Convergence of a large time-step scheme for Mean Curvature Motion

Transcription

1 Outline Convergence of a large time-step scheme for Mean Curvature Motion M. Falcone Dipartimento di Matematica SAPIENZA, Università di Roma in collaboration with E. Carlini (SAPIENZA) and R. Ferretti (Roma Tre) College de France, June 2, 29

2 Outline Outline Introduction

3 Outline Introduction Introduction

4 Level Set Method The Level Set method has had a great success for the analysis of front propagation problems for its capability to handle many different physical phenomena within the same theoretical framework. One can use it for isotropic and anisotropic front propagation, for merging different fronts, for Mean Curvature Motion (MCM) and other situations when the velocity depends on some geometrical properties of the front. It allows to develop the analysis also after the on set of singularities.

5 Level Set Method Our unknown is the representation function u : R n [,T] R and looking at the -level set of u we can back to the front, i.e. Γ t {x : u(x,t) = } The model equation corresponding to the LS method is { ut + c(x) u(x) = x R n [,T] u(x) = u (x) x R n where u must be a representation function for the front (i.e. u >, x R n \ Ω u(x) = x R n () u(x) < x Ω The front Γ = Ω.

6 Front propagation and minimum time problem Let the velocity of the front in the normal direction is a given c : R n R Minimum time problem Target set =Ω Dynamics { ẏ(t) = c(y)a, a B(,) Minimum time function y() = x (2) T(x) inf{t R + : y x (t;a(t)) Ω }

7 Monotone evolution T( ) is the unique viscosity solution of max a B(,) {c(y)a T(x)} = Rn \ Ω with the Dirichlet condition T(x) = on Ω If c does not change sign the evolution is monotone (increasing or decreasing) and we have the following link u(x,t) = T(x) t So we can solve the stationary problem and get any front Γ t, for t >.

8 More General Models In the standard model the normal velocity c : R n R is given, but the same approach applies to other scalar velocities c(x,t) isotropic growth with time varying velocity c(x,η) anisotropic growth, cristal growth c(x,k(x)) Mean Curvature Motion c(x) obtained by convolution (dislocation dynamics)

9 Outline Introduction Introduction

10 SL schemes for MCM u t (x,t) = div u(x,) = u (x) Representation formula (Soner Touzi): ( ) Du(x,t) Du(x,t) Du(x, t) u(x,t) = E{u (y(x,t,t))}, Du { dy(x,t,s) = 2P(y,t,s)dW (s) y(x,t,) = x P(y,t,s) = ( u 2 ) x 2 u x u x2 Du 2 u x u x2 ux 2

11 Soner Touzi formula between t and t + t : u(x,t + t) = E{u(y(x,t + t, t),t)} Brownian dimension reduction 2PdW = 2 Du ( ux2 u x )( ux2 dw Du u ) x dw 2 = σdŵ Du where σ(x,t) = ( 2 ux2 Du u x )

12 we can replace the stochastic Cauchy problem by { dy(x,t,s) = σ(y,t,s)dŵ (s) y(x,t,) = x. i.e. with a -dimensional brownian motion in the direction tangent to the curve.

13 Euler scheme for SDE with Time-discretization { yk+ = y k + 2σ(y,t k,) Ŵk y = x. P( Ŵk = ± t) = 2. u t (x,t n+ ) = 2 u t(x + 2σ(x,t n,) t,t n ) + Fully discrete scheme Du u n+ j = u t(x 2σ(x,t n,) t,t n ). ( I[u n ](x j + σ n j t) + I[u n ](x j σ n j t) )

14 Some references MCM via viscosity solution techniques Evans-Spruck, J. Diff. Geom., 99 Evans - Soner-Souganidis, Comm. Pure Appl. Math, 992,... Kohn-Serfaty, Comm. Pure Appl. Math., 26 Soner-Touzi, J. Eur. Math. Soc, 22 and Ann. Prob. 23 Numerical methods Osher-Sethian, JCP, 988 Merriman-Bence-Osher, AMS LN, 993 Crandall-Lions, Numer. Math., 996 Barles-Georgelin, SIAM J. Num. Anal., 995 Catté-Dibos-Koepfler, SIAM J. Num. Anal., 995 F. Ferretti, ENUMATH 2 Carlini- F. -Ferretti, JCP 25 and Interfaces and Free Boundaries, 29 (submitted)

15 Outline Introduction Introduction

16 Modified scheme with threshold C x s u n+ j = 2 u n+ j = 4 ui n i D(j) [ ] I[u n ](x j + σj n t) + I[u n ](x j σj n t) if D n j > C xs if D n j C xs Consistency error (case Dj n > C xs ) ( ) ( x r x q s τ x, t = O + O t t 2 ) + O( t 2) + O( t)

17 Consistency for small gradients Let us consider the case D n j C xs. Case a: (x j,t n ) is such that Du(x j,t n ) =. This is a stardad computation based on the consistency with the heat equation. Case b : (x j,t n ) is such that Du(x j,t n ) and D j [w] C x s. By the lower semicontinuity of F we have that for any ε > there exists a δ (ε ) such that F(Du,D 2 u)(y,s) F(Du,D 2 u)(x,t) ε for any (y,s) B δ (x,t). (3)

