A Blob Method for the Aggregation Equation

Transcription

1 A Blob Method for the Aggregation Equation Katy Craig University of California, Los Angeles joint with Andrea Bertozzi Univeristy of California, Santa Barbara October 10, / 36

2 Plan aggregation equation numerical methods blob method blob method converges sketch of proof numerics 2 / 36

3 Plan aggregation equation numerical methods blob method blob method converges sketch of proof numerics 3 / 36

4 Aggregation Equation Applied interest: dρ dt + (vρ) = 0 ρ(0, t) = ρ 0(t) 0 v = K ρ K(x) = x a /a x b /b, d < b < a, K(x) = log x /2π, Kernels with low regularity Mathematical interest: non-local blowup social aggregation in biology evolution of vortex densities in superconductors rich structure of steady states gradient flow in the Wasserstein metric: dρ dt = W E(ρ) ( W E(ρ) = ρ δe ), E(ρ) = 1 ρ(x)k(x y)ρ(y)dxdy δρ 2 R d R d 4 / 36

5 Particle Approximation and Wasserstein Gradient Flow Suppose K is radial, continuously differentiable, and convex and we seek a weak solution of the form ρ particle (x, t) = N δ(x X j (t))m j Then the velocity field would be given by N v(x, t) = K(x y)ρ(y, t)dy = K(x X j (t))m j, j=1 j=1 and ρ particle is a weak solution in case d N dt X i(t) = K(X i (t) X j (t))m j j= K(x) = 2x K(x) = 1 2πx / 36

6 Plan aggregation equation numerical methods blob method blob method converges sketch of proof numerics 6 / 36

7 Plan KOLOKOLNIKOV, SUN, UMINSKY, AND BERTOZZI aggregation equation numerical methods Co the fo 1 Aft λφ j = blob method blob method converges sketch of proof numerics where G Ne we ass This l where FIG 1 Top: Minimizers of the energy (1) with force law (3) [Kolokolnikov, Sun, Uminsky, Bertozzi, 2011] Diameter d = max i,j x i x j ranges from 03 in top left to 3 in bottom right Bottom: time evolution of (4) with values a = 8, b = 067 I 1 (m 6 / 36 I 2 (m

8 Numerical Methods: Recent Results particlemethods complement theoretical results: repulsive attractive steady states [Bertozzi, Sun, Kolokolnikov, Uminsky, Von Brecht 2011], [Balagué, Carrillo, Laurent, Raoul 2012] used to prove theoretical results: finite time blowup, confinement [Carrillo, DiFrancesco, Figalli, Laurent, Slepčev 2010, 2011] convergence of particle method [Carrillo, Choi, Hauray 2013] othernumericalmethods developed finite volume method [Carrillo, Chertock, Huang 2014] convergence of finite difference method to measure solutions, 1D [James, Vauchelet 2014] 7 / 36

9 Numerical Methods: Our Goal Develop new numerical method for multidimentional aggregation equation Allow singular and non-singular potentials Prove quantitative estimates on convergence to classical solutions Validate sharpness of estimates with numerical examples 8 / 36

10 Blob Method for the Aggregation Equation Theorem (C, Bertozzi 2014) Let K(x) have power law growth x s, s 2 d (for simplicity of notation d 3, Newtonian potential admissible for d = 2) Suppose ρ : R d [0, T ] R + is a smooth, compactly supported solution The blob method discretizes ρ 0 (x) on a mesh of size h and prescribes approximate particle trajectories X i, approximate density along particle trajectories ρ i, so that for 1 2 q < 1 and m 4 (parameters specifying shape of blobs) X i (t) X i (t) L p Chmq ρ h i (t) ρ i (t) W 1,p Ch mq, h for 1 p < 9 / 36

11 Plan aggregation equation numerical methods blob method blob method converges sketch of proof numerics 10 / 36

12 Euler vs Aggregation: Similarities Vorticity formulation of Euler equations: ω t + v ω = ω ( v) (V) v = K d ω material derivative Dω/Dt = ω ( v) v = K d ω Biot-Savart kernel: K 2 (x) = 1 2π x ( x 2 2, x 1 ), K 3 (x)h = 1 x h 4π x 3 v = 1 ω Aggregation equation: ρ t + (vρ) = 0 (A) v = K ρ material derivative Dρ/Dt = ρ( v) v = K ρ Newtonian potential: K(x) = 1 d(d 2)ω d x 2 d (K(x) = log x /2π when d = 2) v = 1 ρ 11 / 36

