A regularised particle method for linear and nonlinear diffusion

Transcription

1 1/25 A regularised particle method for linear and nonlinear diffusion Francesco Patacchini Department of Mathematical Sciences, Carnegie Mellon University Joint work with J. A. Carrillo (Imperial College London) and K. Craig (University of California, Santa Barbara) Ki-Net Young Researchers Workshop, CSCAMM 12 October 2017

2 2/25 Contents Background Motivation Results Numerical simulations Outlook

3 3/25 Contents Background Motivation Results Numerical simulations Outlook

4 4/25 Background We consider the continuity equation { tµ = div(µ φ), φ = U m µ + V + W µ, on [0, T ] R d, (1) where

5 4/25 Background We consider the continuity equation { tµ = div(µ φ), φ = U m µ + V + W µ, on [0, T ] R d, (1) where µ: [0, T ] P(R d ) is an unknown curve of probability measures,

6 4/25 Background We consider the continuity equation { tµ = div(µ φ), φ = U m µ + V + W µ, on [0, T ] R d, (1) where µ: [0, T ] P(R d ) is an unknown curve of probability measures, U m : [0, ) R, m 1, is a density of internal energy (diffusion), U 0(s) = 0 U 1(s) = s log s U m(s) = sm for all m > 1 m 1 (no diffusion), (diffusion of heat type), (diffusion of porous-medium type),

7 4/25 Background We consider the continuity equation { tµ = div(µ φ), φ = U m µ + V + W µ, on [0, T ] R d, (1) where µ: [0, T ] P(R d ) is an unknown curve of probability measures, U m : [0, ) R, m 1, is a density of internal energy (diffusion), U 0(s) = 0 U 1(s) = s log s U m(s) = sm for all m > 1 m 1 (no diffusion), (diffusion of heat type), (diffusion of porous-medium type), V : R d R is a confinement or external potential,

8 4/25 Background We consider the continuity equation { tµ = div(µ φ), φ = U m µ + V + W µ, on [0, T ] R d, (1) where µ: [0, T ] P(R d ) is an unknown curve of probability measures, U m : [0, ) R, m 1, is a density of internal energy (diffusion), U 0(s) = 0 U 1(s) = s log s U m(s) = sm for all m > 1 m 1 (no diffusion), (diffusion of heat type), (diffusion of porous-medium type), V : R d R is a confinement or external potential, W : R d R is an interaction or aggregation kernel.

9 4/25 Background We consider the continuity equation { tµ = div(µ φ), φ = U m µ + V + W µ, on [0, T ] R d, (1) where µ: [0, T ] P(R d ) is an unknown curve of probability measures, U m : [0, ) R, m 1, is a density of internal energy (diffusion), U 0(s) = 0 U 1(s) = s log s U m(s) = sm for all m > 1 m 1 (no diffusion), (diffusion of heat type), (diffusion of porous-medium type), V : R d R is a confinement or external potential, W : R d R is an interaction or aggregation kernel. For simplicity, V and W belong to C 2 (R d ) and are bounded from below.

10 Background Solutions to (1) are understood in the weak sense: we say µ: [0, T ] P(R d ) is a solution if, for all ψ Cc ((0, T ) R d ), T ( tψ(t, x) + ψ(t, x), φ(t, x) ) dµ t(x) dt = 0. 0 R d 5/25

11 Background Solutions to (1) are understood in the weak sense: we say µ: [0, T ] P(R d ) is a solution if, for all ψ Cc ((0, T ) R d ), T ( tψ(t, x) + ψ(t, x), φ(t, x) ) dµ t(x) dt = 0. 0 R d We are going to restrict to P 2(R d ) := {µ P(R d ) R d x 2 dµ(x) < }, 5/25

12 Background Solutions to (1) are understood in the weak sense: we say µ: [0, T ] P(R d ) is a solution if, for all ψ Cc ((0, T ) R d ), T ( tψ(t, x) + ψ(t, x), φ(t, x) ) dµ t(x) dt = 0. 0 R d We are going to restrict to P 2(R d ) := {µ P(R d ) R d x 2 dµ(x) < }, and equip it with the 2-Wasserstein distance W 2(µ, ν) = ( ) 1/2 min x y 2 dπ(x, y) for all µ, ν P 2(R d ), π Π(µ,ν) R d R d where Π(µ, ν) is the set of probability measures on R d R d with first marginal µ and second marginal ν. 5/25

13 6/25 Background The metric space (P 2(R d ), W 2) has a weak Riemannian structure, on which we can define the gradient of a functional E : P 2(R d ) (, ] by W2 E(µ) = div(µ E µ), where E µ is the first variation density of E at µ.

