A generalized MBO diffusion generated motion for constrained harmonic maps

Transcription

1 A generalized MBO diffusion generated motion for constrained harmonic maps Dong Wang Department of Mathematics, University of Utah Joint work with Braxton Osting (U. Utah) Workshop on Modeling and Simulation of Interface-related Problems IMS, NUS, Singapore, May. 3, 2018

2 Motion by mean curvature 1 Mean curvature flow arises in a variety of physical applications: Related to surface tension A model for the formation of grain boundaries in crystal growth Some ideas for numerical computation: We could parameterize the surface and compute H = 1 2 ˆn If the surface is implicitly defined by the equation F(x, y, z) = 0, then mean curvature can be computed H = 1 ( ) F 2 F

3 MBO diffusion generated motion 2 In 1989, Merriman, Bence, and Osher (MBO) developed an iterative method for evolving an interface by mean curvature. Repeat until convergence: Step 1. Solve the Cauchy problem for the diffusion equation (heat equation) u t = u u(x, t = 0) = χ D, with initial condition given by the indicator function χ D of a domain D until time τ to obtain the solution u(x, τ). Step 2. Obtain a domain D new by thresholding: D new = { x R d : u(x, τ) 1 2 }.

4 How to understand the MBO method? 3 Intuitively, from pictures, one can easily see: diffusion quickly blunts sharp points on the boundary and diffusion has little effect on the flatter parts of the boundary. Formally, consider a point P D. In local polar coordinates, the diffusion equation is given by u t = 1 r u r + 2 u r u 2 r 2 θ. 2 Considering local symmetry, we have u t = 1 r u r + 2 u r 2 = H u r + 2 u r 2. The 1 level set will move in the normal 2 direction with velocity given by the mean curvature, H. Initial t = t = t = 0.01 t = 0.02

5 A variational point of view: Modica+Mortola 4 Define the energy J ε (u) = Ω 1 2 u W (u) du ε2 ( where W (u) = 1 4 u 2 u ) 2 is a double well potential. Theorem (Modica+ Mortola, 1977) A minimizing sequence (u ε ) converges (along a subsequence) to χ D in L 1 for some D Ω. Furthermore, εj ε (u ε ) Hd 1 ( D) as ε 0. Gradient flow. The L 2 gradient flow of J ε gives the Allen-Cahn equation: u t = u 1 ε 2 W (u) in Ω.

6 A variational point of view: 5 Operator/energy splitting. Repeat the following two steps until convergence: Step 1. Solve the diffusion equation until time τ with initial condition u(x, t = 0) = χ D t u = u Step 2*. Solve the (pointwise defined!) equation until time τ: φ t = W (φ)/ε 2, φ(x, 0) = u(x, τ), in Ω. Step 2. Rescaling t = ε 2 t, we have as ε 0, ε 2 τ. So, Step 2* is equivalent to thresholding: { 1 if φ(x, 0) > 1/2 φ(x, ) = 0 if φ(x, 0) < 1/2.

7 Analysis, extensions, applications, connections, and computation 6 Proof of convergence of the MBO method to mean curvature flow [Evans1993, Barles and Georgelin 1995, Chambolle and Novaga 2006, Laux and Swartz 2017, Swartz and Yip 2017, Laux and Yip 2018]. Multi-phase problems with arbitrary surface tensions [Esedoglu and Otto 2015, Laux and Otto 2016] Numerical algorithms [Ruuth 1996, Ruuth 1998, Jiang et al. 2017] Area or volume preserving interface motion [Ruuth and Weston 2003] Image processing [Esedoglu et al. 2006, Merkurjev et al. 2013, Wang et al. 2017, Jacobs 2017] Problems of anisotropic interface motion [Merriman et al. 2000, Ruuth et al. 2001, Bonnetier et al. 2010, Elsey et al. 2016] Diffusion generated motion using signed distance function [Esedoglu et al. 2009] High order geometric motion [Esedoglu et al. 2008] Harmonic map of heat flow [E and Wang 2000, Wang et al. 2001, Ruuth et al. 2002] Nonlocal threshold dynamics method [Caffarelli and Souganidis 2010] Wetting problem on solid surfaces [Xu et al. 2017] Graph partitioning and data clustering [Van Gennip et al. 2013] Auction dynamics [Jacobs et al. 2017] Centroidal Voronoi Tessellation [Du et al. 1999] Quad meshing [Viertel and Osting 2017]

