MBO methods
I We only considered a single matrix valued field that has two "phases”given by when the determinant is positive or negative. It would be veryinteresting to extend this work to the multi-phase problem as wasaccomplished for n = 1 in [Esedoglu+Otto, 2015].
I For O(n) valued fields with n = 2, the motion law for the interface isunknown.
I For n = 2 on a two-dimensional flat torus, further analysis regarding thewinding number of the field is required. Is it possible to determine thefinal solution based on the winding number of the initial field?
I For problems with a non-trivial boundary condition, it not obvious how toadapt the Lyapunov functional.
Thanks! Questions? Email: [email protected]
I B. Osting and D. Wang, A generalized MBO diffusion generated motion for orthogonal matrixvalued fields, submitted, arXiv:1711.01365 (2017).
I D. Wang and B. Osting, A diffusion generated method for computing Dirichlet partitions,submitted, arXiv:1802.02682 (2018).
Dong Wang | Generalized MBO May. 3, 2018 IMS, NUS, Singapore