14 6/25 Background The metric space (P 2(R d ), W 2) has a weak Riemannian structure, on which we can define the gradient of a functional E : P 2(R d ) (, ] by W2 E(µ) = div(µ E µ), where E µ is the first variation density of E at µ. Define E m (µ) = U m (µ) + V (x) dµ(x) + 1 W µ(x) dµ(x) for all µ P 2(R d ), R d 2 R d where { U 0 = 0, U m U (µ) = R m(µ(x)) dµ(x) for all µ P 2,ac(R d ), d + otherwise.

15 6/25 Background The metric space (P 2(R d ), W 2) has a weak Riemannian structure, on which we can define the gradient of a functional E : P 2(R d ) (, ] by W2 E(µ) = div(µ E µ), where E µ is the first variation density of E at µ. Define E m (µ) = U m (µ) + V (x) dµ(x) + 1 W µ(x) dµ(x) for all µ P 2(R d ), R d 2 R d where { U 0 = 0, U m U (µ) = R m(µ(x)) dµ(x) for all µ P 2,ac(R d ), d + otherwise. Under suitable regularity conditions on V, W and µ, we can show that (E m ) µ(x) = U m(µ(x)) + V (x) + W µ(x) for all x R d.

16 Background The metric space (P 2(R d ), W 2) has a weak Riemannian structure, on which we can define the gradient of a functional E : P 2(R d ) (, ] by W2 E(µ) = div(µ E µ), where E µ is the first variation density of E at µ. Define E m (µ) = U m (µ) + V (x) dµ(x) + 1 W µ(x) dµ(x) for all µ P 2(R d ), R d 2 R d where { U 0 = 0, U m U (µ) = R m(µ(x)) dµ(x) for all µ P 2,ac(R d ), d + otherwise. Under suitable regularity conditions on V, W and µ, we can show that (E m ) µ(x) = U m(µ(x)) + V (x) + W µ(x) for all x R d. Thus (1) becomes µ (t) = W2 E m (µ(t)) for a.e. t [0, T ], 6/25 and so our continuity equation is a gradient flow for E m.

17 7/25 Contents Background Motivation Results Numerical simulations Outlook

18 8/25 Motivation For a moment, ignore diffusion and only consider confinement and interaction: µ t = div(µ t( V + W µ t)), µ(0) = µ 0 D(E 0 ). (2)

19 8/25 Motivation For a moment, ignore diffusion and only consider confinement and interaction: µ t = div(µ t( V + W µ t)), µ(0) = µ 0 D(E 0 ). (2) Approximate µ 0 (e.g. using quantisation) as µ 0 µ 0 N = N i=1 miδ x 0 i, mi > 0, (x0 i ) i {1,...,N} R d. Then, under suitable conditions on V and W and if µ 0 N D(E 0 ), the solution µ N to (2) with initial datum µ 0 N stays a combination of point masses, i.e., µ N (t) = N i=1 miδ x i (t), t > 0.

20 8/25 Motivation For a moment, ignore diffusion and only consider confinement and interaction: µ t = div(µ t( V + W µ t)), µ(0) = µ 0 D(E 0 ). (2) Approximate µ 0 (e.g. using quantisation) as µ 0 µ 0 N = N i=1 miδ x 0 i, mi > 0, (x0 i ) i {1,...,N} R d. Then, under suitable conditions on V and W and if µ 0 N D(E 0 ), the solution µ N to (2) with initial datum µ 0 N stays a combination of point masses, i.e., µ N (t) = N i=1 miδ x i (t), t > 0. We may define a particle method simply by solving the ODE system ẋ i(t) = V (x i(t)) + N j=1 mjw (xi(t) xj(t)). We can summarise this property as particles remain particles.

21 Motivation For a moment, ignore diffusion and only consider confinement and interaction: 8/25 µ t = div(µ t( V + W µ t)), µ(0) = µ 0 D(E 0 ). (2) Approximate µ 0 (e.g. using quantisation) as µ 0 µ 0 N = N i=1 miδ x 0 i, mi > 0, (x0 i ) i {1,...,N} R d. Then, under suitable conditions on V and W and if µ 0 N D(E 0 ), the solution µ N to (2) with initial datum µ 0 N stays a combination of point masses, i.e., µ N (t) = N i=1 miδ x i (t), t > 0. We may define a particle method simply by solving the ODE system ẋ i(t) = V (x i(t)) + N j=1 mjw (xi(t) xj(t)). We can summarise this property as particles remain particles. The convergence of this particle method and other properties of (2) have been widely studied: Laurent (2007); Bertozzi Laurent Rosado (2011); Carrillo Di Francesco Figalli Laurent Slepčev (2011); Carrillo Choi Hauray (2014); Jabin (2014),...