8 Generalized energies 7 Let Ω R d be a bounded domain with smooth boundary. Let T R k be "target set" and f : R k R + be a smooth function such that T = f 1 (0). T is the set of global minimizers of f. Roughly, we want f (x) dist 2 (x; T ). Consider a generalized variational problem, inf E(u) where E(u) = 1 u 2 dx u:ω T 2 Ω Relax the energy to obtain: 1 min E ε(u) where E ε(u) = u H 1 (Ω,R k ) Ω 2 u f (u(x)) dx. ε2 Examples. k T f(x) comment 1 1 {±1} 4 (x 2 1) 2 Allen-Cahn 2 S ( x 2 1) 2 Ginzburg-Landau k S k ( x 2 1) 2 n 2 1 O(n) 4 x t x I n 2 F Orthogonal matrix valued fields 1 k coordinate axes, Σ k 4 i j x i 2 xj 2 Dirichlet partitions RP 2 Landau-de Gennes model.

9 Orthogonal matrix valued fields 8 Let O n M n = R n n be the group of orthogonal matrices. inf E(A), where E(A) := 1 A 2 F dx. A:Ω O n 2 Relaxation: min E ε(a), where E ε(a) := A H 1 (Ω,M n) Ω Ω 1 2 A 2 F + 1 4ε 2 At A I n 2 F dx. The penalty term can be written: 1 4ε 2 At A I n 2 F = 1 n W (σ ε 2 i (A)), where W (x) = 1 4 i=1 Gradient Flow. The gradient flow of E ε is Special cases. ta = A E 2,ε (A) = A ε 2 A(A t A I n). For n = 1, we recover Allen-Cahn equation. For n = 2, if the initial condition is taken to be in SO(2) = S 1, we recover the complex Ginzburg-Landau equation. ( x 2 1) 2.

10 Diffusion generated motion for O n valued fields. 9 E ε(a) := Ω 1 2 A 2 F + 1 4ε 2 At A I n 2 F dx. k T f(x) comment n 2 O(n) 1 x I 4 n 2 F Orthogonal matrix valued fields Lemma. The nearest-point projection map, Π T : R n n T, for T = O n is given by Π T A = A(A t A) 1 2 = UV t, where A has the singular value decomposition, A = UΣV t. Diffusion generated method. For i = 1, 2,..., Step 1. Solve the diffusion equation until time τ with initial value given by A i (x): tu = u, u(x, t = 0) = A i (x). Step 2. Point-wise, apply the nearest-point projection map: A i+1 (x) = Π T u(x, τ).

11 Computational Example I: flat torus, n = 2 10 closed line defect vol. const. closed line defect parallel lines defect parallel lines defect O(n) = SO(n) SO (n), SO(2) = S 1 x is yellow det(a(x)) = 1 A(x) SO(n) x is blue det(a(x)) = 1 A(x) SO (n).

12 Computational Example II: sphere, n = 3 11 shrinking on sphere vol. const. on sphere

13 Computational Example III: peanut, n = 3 12 x(t, θ) = (3t t 3 ), y(t, θ) = 1 (1 + x 2 2 )(4 x 2 ) cos(θ), z(t, θ) = 1 (1 + x 2 2 )(4 x 2 ) sin(θ) peanut with closed geodesic

14 Lyapunov functional for MBO iterates 13 Motivated by [Esedoglu + Otto, 2015], we define the functional E τ : H 1 (Ω, M n ) R, given by E τ (A) := 1 n A, e τ A F dx τ Ω Here, e τ A denotes the solution to the heat equation at time τ with initial condition at time t = 0 given by A = A(x). Denoting the spectral norm by A 2 = σ max (A), the convex hull of O n is K n = conv O n = {A M n : A 2 1}.