13 Euler vs Aggregation: Differences Euler Equations velocity is divergence free Biot Savart kernel 2 and 3 dimensions Aggregation Equation mass is conserved Newtonian, Riesz, and non-singular kernels (growth at infinity) d 1 12 / 36

14 Aggregation equation: Lagrangian perspective d dtx(α, t) = K ρ(x(α, t), t) Particle trajectories: X(α, 0) = α d dtρ(x(α, t), t) = ( K ρ(x(α, t), t)) ρ(x(α, t), t) Density along trajectories: ρ(x(α, 0), 0) = ρ 0 (α) By conservation of mass, ρ(x(β, t), t)j(β, t) = ρ 0 (β), K(x y)ρ(y, t)dy = K(x X(β, t))ρ(x(β, t), t)j(β, t)dβ = K(x X(β, t))ρ 0 (β)dβ and similarly for K ρ(x(α, t), t) 13 / 36

15 Steps for blob method 1 Remove the singularity of K by convolution with a mollifier, K δ = K ψ δ 2 Replace ρ 0 with a particle approximation on the grid hz d ρ particle 0 (y) = j Z d δ(y jh)ρ 0j h d 14 / 36

16 Blob method for the aggregation equation Exact Particle Trajectories: d dt X(α, t) = K(X(α, t) X(β, t))ρ 0 (β)dβ X(α, 0) = α Exact Density : d dt ρ(x(α, t), t) = ρ(x(α, t), t) K(X(α, t) X(β, t))ρ 0 (β)dβ ρ(α, 0) = ρ 0 (α) Approx Particle Trajectories: Approx Density : d dt X i (t) = j K δ( X i (t) X j (t))ρ 0j h d X i (0) = ih ( d dt ρ i(t) = ρ i (t) j K δ( X i (t) X ) j (t))ρ 0j h d ρ i (0) = ρ 0 (ih) (For pure particle method, take δ = 0) 15 / 36

17 Heuristic interpretation of blob method When K is the Newtonian potential, v = K ρ implies ρ = v Applying this to the approximate velocity ṽ ρ alt (x, t) = j K δ (x X j (t))ρ 0j h d = j ψ δ (x X j (t))ρ 0j h d / 36

18 Advantages of blob method Avoids main source of numerical diffusion Only requires computational elements on support of density Inherently adaptive Accommodates singular kernels, up to and including the Newtonian potential Arbitrarily high order rates of convergence, depending on the accuracy of the mollifier and the widths of the blobs Without regularization: fewer admissible potentials, slower rates of convergence These agree with the rate of O(h 2 ϵ ) for the Euler equations [Goodman, Hou, Lowengrub 1990] 17 / 36

19 Plan aggregation equation numerical methods blob method blob method converges sketch of proof numerics 18 / 36

20 Mollifier Assumption Assume ψ is radial, ψ = 1, and for some m 4, L d Accuracy: x γ ψ(x)dx = 0 for 1 γ m 1 2 Regularity: ψ C L (R d ) 3 Decay: x n β ψ(x) C for all n 0 1 ensures convolution with ψ preserves polynomials of order α m 1, (x y) α ψ(y)dy = α k=0 ( α )x α k k y k ψ(y)dy = x α ψ(y)dy = x α 2 and 3 ensure K δ, K δ C L (R d ) 19 / 36

21 Mollifier Assumption Assume ψ is radial, ψ = 1, and for some m 4, L d Accuracy: x γ ψ(x)dx = 0 for 1 γ m 1 2 Regularity: ψ C L (R d ) 3 Decay: x n β ψ(x) C for all n 0 Example: d = 1, m = 4, L = +, ψ(x) = π e x π e (x/2) Mollifier ψ δ (x), m = 4 δ = 1 δ = 4 δ = / 36

22 Kernel Assumption Suppose that K(x) = N n=1 K n(x) For each K n (x), there exists S n 2 d such that β K n (x) C x Sn β, x R d \ {0}, β 0 If S n = 2 d, we additionally require that K n (x) is a constant multiple of the Newtonian potential Let s = min n S n be the smallest power of the kernel Example: K(x) = x a /a x b /b, 2 d b < a s = b 21 / 36

23 Discrete L p norms Definition For 1 p, u i L p = 1/p u h i p h d i Z d u i W 1,p h = u i p + L p h d D + j u i p L p h j=1 1/p (u i, g i ) h = i Z d u i g i h d u i W 1,p h u i, g i = sup g {g i } W 1,p i h W 1,p h D + j is the forward difference operator in the jth coordinate direction We measure the convergence of X and v in L p h and we measure the convergence of ρ in W 1,p h This is because, in the most singular case when K is the Newtonian potential, v = K ρ = ρ = v 22 / 36