18 Consistency for small gradients The upper semicontinuity of F implies that for any ε 2 > there exists a δ 2 (ε 2 ) such that F(Du,D 2 u)(y,s) F(Du,D 2 u)(x,t)+ε 2 for any (y,s) B δ2 (x,t). (4) The interesting case is when for ε max(ε,ε 2 ) there exists (y,s) B δ (x,t) B δ2 (x,t) such that Du(y,s) =. For (x,t) = (x j,t n ), we can apply both (3) and (4) getting F(Du,D 2 u)(x j,t n ) F(Du,D 2 u)(x j,t n ) + 2ε (5)

19 Consistency for small gradients In fact, we have = liminf t F(Du,D 2 u)(x j,t n ) 2 D 2 u(y,s) + ε ε = u(x j,t n) H(w;j) t + ε = limsup t u(x j,t n) H(w;j) t 2 D 2 u(y,s) + ε F(Du,D 2 u)(x j,t n ) + 2ε and, since ε and ε 2 are arbitrary, we obtain the result. + ε

20 Monotonicity The standard scheme presented at the beginning is not monotone even local monotone interpolation operators (linear, bilinear). We modify it in order to satisfy a relaxed monotonicity property. u n+ j u n j t = ρ 2 ( 2 I[un ](x j + σ n j ρ)+ 2 I[un ](x j σ n j ρ) un j ) u n+ j = 4 i D(j) In compact form we write u n i if D n j C xs. u n+ j = H ρ (u n ;j).

21 Monotonicity Since for this scheme H ρ we have H ρ (u n ;j) u n i 4 tl I[u n ],G Cρ x s+ we need to compensate the negative bound and we add a small viscosity P H ρ (u n ;j) = H ρ (u n ;j) + t W x i D(j) un i 4un j ρ x, if D n s x 2 j > C xs u n+ j = 4 i D(j) where W is a positive constant. u n i if D n j C xs.

22 Monotonicity Differentiating H ρ we get: H ρ (u n ;j) uj n t 4W t ρ2 ρ x +s H ρ (u n ;j) = ψ i (x j + σj n ρ) for i S(j) \ (D(j) {j}) u n i H ρ (u n ;j) u n i 4 tl I[u n ],G W t + Cρ xs+ ρ x +s Then H ρ (u n ;j) is monotone if t 4W t ρ2 ρ x +s 4L I[u n ],G < W. C for i D(j). (6) (7)

23 Weak monotonicity property Theorem Under the above assumption, the scheme H ρ satisfies H ρ H ρ (η;j) + o( t).

24 Outline Introduction Introduction

25 Let us consider now a general scheme that is supposed to approximate our HJ equation. Its abstract form on a lattice will be given by u n+ j = S Dt (u n ;j) for j Z 2 n =,...,N u j = u (x j ) for j Z 2 (HJ Dt ) where S Dt : B(G x ) R and B(D) is the space of the bounded functions defined on D.

26 The scheme has to satisfy the following conditions: A - Invariance with respect to the addition of constants For any k R and j Z 2, S Dt (v + k;j) = S Dt (v;j) + k (A) A2 - Weak consistency Let us define F(Dφ,D 2 φ)(x,t) = liminf (y,s) (x,t) F(Dφ,D2 φ)(y,s), F(Dφ,D 2 φ)(x,t) = limsup F(Dφ,D 2 φ)(y,s). (y,s) (x,t)

27 Consistency The consistency assumption (A2) requires φ t (x,t) + F(Dφ,D 2 φ)(x,t) φ(x j,t n ) S Dt (φ n ;j) liminf (x j,t n) (x,t) t t φ(x j,t n ) S Dt (φ n ;j) lim sup (x j,t n) (x,t) t t φ t (x,t) + F(Dφ,D 2 φ)(x,t) where φ C (R 2 (,T]) and φ n = (φ(x j,t n )) xj G x. Note that F(Dφ,D 2 φ), F(Dφ,D 2 φ)(x,t) are respectively lower and upper semicontinuous extension of F.

28 Note that if F is continuous, then the liminf and the limsup must coincide, and the definition reduces to the usual definition of consistency. A3 - Generalized monotonicity v j φ n j for j Z 2 implies S Dt (v;j) S Dt (φ n ;j)+o(dt) where v B(G x ) and S Dt is a (possibly different) scheme weakly consistent.

29 Then, consider u n = (uj n) j Z 2 with un j solution of (HJ Dt ) and its piecewise constant (in time) interpolation u Dt defined as: { u Dt I[u n ](x) if t [t n,t n+ ), (x,t) = u (x) if t [, t), where I[ ] : B(G x ) R is a general interpolation operator I[u n ](x) ψ l (x)ul n (8) l I(x) where ψ l (x) are basis functions in R 2 and I(x) is the set of indices corresponding to the vertices of the cell containing x.