22 9/25 Motivation For most of this talk we only focus on the diffusion part, i.e., µ t = div(µ t (U m µ t)), µ 0 D(U m ). (3) Then particles do not remain particles. (Think of the heat equation with initial datum µ 0 = δ 0 for example.)

23 9/25 Motivation For most of this talk we only focus on the diffusion part, i.e., µ t = div(µ t (U m µ t)), µ 0 D(U m ). (3) Then particles do not remain particles. (Think of the heat equation with initial datum µ 0 = δ 0 for example.) Several ways to cope with this issue have been developed.

24 9/25 Motivation For most of this talk we only focus on the diffusion part, i.e., µ t = div(µ t (U m µ t)), µ 0 D(U m ). (3) Then particles do not remain particles. (Think of the heat equation with initial datum µ 0 = δ 0 for example.) Several ways to cope with this issue have been developed. Stochastics. Cottet Koumoutsakis (2000), Jabin Wang (2017), Liu Wang (2017). Main disadvantage: one must usually average the results over a large number of runs to compensate for inherent randomness.

25 9/25 Motivation For most of this talk we only focus on the diffusion part, i.e., µ t = div(µ t (U m µ t)), µ 0 D(U m ). (3) Then particles do not remain particles. (Think of the heat equation with initial datum µ 0 = δ 0 for example.) Several ways to cope with this issue have been developed. Stochastics. Cottet Koumoutsakis (2000), Jabin Wang (2017), Liu Wang (2017). Main disadvantage: one must usually average the results over a large number of runs to compensate for inherent randomness. Velocity regularization. Russo (1990), Degond Mustieles (1990), Mas-Gallic (2001), Lions Mas-Gallic (2002). Main disadvantage: one loses the gradientflow structure (except in the case m = 2).

26 9/25 Motivation For most of this talk we only focus on the diffusion part, i.e., µ t = div(µ t (U m µ t)), µ 0 D(U m ). (3) Then particles do not remain particles. (Think of the heat equation with initial datum µ 0 = δ 0 for example.) Several ways to cope with this issue have been developed. Stochastics. Cottet Koumoutsakis (2000), Jabin Wang (2017), Liu Wang (2017). Main disadvantage: one must usually average the results over a large number of runs to compensate for inherent randomness. Velocity regularization. Russo (1990), Degond Mustieles (1990), Mas-Gallic (2001), Lions Mas-Gallic (2002). Main disadvantage: one loses the gradientflow structure (except in the case m = 2). Gradient-flow approach. Osberger Matthes (2014), Carrillo, P. Sternberg Wolansky (2016), Carrillo Huang P. Wolansky (2017). Main disadvantege: only works in one dimension and extension to higher dimensions is hard.

27 9/25 Motivation For most of this talk we only focus on the diffusion part, i.e., µ t = div(µ t (U m µ t)), µ 0 D(U m ). (3) Then particles do not remain particles. (Think of the heat equation with initial datum µ 0 = δ 0 for example.) Several ways to cope with this issue have been developed. Stochastics. Cottet Koumoutsakis (2000), Jabin Wang (2017), Liu Wang (2017). Main disadvantage: one must usually average the results over a large number of runs to compensate for inherent randomness. Velocity regularization. Russo (1990), Degond Mustieles (1990), Mas-Gallic (2001), Lions Mas-Gallic (2002). Main disadvantage: one loses the gradientflow structure (except in the case m = 2). Gradient-flow approach. Osberger Matthes (2014), Carrillo, P. Sternberg Wolansky (2016), Carrillo Huang P. Wolansky (2017). Main disadvantege: only works in one dimension and extension to higher dimensions is hard. Goal. Derive a deterministic particle method approximating (3) that respects the underlying gradient-flow structure and which works naturally in higher dimensions.