24 Convergence Theorem ( C, Bertozzi 2014 ) Suppose ψ C L (R d ) for L > s + d, ρ : R d [0, T ] R + is a smooth, compactly supported solution, 0 h q δ 1/2 for some 1 2 < q < 1 Then for 1 p <, X i (t) X i (t) L p h C(δm + δ (L s d) h L ) ρ i (t) ρ i (t) W 1,p h provided that for some ϵ > 0, C(δ m + δ (L+1 s d) h L ), C(δ m + δ (L+1 s d) h L ) < δ 2 h 1+ϵ /2 23 / 36

25 Convergence of arbitrarily high order Take δ = h q for 1 2 < q < 1 Then the technical condition C(δ m + δ (L+1 s d) h L ) < δ 2 h 1+ϵ /2 holds By the previous theorem X i (t) X i (t) L p h C(δm + δ (L s d) h L ) Cδ m ρ i (t) ρ i (t) W 1,p h Theorem ( C, Bertozzi 2014 ) C(δ m + δ (L+1 s d) h L ) Let δ = h q If L is sufficiently large, then for 1/2 q < 1, m 4, Cδ m }{{} for L sufficiently large X i (t) X i (t) L p Chmq h ρ i (t) ρ i (t) W 1,p h Ch mq Benefit of blob methods: arbitrarily high order of convergence 24 / 36

26 Plan aggregation equation numerical methods blob method sketch of proof numerics 25 / 36

27 Sketch of proof: convergence of particle trajectories Velocity Exact v(x, t) = K(x X(β, t))ρ 0 (β)dβ Approx ṽ(x, t) = j K δ(x X j (t))ρ 0j h d Approx along exact traj v h (x, t) = j K δ(x X j (t))ρ 0j h d Main Steps: 1 Control difference between exact and approximate velocity by separately estimating consistency and stability, v(x, t) ṽ(x, t) v(x, t) v h (x, t) + v h (x, t) ṽ(x, t) 2 Use Gronwall's inequality to deduce control of particle error 26 / 36

28 Consistency Proposition (Consistency) (C, Bertozzi 2014 ) v v h L h C ( δ m + δ (L s d) h L) v(x, t) v h (x, t) = v(x, t) K δ ρ(x, t) + K δ ρ(x, t) v h (x, t) = K ρ(x, t) K δ ρ(x, t) + K δ ρ(x, t) j K δ (x X j (t))ρ 0j h d K ρ(x, t) K ρ ψ δ (x, t) + C K δ }{{} W 1,L (B R )h L }{{} ψ is accurate of order m quadrature, kernel estimates Cδ m + Cδ (L s d) h L Lemma (Regularized Kernel Estimates) (C, Bertozzi 2014) For L > s + d, K δ W 1,L (B R ) Cδ (L s d) 27 / 36

29 Stability Proposition (Stability) (C, Bertozzi 2014 ) If X(t) X(t) L h δ, then for 1 < p < v h i ṽ i = j + j v h (t) ṽ(t) L p C X(t) X(t) h L p h K δ (X i X j )ρ 0j h d j K δ ( X i X j )ρ 0j h d j K δ (X i X j )ρ 0j h d K δ (X i X j )ρ 0j h d use mean value theorem to pull out X X = j D 2 K δ (X i X j + y (1) ij )(X j X j )ρ 0j h d + ( X i X i ) j D 2 K δ (X i X j + y (2) ij )ρ 0jh d discrete continuous convolution and L p h Lp v h ṽ L p C D2 K h δ [(X X)ρ] L p + C (X X)(D 2 K δ ρ) L p C X X L p C X X L p h apply Calderón- Zygmund or Young 28 / 36

30 Convergence Therefore, for X(t) X(t) L h δ, v(t) ṽ(t) L p v(t) vh (t) h L p h + v h (t) ṽ(t) L p h C(δ m + δ (L s d) h L ) + C X(t) X(t) L p h (B R 0 ) With Gronwall's inequality and a bootstrap argument, we obtain the result: X(t) X(t) L p h C(δm + δ (L s d) h L ) 29 / 36