30 Let us note that we can always couple the choice of Dt and Dx according to Dt = CDx γ, with C a positive constant, in order to deal with a unique discretization parameter. We assume γ (this parameter has to be tuned to guarantee monotonicity). The interpolation operator I[ ] has to verify a relaxed monotonicity property: if v j η j for any j I(x) then I[v](x) I[η](x)+o(Dt) (A4) with v B(G x ) and η = (f (x j )) xj G x,where f (x) is a smooth function.

31 Moreover I[ ] satisfies I[η](x) f (x) = o(dt) for any x R 2. (A5) Theorem Assume (A) (A5) and let u(x,t) be the unique viscosity solution of (HJ). Then u Dt (x,t) u(x,t) locally uniformly on R 2 [,T] as Dt.

32 Outline Introduction Introduction

33 TEST: a shrinking circle t = O( x 3 4), s = 4. x t L order L order Errors for the SL scheme

34 TEST: Development of non empty interior Fattening: evolution of the level curves u =.95,,.5

35 The torus evolving into a sphere, R < r

36 The torus collapsing in a circle, R > r

37 Dumb-bell: topology change in R 3

38 Crandall-Lions scheme In the scheme proposed by such problems are avoided by replacing the matrix Θ in the equation of MCM by the following one: with ǫ >. Θ ǫ (Du) = I Du Du Du 2 + ǫ,

39 Crandall-Lions scheme cntd Denoting by e j (j =,2) the canonical base of R 2, we can write the CL scheme as: = H CL (u n ;j) where u n+ j H CL (u n ;j) = u n j + t ρ 2 (I[un ](x j + ρθ ǫ (D j [u n ])e )+ +I[u n ](x j + ρθ ǫ (D j [u n ])e 2 ) + +I[u n ](x j ρθ ǫ (D j [u n ])e ) + + I[u n ](x j ρθ ǫ (D j [u n ])e 2 ) uj n )

40 The scheme for which convergence is proved is u n+ j = H CL (u n ;j) + tk ui n 4uj n. ρ x i D(j)

41 Kohn-Serfaty scheme It has been proved that we can write ( ) Du(x,t) { } div Du(x, t) = min a T D 2 u(x,t)a, (9) Du(x, t) a S,a Du= By penalization we can write min {a max T D 2 u(x,t)a ε a Du,aT D 2 u(x,t)a + ε } a Du a S

42 Kohn-Serfaty scheme We write the scheme as u n+ j = H KS (u n ;j), where { ( H KS (w;j) = min max I[w] x j + ) ( 2 t a,i[w] x j )} 2 t a a S The average of the (SL) scheme is replaced by a min-max operator. This scheme is more expensive and difficult to extend to higher dimension.

43 Test : Evolution of a circle (shrinking) Errors for the SL scheme x t order order CPU s s s

44 Test : Evolution of a circle (shrinking) Errors for the min max scheme x t order order CPU s s m29s ms

45 Outline Introduction Introduction

46 Cuves evolving in the space Let us consider the evolution of a curve C in R 3. There are two ways to handle the problem: The curve is described as the intersection of two surfaces This leads to a system of HJ equation (Osher at alia) The curve is replaced by an ε-tube centered at the curve C. We study the evolution of the surface and get back to the curve in the limit for ε. Following the second characterization, the (SL) scheme has been extended to codimension-2 problems.

47 Cuves evolving in the space { u t = F(D 2 u,du) R 3 [, ) u(x,) = 2d(x, C)2 where F(A,p) = inf ν N(p) {trace[apν ]} and N(p) = {ν S 2 : P ν p = }, and P ν = νν T Soner Touzi formula u(x,t) = inf ν U {E{u(y ν(x,t,t),)}} with U = {ν : [,T] S 2 R 3 : ν Du(x,t) = }

48 The generalized characteristics curves solve { dy ν (x,t,s) = 2ν(s)dŴ(s) y ν (x,t,) = x L.Ambrosio, M.Soner, Level Set Approach to Mean Curvature Flow in Arbitrary Codimension, J.Differential Geometry, 43 (996),

49 MCM cod 2: Numerical approximation Time-discrete scheme u t (x,t n+ ) = 2 inf ν n b U Fully-discrete scheme { u t (x + 2 tν n,t n ) + u t (x } 2 tν n,t n ) u n j = min ν n R 3{ 2 I[un ](x + 2 tν n ) + 2 I[un ](x 2 tν n ) + + ( Dn j νn ) 2 ǫ + ( νn ) 2 ǫ 2 }.

50 Evolution of ǫ helical surface,ǫ =.8.

51 Evolution of two linked circles in R 3 (ǫ-tube).

52 Optimal Trajectory Algorithm s := x jmin ν n j point of the curve = argmin{u n (s j + 2 sνj n )} j =,...,ĵ νj n Û s j+ = s j + 2 sνj n s sĵ ǫ j =,...,ĵ

53 Evolution of a helix in R 3.

54 Evolution of two linked circles in R 3.