28 Motivation Our approach. Analogous to Craig Bertozzi and Craig Topaloglu (2016). 10/25

29 10/25 Motivation Our approach. Analogous to Craig Bertozzi and Craig Topaloglu (2016). Choose a mollifier ϕ: R d R C (R d ) and define, for all ε > 0, ( ϕ ε(x) = ε d x ) ϕ for all x R d. ε

30 10/25 Motivation Our approach. Analogous to Craig Bertozzi and Craig Topaloglu (2016). Choose a mollifier ϕ: R d R C (R d ) and define, for all ε > 0, ( ϕ ε(x) = ε d x ) ϕ for all x R d. ε Define F m : (0, ) R by and F m(s) = Um(s) s for all s (0, ), Fε m (µ) = Uε m (µ) = F m(ϕ ε µ(x)) dµ(x) for all µ P 2(R d ). R d (Note that D(F m ε ) = P 2(R d ).)

31 10/25 Motivation Our approach. Analogous to Craig Bertozzi and Craig Topaloglu (2016). Choose a mollifier ϕ: R d R C (R d ) and define, for all ε > 0, ( ϕ ε(x) = ε d x ) ϕ for all x R d. ε Define F m : (0, ) R by and F m(s) = Um(s) s for all s (0, ), Fε m (µ) = Uε m (µ) = F m(ϕ ε µ(x)) dµ(x) for all µ P 2(R d ). R d (Note that D(F m ε ) = P 2(R d ).) This energy is of a novel form and mixes features from the classical internal and interaction energies.

32 10/25 Motivation Our approach. Analogous to Craig Bertozzi and Craig Topaloglu (2016). Choose a mollifier ϕ: R d R C (R d ) and define, for all ε > 0, ( ϕ ε(x) = ε d x ) ϕ for all x R d. ε Define F m : (0, ) R by and F m(s) = Um(s) s for all s (0, ), Fε m (µ) = Uε m (µ) = F m(ϕ ε µ(x)) dµ(x) for all µ P 2(R d ). R d (Note that D(F m ε ) = P 2(R d ).) This energy is of a novel form and mixes features from the classical internal and interaction energies. Solve the gradient flow for F m ε : µ t = W2 F m ε (µ t). (4) We have (Fε m ) µ = ϕ ε (µf m (ϕ ε µ)) + F m (ϕ ε µ), and so (4) becomes ( ) µ t = div µ t ϕ ε (µ tf m (ϕ ε µ t)) + µ t( ϕ ε µ t)f m (ϕ ε µ t).

33 11/25 Motivation ( ) µ t = div µ t ϕ ε (µ tf m (ϕ ε µ t))+µ t( ϕ ε µ t)f m (ϕ ε µ t), µ 0 P 2(R d ). We can show that particles do remain particles.

34 11/25 Motivation ( ) µ t = div µ t ϕ ε (µ tf m (ϕ ε µ t))+µ t( ϕ ε µ t)f m (ϕ ε µ t), µ 0 P 2(R d ). We can show that particles do remain particles. Thus our particle method is defined as the ODE system ( ( N N ) ẋ i = m j ϕ ε(x i x j) m k ϕ ε(x j x k ) j=1 F m k=1 ( N ) ) + F m m k ϕ ε(x i x k ). k=1

35 Motivation ( ) µ t = div µ t ϕ ε (µ tf m (ϕ ε µ t))+µ t( ϕ ε µ t)f m (ϕ ε µ t), µ 0 P 2(R d ). We can show that particles do remain particles. Thus our particle method is defined as the ODE system ( ( N N ) ẋ i = m j ϕ ε(x i x j) m k ϕ ε(x j x k ) j=1 F m k=1 ( N ) ) + F m m k ϕ ε(x i x k ). k=1 The case m = 2 is very particular. Indeed, F 2(s) = 1, µ t = 2 div(µ t ϕ ε µ t), which is an interaction equation with kernel 2ϕ ε.

36 11/25 Motivation ( ) µ t = div µ t ϕ ε (µ tf m (ϕ ε µ t))+µ t( ϕ ε µ t)f m (ϕ ε µ t), µ 0 P 2(R d ). We can show that particles do remain particles. Thus our particle method is defined as the ODE system ( ( N N ) ẋ i = m j ϕ ε(x i x j) m k ϕ ε(x j x k ) j=1 F m k=1 ( N ) ) + F m m k ϕ ε(x i x k ). k=1 The case m = 2 is very particular. Indeed, F 2(s) = 1, µ t = 2 div(µ t ϕ ε µ t), which is an interaction equation with kernel 2ϕ ε. This is the same equation as Lions Mas-Gallic studied; our method is therefore a generalisation of theirs to any porous medium equation.

