A generalized MBO diffusion generated method for constrained - PowerPoint PPT Presentation

A generalized MBO diffusion generated method for constrained harmonic maps Braxton Osting University of Utah February 10, 2018 Inverse Problems and Machine Learning Based on joint work with Dong Wang and Ryan Viertel 1/ 28

Motion by mean curvature 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 ∇ · ˆ n ◮ If the surface is implicitly defined by the equation F ( x , y , z ) = 0, then mean curvature can be computed � ∇ F H = − 1 � 2 ∇ · |∇ F | 2/ 28

MBO diffusion generated motion In 1992, 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: � x ∈ R d : u ( x , τ ) ≥ 1 � D new = . 2 3/ 28

How to understand the MBO method? 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 with the origin at P , the diffusion equation is given by ∂ r + ∂ 2 u ∂ 2 u ∂ u ∂ u ∂ t = 1 ∂ r 2 + 1 ∂θ 2 . r 2 r Considering local symmetry, we have ∂ r + ∂ 2 u ∂ u ∂ t = 1 ∂ u ∂ r 2 r ∂ r + ∂ 2 u = H ∂ u ∂ r 2 . The 1 2 level set will move in the normal direction with Initial velocity given by the mean curvature, H . t = 0.0025 t = 0.005 t = 0.01 t = 0.02 4/ 28

A variational point of view: Modica+Mortola, Allen+Cahn, Ginzburg+Landau Define the energy � 1 2 |∇ u ( x ) | 2 + 1 J ε ( u ) = ε 2 W ( u ( x )) dx Ω � 2 is a double well potential. u 2 − 1 where W ( u ) = 1 � 4 Theorem (Modica+ Mortola, 1977) A minimizing sequence ( u ε ) converges (along a subsequence) to χ D − χ Ω \ D in L 1 for some D ⊂ Ω . Furthermore, √ ε J ε ( u ε ) → 2 2 H d − 1 ( ∂ D ) as ε → 0 . 3 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 . 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 . t = ε − 2 t , we have as ε → 0, ε − 2 τ → ∞ . So, Step 2 is equivalent to ◮ Step 2*. Rescaling ˜ thresholding: � if φ ( x , 0 ) > 1 / 2 1 φ ( x , ∞ ) = if φ ( x , 0 ) < 1 / 2 . 0 5/ 28

Analysis, extensions, applications, connections, and computation ◮ 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]. ◮ Multi-phase problems with arbitrary surface tensions [Esedoglu and Otto 2015, Laux and Otto 2016] ◮ Numerical algorithms [Ruuth 1996, Ruuth 1998] ◮ Adaptive methods based on NUFFT [Jiang et. al. 2017] ◮ Area or volume preserving interface motion [Ruuth 2003] ◮ Image processing [Esedoglu et al. 2006, Merkurjev et al. 2013, Wang et. al. 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 2008] ◮ 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 1999] 6/ 28

Generalized energies Let Ω ⊂ R d be a bounded domain with smooth boundary. Let T ⊂ R k be the “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 the generalized variational problem, E ( u ) = 1 � |∇ u | 2 dx u : Ω → T E ( u ) inf where 2 Ω Relax the energy to obtain: � 1 1 2 |∇ u | 2 + E ε ( u ) E ε ( u ) = ε 2 f ( u ( x )) dx . min where u ∈ H 1 (Ω; R k ) Ω Examples. k T f(x) comment 4 ( x 2 − 1 ) 2 {± 1 } 1 1 Allen-Cahn 4 ( | x | 2 − 1 ) 2 S 1 1 2 Ginzburg-Landau n 2 4 � x t x − I n � 2 O ( n ) 1 orthogonal matrix valued fields F 1 i � = j x 2 i x 2 coordinate axes, Σ k � k Dirichlet partitions 4 j 4 ( | x | 2 − 1 ) 2 S k − 1 1 k RP 2 Landau-de Gennes model for nematic liquid crystals . . . 7/ 28