31 Plan aggregation equation numerical methods blob method sketch of proof numerics 30 / 36

32 Newtonian potential, one dimension Time Particle Trajectories Position 10 Particle Trajectories Height Density approx exact Position L 1 h Error h = 004 q = 09 m = 4 ρ 0(x) = (1 x 2 ) 20 + blowup: t = 1 Time Position Error density slope 355 particle slope h approximate particle trajectories bend to avoid collision convergence of method agrees with theoretically predicted 36 = m q 31 / 36

33 Newtonian potential, one dimension Time 10 Particle Trajectories 05 Height 5 Density approx exact h = 004 q = 09 m = 4 ρ 0 (x) = (1 x 2 ) Position 10 Particle Trajectories Position L 1 h Error blowup: t = 1 Time Position Error density slope 355 particle slope h approximate particle trajectories bend to avoid collision convergence of method agrees with theoretically predicted 36 = m q 31 / 36

34 Various kernels, one dimension: blob vs particle 10 4 particle m = 4 m = 6 Lh 1 Error K(x) = ( ) 1 Error particle slope 200 density slope 356 particle slope 359 density slope 440 particle slope 530 K(x) = x 3 /3 density slope 201 particle slope h density slope 356 particle slope 358 density slope 518 particle slope 479 Blob method is more beneficial for more singular kernels 32 / 36

35 two dimensions, aggregation K(x) = log x /2π K(x) = x 2 /2 K(x) = x 3 / Time Position finite vs infinite time collapse delta function vs delta ring h = 004, q = 09, m = 4 33 / 36

36 two dimensions: repulsive-attractive kernels K(x) = x 4 /4 log x /2π K(x) = x 4 /4 x 3/2 /(3/2) K(x) = x 7 /7 x 3/2 /(3/2) t = 250 t = 12 t = 225 t = 150 t=8 t = 175 t = 05 t=4 t = 75 t = 00 t=0 t=0 d =000 d =032 d =000 d =032 d =000 d =032 large δ affects steady state behavior illustrates role of kernel's regularity in dimensionality of steady states [Balagué, Carrillo, Laurent, Raoul 2013] 34 / 36

37 Future Work Keller-Segel equation [Yao, Bertozzi 2013] Interplay between particle methods and theoretical results Finite time blowup, confinement [Carrillo, DiFrancesco, Figalli, Laurent, Slepčev 2010, 2011] Existence of weak measure solutions [Lin, Zhang 2000] ongoing work with Ihsan Topaloglu (Fields Institute): Γ-convergence of regularized interaction energy; convergence of blob method to steady states E δ (ρ) = 1 2 R d R d ρ(x)k δ (x y)ρ(y)dxdy ongoing work with Andrea Bertozzi: long time error estimates for repulsive attractive kernels? 35 / 36

38 Thank you! 36 / 36

39 Backup 37 / 36

40 Associated particle system and gradient flow Given the blob method particle trajectories d X dt i (t) = j K δ( X i (t) X j (t))ρ 0j h d X i (0) = ih, we may define the corresponding particle measure ˆρ(x, t) = j δ(x X j (t))ρ 0j h d This is energy decreasing formally Wasserstein gradient flow for the regularized energy functional E δ (ρ) = 1 ρ(x)k δ (x y)ρ(y)dxdy 2 R d R d (For pure particle method, take δ = 0) 38 / 36

41 Blowup For which kernels does finite time blowup occur? Intuition from particle approximation: ρ(x, t) = N j=1 δ(x X j(t))m j v(x, t) = d dt X i(t) = K(x y)ρ(y, t)dy = N K(X i (t) X j (t))m j j=1 N K(x X j (t))m j j= K(x) = 2x K(x) = 1 2πx / 36

42 Osgood Condition Simple case: particle moving toward minimum of attractive potential: K(x) = k( x ) To move a distance dr, it takes time d dt X(t) = K(X(t)) X(0) = x 0 d dt r(t) = k (r(t)) r(0) = R 0 dr k (r) Thus, the particle reaches the origin at time T = R0 0 dr k (r) 40 / 36

43 Osgood Condition Theorem (Osgood Condition) (Bertozzi, Carrillo, Laurent 2009) A kernel K satisfies the Osgood condition in case R0 This is a necessary and sufficient condition for finite time blowup 0 dr k (r) < K(x) = x α : α 2 = no finite time blowup, α < 2 = finite time blowup K(x) = 2x K(x) = 1 2πx / 36