37 12/25 Contents Background Motivation Results Numerical simulations Outlook

38 13/25 Results We have proved the following. Fε m : P 2(R d ) R has good basic properties necessary for the use of gradientflow theory (Ambrosio Gigli Savaré, 2008):

39 13/25 Results We have proved the following. Fε m : P 2(R d ) R has good basic properties necessary for the use of gradientflow theory (Ambrosio Gigli Savaré, 2008): Fε m is narrowly lower semicontinuous for all m > 1; if ϕ is Gaussian, then is 2-Wasserstein lower semicontinuous; F 1 ε

40 13/25 Results We have proved the following. Fε m : P 2(R d ) R has good basic properties necessary for the use of gradientflow theory (Ambrosio Gigli Savaré, 2008): Fε m is narrowly lower semicontinuous for all m > 1; if ϕ is Gaussian, then Fε 1 is 2-Wasserstein lower semicontinuous; Fε m is differentiable on generalised geodesics;

41 13/25 Results We have proved the following. Fε m : P 2(R d ) R has good basic properties necessary for the use of gradientflow theory (Ambrosio Gigli Savaré, 2008): Fε m is narrowly lower semicontinuous for all m > 1; if ϕ is Gaussian, then Fε 1 is 2-Wasserstein lower semicontinuous; Fε m is differentiable on generalised geodesics; Fε m is semiconvex along generalised geodesics for every ε > 0 with constant λ m,ε = 4 ( ) D 2 ϕ L ε F (R d ) m ϕ ε L (R d ) ;

42 13/25 Results We have proved the following. Fε m : P 2(R d ) R has good basic properties necessary for the use of gradientflow theory (Ambrosio Gigli Savaré, 2008): Fε m is narrowly lower semicontinuous for all m > 1; if ϕ is Gaussian, then Fε 1 is 2-Wasserstein lower semicontinuous; Fε m is differentiable on generalised geodesics; Fε m is semiconvex along generalised geodesics for every ε > 0 with constant λ m,ε = 4 ( ) D 2 ϕ L ε F (R d ) m ϕ ε L (R d ) ; the subdifferential of Fε m can be characterised: given µ P 2(R d ), v F m ε (µ) Tan µp 2(R d ) v = (F m ε ) µ.

43 13/25 Results We have proved the following. Fε m : P 2(R d ) R has good basic properties necessary for the use of gradientflow theory (Ambrosio Gigli Savaré, 2008): Fε m is narrowly lower semicontinuous for all m > 1; if ϕ is Gaussian, then Fε 1 is 2-Wasserstein lower semicontinuous; Fε m is differentiable on generalised geodesics; Fε m is semiconvex along generalised geodesics for every ε > 0 with constant λ m,ε = 4 ( ) D 2 ϕ L ε F (R d ) m ϕ ε L (R d ) ; the subdifferential of Fε m can be characterised: given µ P 2(R d ), F m ε Γ-converges to F m as ε 0. v F m ε (µ) Tan µp 2(R d ) v = (F m ε ) µ.

44 13/25 Results We have proved the following. Fε m : P 2(R d ) R has good basic properties necessary for the use of gradientflow theory (Ambrosio Gigli Savaré, 2008): Fε m is narrowly lower semicontinuous for all m > 1; if ϕ is Gaussian, then Fε 1 is 2-Wasserstein lower semicontinuous; Fε m is differentiable on generalised geodesics; Fε m is semiconvex along generalised geodesics for every ε > 0 with constant λ m,ε = 4 ( ) D 2 ϕ L ε F (R d ) m ϕ ε L (R d ) ; the subdifferential of Fε m can be characterised: given µ P 2(R d ), F m ε Γ-converges to F m as ε 0. v F m ε (µ) Tan µp 2(R d ) v = (F m ε ) µ. The gradient flow for Fε m is well-posed; i.e., there exists a unique solution (so particles do remain particles).

45 13/25 Results We have proved the following. Fε m : P 2(R d ) R has good basic properties necessary for the use of gradientflow theory (Ambrosio Gigli Savaré, 2008): Fε m is narrowly lower semicontinuous for all m > 1; if ϕ is Gaussian, then Fε 1 is 2-Wasserstein lower semicontinuous; Fε m is differentiable on generalised geodesics; Fε m is semiconvex along generalised geodesics for every ε > 0 with constant λ m,ε = 4 ( ) D 2 ϕ L ε F (R d ) m ϕ ε L (R d ) ; the subdifferential of Fε m can be characterised: given µ P 2(R d ), F m ε Γ-converges to F m as ε 0. v F m ε (µ) Tan µp 2(R d ) v = (F m ε ) µ. The gradient flow for Fε m is well-posed; i.e., there exists a unique solution (so particles do remain particles). Convergence of gradient flows. When m = 2 and under suitable regularity assumptions on the regularised gradient flows (µ ε) ε, the gradient flow for Fε m converges to that for F m in the Sandier Serfaty sense.