A diffusion generated method for the Ginzburg-Landau model � 1 1 � 2 2 |∇ u ( x ) | 2 + � | u ( x ) | 2 − 1 E ε ( u ) = dx . 4 ε 2 Ω k T f(x) comment 4 ( | x | 2 − 1 ) 2 1 S 1 2 Ginzburg-Landau The nearest-point projection map, Π T : R 2 → T , for T = S 1 is given by Π T x = x | x | . ◮ S. J. Ruuth, B. Merriman, J. Xin, and S. Osher, Diffusion-Generated Motion by Mean Curvature for Filaments, J. Nonlinear Sci. 11 (2001). Diffusion generated method. For i = 1 , 2 , . . . , ◮ Step 1. Solve the diffusion equation until time τ ∂ t u = ∆ u u ( x , t = 0 ) = φ i ◮ Step 2. Point-wise, apply the nearest-point projection map: φ i + 1 ( x ) = Π T u ( x , τ ) . 8/ 28

Application: Quad meshing, joint work with Ryan Viertel (U. Utah) Theorem [ Viertel + O. (2017)] If no separatrix of u converges to a limit cycle, then the separatrices of U , along with ∂ D partition D into a 4 sided partition. 9/ 28

Examples of quad meshes 10/ 28

Orthogonal matrix valued fields — joint work with Dong Wang (U. Utah) Let O n ⊂ M n = R n × n be the group of orthogonal matrices. � where E ( A ) := 1 �∇ A � 2 inf E ( A ) , F dx . A : Ω → O n 2 Ω Relaxation: � 1 1 2 �∇ A � 2 4 ε 2 � A t A − I n � 2 E ε ( A ) , where E ε ( A ) := F + F dx . min A ∈ H 1 (Ω , M n ) Ω The penalty term can be written: n 1 F = 1 where W ( x ) = 1 � 2 � x 2 − 1 � 4 ε 2 � A t A − I n � 2 W ( σ i ( A )) , . ε 2 4 i = 1 Gradient Flow. The gradient flow of E ε is ∂ t A = −∇ A E ε ( A ) = ∆ A − ε − 2 A ( A t A − I n ) . Special cases. ◮ 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. 11/ 28

Diffusion generated method for O n valued fields � 1 1 2 �∇ A � 2 4 ε 2 � A t A − I n � 2 E ε ( A ) := F + F dx . Ω k T f(x) comment n 2 4 � x t x − I n � 2 1 O ( n ) orthogonal matrix valued fields F 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 = U Σ V t . Diffusion generated method. For i = 1 , 2 , . . . , ◮ Step 1. Solve the diffusion equation until time τ ∂ t u = ∆ u u ( x , t = 0 ) = φ i ◮ Step 2. Point-wise, apply the nearest-point projection map: φ i + 1 ( x ) = Π T u ( x , τ ) . 12/ 28

Computational Example I: flat torus, n = 2 closed line defect vol. const. closed line defect parallel lines defect parallel lines defect O ( n ) = SO ( n ) ∪ SO − ( n ) , SO ( 2 ) ∼ = S 1 ⇐ ⇒ det ( A ( x )) = 1 ⇐ ⇒ A ( x ) ∈ SO ( n ) x is yellow A ( x ) ∈ SO − ( n ) . x is blue ⇐ ⇒ det ( A ( x )) = − 1 ⇐ ⇒ ind ( γ ) := 1 2 π [ arg v ( γ ( 1 )) − arg v ( γ ( 0 ))] 13/ 28

Computational Example II: sphere, n = 3 shrinking on sphere vol. const. on sphere 14/ 28

Computational Example III: peanut, n = 3 x ( t , θ ) = ( 3 t − t 3 ) , y ( t , θ ) = 1 � ( 1 + x 2 )( 4 − x 2 ) cos ( θ ) , 2 z ( t , θ ) = 1 � ( 1 + x 2 )( 4 − x 2 ) sin ( θ ) 2 peanut with closed geodesic 15/ 28

Lyapunov function for MBO iterates Let Ω be a closed surface. 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 } . Lemma. The functional E τ has the following elementary properties. (i) For A ∈ L 2 (Ω , O n ) , E τ ( A ) = E ( A ) + O ( τ ) . (ii) E τ ( A ) is concave. (iii) We have E τ ( A ) = E τ ( A ) . min min A ∈ L 2 (Ω , O n ) A ∈ L 2 (Ω , K n ) (iv) E τ ( A ) is Fr´ echet differentiable with derivative L τ A : L ∞ (Ω , M n ) → R at A in the direction B given by A ( B ) = − 2 � � e ∆ τ A , B � F dx . L τ τ Ω 16/ 28