44 K = Newtonian potential Rewriting the aggregation equation in terms of the material derivative D Dt = t + v, ρ t + (vρ) = 0 v = K ρ material derivative Dρ Dt = ρ( v) v = K ρ When K is the Newtonian potential, v = K ρ implies ρ = v, so Dρ Dt = ρ2 If X(α, t) denotes the particle trajectories induced by the velocity field v, Hence, d dt ρ(x(α, t), t) = ρ(x(α, t), t)2 ( ) 1 1 ρ ρ(x(α, t), t) = t 0(α) if ρ 0 (α) 0 0 if ρ 0 (α) = 0 42 / 36

45 K = Newtonian potential ( ) 1 1 ρ ρ(x(α, t), t) = t 0(α) if ρ 0 (α) 0 0 if ρ 0 (α) = 0 blowup: If ρ 0 (α) > 0 for any α, the first blowup occurs at time t = ρ 0 1 L patch solutions: If ρ 0 (α) = 1 Ω (α), for Ω t := X t (Ω), ρ(x(α, t), t) = (1 t) 1 1 Ω (α) = (1 t) 1 1 Ωt (X(α, t)) Patch solutions collapse onto a set of Lebesgue measure zero at t = 1 [Bertozzi, Laurent, Léger 2012] 43 / 36

46 2D Euler equations: Lagrangian perspective For simplicity of notation, write K = K 2 d dtx(α, t) = K ω(x(α, t), t) Particle trajectories: X(α, 0) = α Since the velocity field is divergence free and ω(x(β, t), t) = ω 0 (β), K(x y)ω(y, t)dy = K(x X(β, t))ω(x(β, t), t)dβ = K(x X(β, t))ω 0 (β)dβ Thus the particle trajectories evolve according to d dtx(α, t) = K(X(α, t) X(β, t))ω 0 (β)dβ X(α, 0) = α 44 / 36

47 Blob method for the 2D Euler equations Steps for blob method: 1 Remove the singularity of K by convolution with a mollifier Write K δ = K ψ δ 2 Replace ω 0 with a particle approximation on the grid hz d ω particle 0 (y) = j Z d δ(y jh)ω 0j h d Exact Particle Trajectories: d dtx(α, t) = K(X(α, t) X(β, t))ω 0 (β)dβ X(α, 0) = α d X dt i (t) = j Approx Particle Trajectories: K δ( X i (t) X j (t))ω 0j h d X i (0) = ih 45 / 36

48 Blob Method for the 2D Euler Equations First used by [Chorin,1973] [Hald, Del Prete 1978] proved 2D convergence [Hald, 1979] proved second order convergence in 2D for arbitrary time intervals [0, T ] [Beale, Majda 1982] proved convergence in 2D and 3D with arbitrarily high-order accuracy [Cottet, Raviart 1984] simplified 2D and 3D convergence proofs [Anderson, Greengard 1985] modified the 3D blob method, considered time discretization 148 M Nitsche and R Krasny (b) 51 I FIGURE 5 at t = 145 (a) Experiment, from Didden (1979) (b) Simulation, S Comparison at t = 145 (a) Experiment, from Didden (1979) (b) Simulation, 32 Flow field δ = 02 Figure 7 [Nitsche, shows the computed Krasny velocity 1994] field for 6 = 02, before and after the shutoff time The vortex sheet and material line appear as a dashed curve in th half-plane At t = tiff, the ring has moved away from the edge and the velocity the tube exit-plane resembles a slug-flow profile A small region of negative velocity occurs near the edge, arising from the ring-induced circulatory flow result, the vortex sheet appears to leave the edge at a slight inward angle A feature can be seen in the experiment (figure 5a) At t = t&, the velocity field n edge is completely changed The vortex ring induces a starting flow around th from outside to inside the tube, leading to the formation of a counter-rotating r t tow 46 / 36

49 Newtonian potential, one dimension: patch initial data Time Time Particle Trajectories Position Particle Trajectories Position Height Error Density approx exact Position particle trajectories bend, densities round L 1 h Error density slope 089 particle slope h h = 004 q = 09 m = 4 ρ 0 (x) = 1 [ 1,1] blowup: t = 1 lower order accuracy ( 09) compared to regular initial data ( 36) 47 / 36

50 Newtonian potential, one dimension: blob vs particle 10 Particle Trajectories 10 5 L 1 h Error h = 004 q = 09 m = 4 Time 05 Error ρ 0 (x) = (1 x 2 ) Position density slope 355 particle slope h blowup: t = 1 10 Particle Trajectories 10 4 L 1 h Error 10 6 Time 05 Error Position particle slope h blob has higher order accuracy ( 36) compared to particle ( 2) trajectories computed by pure particle method collide at blowup time 48 / 36