46 14/25 Results Assumptions on the mollifier. Let ζ C 2 (R d [0, + )) be even, ζ L 1 (R d ) = 1, and assume there exist C ζ, C ζ > 0, q > d + 1, and q > d such that ζ(x) C ζ x q, ζ(x) C ζ x q for all x R d. Choose ϕ = ζ ζ as mollifier.

47 14/25 Results Assumptions on the mollifier. Let ζ C 2 (R d [0, + )) be even, ζ L 1 (R d ) = 1, and assume there exist C ζ, C ζ > 0, q > d + 1, and q > d such that ζ(x) C ζ x q, ζ(x) C ζ x q for all x R d. Choose ϕ = ζ ζ as mollifier. Write ϕ ε = ε d ϕ( /ε) and ζ ε := ε d ζ( /ε).

48 14/25 Results Assumptions on the mollifier. Let ζ C 2 (R d [0, + )) be even, ζ L 1 (R d ) = 1, and assume there exist C ζ, C ζ > 0, q > d + 1, and q > d such that ζ(x) C ζ x q, ζ(x) C ζ x q for all x R d. Choose ϕ = ζ ζ as mollifier. Write ϕ ε = ε d ϕ( /ε) and ζ ε := ε d ζ( /ε). Theorem (Serfaty, 2011). Let m 2. Suppose that, for all ε > 0, µ ε is a gradient flow for Fε m with well-prepared initial data, i.e., µ ε(0) µ 0 narrowly, lim ε 0 F m ε (µ ε(0)) = F m (µ(0)), µ 0 P 2(R d ). Suppose further that there exists a curve µ: [0, T ] P 2(R d ) such that, for almost every t [0, T ], µ ε(t) µ(t) narrowly and (1) lim inf ε 0 (2) lim inf ε 0 (3) lim inf ε 0 t 0 µ ε (s) 2 ds t F m ε (µ ε(t)) F m (µ(t)), t 0 0 µ (s) 2 ds, t (Fε m ) 2 µ ds ε(s) (F m ) 2 L 2 (µ ε(s);r d ) µ(s) ds. L 2 (µ(s);r d ) Then µ is a gradient flow for F m. 0

49 15/25 Results Things to prove. Existence of well-prepared initial data given µ 0 P 2(R d ).

50 15/25 Results Things to prove. Existence of well-prepared initial data given µ 0 P 2(R d ). Follows from the Γ-convergence of F m ε to F m as ε 0.

51 15/25 Results Things to prove. Existence of well-prepared initial data given µ 0 P 2(R d ). Follows from the Γ-convergence of F m ε to F m as ε 0. Existence of a limiting µ: [0, T ] P 2(R d ).

52 15/25 Results Things to prove. Existence of well-prepared initial data given µ 0 P 2(R d ). Follows from the Γ-convergence of F m ε to F m as ε 0. Existence of a limiting µ: [0, T ] P 2(R d ). Follows from a compactness argument on the (m 1)th moments of the regularised gradient flows (µ ε) ε.

53 15/25 Results Things to prove. Existence of well-prepared initial data given µ 0 P 2(R d ). Follows from the Γ-convergence of F m ε to F m as ε 0. Existence of a limiting µ: [0, T ] P 2(R d ). Follows from a compactness argument on the (m 1)th moments of the regularised gradient flows (µ ε) ε. Prove (1) and (2) in Serfaty s theorem.

54 15/25 Results Things to prove. Existence of well-prepared initial data given µ 0 P 2(R d ). Follows from the Γ-convergence of F m ε to F m as ε 0. Existence of a limiting µ: [0, T ] P 2(R d ). Follows from a compactness argument on the (m 1)th moments of the regularised gradient flows (µ ε) ε. Prove (1) and (2) in Serfaty s theorem. Follow from the facts that W 2 and Fε m on P 2(R d ). are narrowly lower semicontinuous