15 Lyapunov functional for MBO iterates 14 Lemma. The functional E τ has the following elementary properties. (i) For A L 2 (Ω, O n ), (ii) E τ (A) is concave. (iii) We have E τ (A) = E(A) + O(τ). min E τ (A) = min E τ (A). A L 2 (Ω,O n) A L 2 (Ω,K n) (iv) E τ (A) is Fréchet differentiable with derivative L τ A : L (Ω, M n ) R at A in the direction B given by L τ A(B) = 2 e τ A, B F dx. τ Ω

16 Stability 15 The sequential linear programming approach to minimizing E τ (A) subject to A L (Ω, K n ) is to consider a sequence of functions {A s } s=0 which satisfies A s+1 = arg min A L (Ω,K Lτ A s (A), n) A 0 L (Ω, O n ) given. Lemma. If e Asτ = UΣV t, the solution to the linear optimization problem, min A L (Ω,K n) Lτ A s (A). is attained by the function A = UV t L (Ω, O n ). Thus, A s L (Ω, O n ) for all s 0 and these are precisely the iterations generated by the generalized MBO diffusion generated motion! Theorem (Stability). [Osting + W., 2017] The functional E τ is non-increasing on the iterates {A s } s=1, i.e., E τ (A s+1 ) E τ (A s ).

17 Convergence 16 We consider a discrete grid Ω = {x i } Ω i=1 Ω and a standard finite difference approximation of the Laplacian,, on Ω. For A: Ω O n, define the discrete functional Ẽ τ (A) = 1 τ 1 A i, (e τ A) i F x i Ω and its linearization by L τ A(B) = 2 τ B i, (e τ A) i F. x i Ω

18 Convergence 17 Theorem (Convergence for n = 1.) [Osting + W., 2017] Let n = 1. Non-stationary iterations of the generalized MBO diffusion generated motion strictly decrease the value of Ẽ τ and since the state space is finite, {±1} Ω, the algorithm converges in a finite number of iterations. Furthermore, for m := e τ, each iteration reduces the value of J by at least 2m, so the total number of iterations is less than Ẽ τ (A 0 )/2m. Theorem (Convergence for n 2.) [Osting + W., 2017] Let n 2. The non-stationary iterations of the generalized MBO diffusion generated motion strictly decrease the value of Ẽ τ. For a given initial condition A 0 : Ω O n, there exists a partition Ω = Ω + Ω and an S N such that for s S, { +1 x i det A s (x i ) = Ω +. 1 x i Ω Lemma. dist (SO(n), SO (n)) = 2.

19 Volume constrained problem 18 Definitions: D n : A diagonal matrix with diagonal entries 1 everywhere except in the n th position, where it is 1. T + (A) = { UV t, if det A > 0, UD n V t, if det A < 0. T (A) = { UDn V t, if det A > 0, UV t, if det A < 0. E(x) = T + (A(τ, x)) T (A(τ, x)), A(τ, x) F, Ω λ + = {x : E(x) λ}, Ω λ = Ω \ Ω λ +.

20 Volume constrained problem 19 Similar as that in [Ruuth+Weston, 2003], we treat the volume of Ω λ + as a function 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 λ 0 satisfies f (λ 0 ) = V. Then, { B T + (A(τ, x)) if E(x) λ 0 =. T (A(τ, x)) if E(x) < λ 0 attains the minimum in min L τ A(B) B L (Ω,O n) s.t. vol({x : B(x) SO(n)}) = V. Theorem (Stability). [Osting+W., 2017] For any τ > 0, the functional E τ, is non-increasing on the volume-preserving iterates {A s} s=1, i.e., E τ (A s+1 ) E τ (A s).