55 15/25 Results Things to prove. Existence of well-prepared initial data given µ 0 P 2(R d ). Follows from the Γ-convergence of F m ε to F m as ε 0. Existence of a limiting µ: [0, T ] P 2(R d ). Follows from a compactness argument on the (m 1)th moments of the regularised gradient flows (µ ε) ε. Prove (1) and (2) in Serfaty s theorem. Follow from the facts that W 2 and Fε m on P 2(R d ). Prove (3) in Serfaty s theorem. are narrowly lower semicontinuous

56 15/25 Results Things to prove. Existence of well-prepared initial data given µ 0 P 2(R d ). Follows from the Γ-convergence of F m ε to F m as ε 0. Existence of a limiting µ: [0, T ] P 2(R d ). Follows from a compactness argument on the (m 1)th moments of the regularised gradient flows (µ ε) ε. Prove (1) and (2) in Serfaty s theorem. Follow from the facts that W 2 and Fε m on P 2(R d ). are narrowly lower semicontinuous Prove (3) in Serfaty s theorem. That is the tough one: we need some extra regularity assumptions on the regularised gradient flows (µ ε) ε.

57 16/25 Results Write M p(µ) := R d x p dµ(x) for all p 0, and

58 16/25 Results Write M p(µ) := R d x p dµ(x) for all p 0, and µ BV m ε := R d R d ζ ε(x y) ( ζ ε p ε)(x) + ( ζ ε µ)(x)f m(ϕ ε µ(y)) dµ(y) dx where p ε := µf m (ϕ ε µ). Note µ BV m ε (Fε m ) µ. L 1 (µ;r d )

59 16/25 Results Write M p(µ) := R d x p dµ(x) for all p 0, and µ BV m ε := R d R d ζ ε(x y) ( ζ ε p ε)(x) + ( ζ ε µ)(x)f m(ϕ ε µ(y)) dµ(y) dx where p ε := µf m (ϕ ε µ). Note µ BV m ε (Fε m ) µ. L 1 (µ;r d ) Theorem (Carrillo Craig P., 2017). Let m 2, and µ ε : [0, T ] P 2(R d ) be a gradient flow for Fε m for all ε > 0 with well-prepared initial data with respect to µ 0 P 2(R d ). Furthermore, suppose that the following hold: (1) sup ε>0 T 0 Mm 1(µε(t)) dt <, T (2) sup ε>0 0 µε(t) BV ε m dt <, { ζε µ ε(t) µ(t) in L 1 ([0, T ]; L m loc(r d )) as ε 0, (3) T sup ε>0 0 ζε µε(t) m L m (R d ) dt <. Then µ ε(t) µ(t) narrowly for almost every t [0, T ] for some µ: [0, T ] P 2(R d ), and µ is the gradient flow for F m with initial datum µ 0.

60 17/25 Contents Background Motivation Results Numerical simulations Outlook

61 18/25 Numerical simulations We now reintroduce the potentials V and W. Recall that our particle method is based on the ODE system ẋ i = ( N m j ϕ ε(x i x j) j=1 F m ( N ) m k ϕ ε(x j x k ) k=1 ( N ) ) N + F m m k ϕ ε(x i x k ) V (x i) m j W (x i x j). k=1 j=1

62 18/25 Numerical simulations We now reintroduce the potentials V and W. Recall that our particle method is based on the ODE system ẋ i = ( N m j ϕ ε(x i x j) j=1 F m ( N ) m k ϕ ε(x j x k ) k=1 ( N ) ) N + F m m k ϕ ε(x i x k ) V (x i) m j W (x i x j). k=1 j=1 We have proved that if we initially place our N particles (x 0 i ) i on a grid with spacing h = N 1/d and if the assumptions of our previous theorem hold, then our particle methods converges to the continuity equation provided h = o(ε).

63 18/25 Numerical simulations We now reintroduce the potentials V and W. Recall that our particle method is based on the ODE system ẋ i = ( N m j ϕ ε(x i x j) j=1 F m ( N ) m k ϕ ε(x j x k ) k=1 ( N ) ) N + F m m k ϕ ε(x i x k ) V (x i) m j W (x i x j). k=1 j=1 We have proved that if we initially place our N particles (x 0 i ) i on a grid with spacing h = N 1/d and if the assumptions of our previous theorem hold, then our particle methods converges to the continuity equation provided h = o(ε). For example, take ε = h 0.99.

64 18/25 Numerical simulations We now reintroduce the potentials V and W. Recall that our particle method is based on the ODE system ẋ i = ( N m j ϕ ε(x i x j) j=1 F m ( N ) m k ϕ ε(x j x k ) k=1 ( N ) ) N + F m m k ϕ ε(x i x k ) V (x i) m j W (x i x j). k=1 j=1 We have proved that if we initially place our N particles (x 0 i ) i on a grid with spacing h = N 1/d and if the assumptions of our previous theorem hold, then our particle methods converges to the continuity equation provided h = o(ε). For example, take ε = h To visualise numerically our particle solution µ ε(h), we convolve it with the mollifier ζ ε; i.e., we plot µ ε(h) = ζ ε µ ε(h).

65 19/25 Numerical simulations One-dimensional heat and porous medium equations: fundamental solutions Height Height Exact vs numerical solution, h = 0.02, varying m m= Position Times Position m= Position m= Exact vs numerical solution, varying h, m = 3 h=.02 Times h= Position h=

66 20/25 Numerical simulations 10 1 Convergence analysis: one-dimensional diffusion m=1 m=2 m=3 W2 Error slope 1.00 slope 1.00 slope L 1 Error slope 2.00 slope 2.03 slope L Error slope 2.00 slope 1.13 slope h h h

67 21/25 Numerical simulations Convergence analysis: two-dimensional diffusion m = 1 m = 2 m = 3

68 22/25 Numerical simulations Second Moment Modified one-dimensional Keller Segel equation: blow-up with χ = 1.5, h = blob method DGF particle exact Time Time Position F m(s) = F 1(s) = log s, W (x) = 2χ log x, V (x) = 0. (DGF: see Carrillo Huang P. Wolansky (2017).)

69 Numerical simulations Particle Trajectories Two-dimensional Keller Segel equation: blow-up with supercritical mass 9π, h = numerical exact Second Moment Time Time Fm (s) = F1 (s) = log s, 23/ W (x) = 1/(2π) log x, 0 Position V (x) = 0. 1

70 24/25 Contents Background Motivation Results Numerical simulations Outlook

71 25/25 Outlook Extensions.

72 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R.

73 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W.

74 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W. When V and W are present, then minimisers of Eε m E m as ε 0. converge to minimisers of

75 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W. When V and W are present, then minimisers of Eε m E m as ε 0. Open questions. converge to minimisers of

76 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W. When V and W are present, then minimisers of Eε m E m as ε 0. converge to minimisers of Open questions. be improved so that it does not degen- Can the semiconvexity estimate of Fε m erate as ε 0?

77 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W. When V and W are present, then minimisers of Eε m E m as ε 0. Open questions. be improved so that it does not degen- Can the semiconvexity estimate of Fε m erate as ε 0? converge to minimisers of Can we extend the convergence of the gradient flows to m [1, 2)?

78 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W. When V and W are present, then minimisers of Eε m E m as ε 0. Open questions. be improved so that it does not degen- Can the semiconvexity estimate of Fε m erate as ε 0? converge to minimisers of Can we extend the convergence of the gradient flows to m [1, 2)? Can we remove the regularity conditions on (µ ε) ε in the convergence of the gradient flows?

79 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W. When V and W are present, then minimisers of Eε m E m as ε 0. Open questions. be improved so that it does not degen- Can the semiconvexity estimate of Fε m erate as ε 0? converge to minimisers of Can we extend the convergence of the gradient flows to m [1, 2)? Can we remove the regularity conditions on (µ ε) ε in the convergence of the gradient flows? Can we improve the method by using other mollifiers?

80 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W. When V and W are present, then minimisers of Eε m E m as ε 0. Open questions. be improved so that it does not degen- Can the semiconvexity estimate of Fε m erate as ε 0? converge to minimisers of Can we extend the convergence of the gradient flows to m [1, 2)? Can we remove the regularity conditions on (µ ε) ε in the convergence of the gradient flows? Can we improve the method by using other mollifiers? Can we find rate estimates for the convergence of gradient flows? Γ-convergence tools.) (Not via

81 25/25 Outlook Extensions. Most of the basic properties of F m ε extend to general functions F : (0, ) R. Our results extend to E m, i.e., to confinement and interaction for semiconvex, smooth potentials V and W. When V and W are present, then minimisers of Eε m E m as ε 0. Open questions. be improved so that it does not degen- Can the semiconvexity estimate of Fε m erate as ε 0? converge to minimisers of Can we extend the convergence of the gradient flows to m [1, 2)? Can we remove the regularity conditions on (µ ε) ε in the convergence of the gradient flows? Can we improve the method by using other mollifiers? Can we find rate estimates for the convergence of gradient flows? Γ-convergence tools.) THANK YOU! (Not via