21 Dirichlet partitions 20 Let U R d be either an open bounded domain with Lipschitz boundary. Dirichlet k-partition: A collection of k disjoint open sets, U 1, U 2,..., U k U, attains inf U l U U l U m= k l=1 λ 1 (U l ) where λ 1 (U) := min E(u). u H0 1 (U) u L 2 (U) =1 E(u) = U u 2 dx is the Dirichlet energy and u L 2 (U) := ( U u2 (x)dx) 1 2. λ 1 (U) is the first Dirichlet eigenvalue of the Laplacian,. Monotonicity of eigenvalues = U = k l=1u l.

22 A mapping formulation of Dirichlet partitions [Cafferelli and Lin 2007] 21 Consider vector valued functions u = (u 1, u 2,..., u k ), that takes values in the singular space, Σ k, given by the coordinate axes, k Σ k := x Rk : xi 2 xj 2 = 0. i j The Dirichlet partition problem for U is equivalent to the mapping problem { } min E(u): u = (u 1,..., u k ) H0 1 (U; Σ k ), ul 2 (x) dx = 1 l [k], U where E(u) := k l=1 U u l 2 dx is the Dirichlet energy of u and H0 1(U; Σ k ) = {u H0 1(U, Rk ) : u Σ k a.e.}. Refer to minimizers u as ground states, and WLOG take u > 0 and quasi-continuous. U = l u is a ground state U l with U l = u 1 l ((0, )) for l [k] is a Dirichlet partition.

23 Diffusion generated method for computing Dirichlet partitions 22 Relaxed problem: 1 E ε(u) := Ω 2 u 2 F + 1 u 2 4ε 2 i (x)u 2 j (x) dx. i j k T f(x) comment 1 k coordinate axes, Σ k 4 i j x 2 i x 2 j Dirichlet partitions min E ε(u) s.t. u j L 2 (Ω) = 1. u H1(Ω;R k ) The nearest-point projection map, Π T : R k T, for T = Σ k is given by { x (Π T x) i = i x i = max j x j 0 otherwise Diffusion generated method. For i = 1, 2,..., Step 1. Solve the diffusion equation until time τ t φ = φ, φ(x, t = 0) = u i (x). Step 2. Point-wise, apply the nearest-point projection map: ũ i+1 (x) = Π T φ(x, τ). Step 3. Nomarlize: u i+1 (x) = ũi+1(x) ũ i+1 L 2 (Ω).

24 Results for 2D flat tori, k = 3 9, 11, 12, 15, 16, and 20 23

25 Results for 3D flat tori, k = 2 24

26 Results for 3D flat tori, k = 4, tessellation by rhombic dodecahedra 25

27 Results for 3D flat tori, k = 8, Weaire-Phelan structure 26

28 Results for 3D flat tori, k = 12, Kelvin s structure composed of truncated octahedra 27

29 Results for 4D flat tori, k = 8, 24-cell honeycomb 28 Dong Wang Generalized MBO May. 3, 2018 IMS, NUS, Singapore

30 Discussion and future directions for generalized MBO methods 29 We only considered a single matrix valued field that has two "phases given by when the determinant is positive or negative. It would be very interesting to extend this work to the multi-phase problem as was accomplished for n = 1 in [Esedoglu+Otto, 2015]. For O(n) valued fields with n = 2, the motion law for the interface is unknown. For n = 2 on a two-dimensional flat torus, further analysis regarding the winding number of the field is required. Is it possible to determine the final solution based on the winding number of the initial field? For problems with a non-trivial boundary condition, it not obvious how to adapt the Lyapunov functional. Thanks! Questions? dwang@math.utah.edu B. Osting and D. Wang, A generalized MBO diffusion generated motion for orthogonal matrix valued fields, submitted, arxiv: (2017). D. Wang and B. Osting, A diffusion generated method for computing Dirichlet partitions, submitted, arxiv: (2018).