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A blob method for diffusion

José Antonio Carrillo, Katy Craig, Francesco S. Patacchini

arXiv:1709.09195v4math.APmath.NA

TL;DR

Diffusion causes Dirac-mass particle approximations to lose their particle form, motivating deterministic methods that preserve particles while respecting Wasserstein gradient-flow structure. The paper introduces a mollifier-based blob method and proves convergence of regularized energies, gradient flows, and numerical particle solutions under stated regularity assumptions.

  • Problem

    Diffusion immediately smooths Dirac-mass initial data, limiting basic deterministic particle methods and motivating alternatives to stochastic particle simulations.

    Source: §1 · p. 3

  • Method

    The paper convolves internal energies with a mollifier to preserve the gradient-flow structure and construct particle-preserving regularized equations and a deterministic blob method.

    Source: §1 · p. 3, §3 · pp. 9–10

  • Results

    Regularized energies Γ-converge to the original energies, their gradient flows and blob-method particle solutions converge to diffusion solutions under sufficient regularity, and critical mass converges numerically.

    Source: §1 · pp. 4, 6

  • Takeaways & Limitations

    The method provides a deterministic particle framework for numerically simulating Wasserstein-gradient-flow diffusion equations while preserving qualitative behavior such as critical mass.

    Source: §1 · pp. 4, 6, §6.1 · p. 27

    Convergence for m > 2 or general initial data requires regularity controls that remain unresolved, while the approach for 1 ≤ m < 2 is unclear.

    Source: §1 · p. 6

Abstract

from arXiv · show

As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocity field that ensures particles do remain particles, and we apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient flow type, including the heat equation, the porous medium equation, the Fokker-Planck equation, the Keller-Segel equation, and its variants. Our choice of regularization is guided by the Wasserstein gradient flow structure, and the corresponding energy has a novel form, combining aspects of the well-known interaction and potential energies. In the presence of a confining drift or interaction potential, we prove that minimizers of the regularized energy exist and, as the regularization is removed, converge to the minimizers of the unregularized energy. We then restrict our attention to nonlinear diffusion of porous medium type with at least quadratic exponent. Under sufficient regularity assumptions, we prove that gradient flows of the regularized energies converge to solutions of the porous medium equation. As a corollary, we obtain convergence of our numerical blob method, again under sufficient regularity assumptions. We conclude by considering a range of numerical examples to demonstrate our method's rate of convergence to exact solutions and to illustrate key qualitative properties preserved by the method, including asymptotic behavior of the Fokker-Planck equation and critical mass of the two-dimensional Keller-Segel equation.

1. Introduction

The paper introduces a deterministic blob method for diffusive Wasserstein gradient flows by regularizing internal energy, preserving particles and the gradient-flow structure. It proves convergence of energies, gradient flows, and particle solutions, and demonstrates convergence rates and qualitative behavior numerically.

Motivation

  • Stochastic particle methods require averaging and do not readily extend beyond linear diffusion, while existing deterministic methods generally fail to preserve the gradient-flow structure.Diffusion causes Dirac particles to spread instantly; the paper addresses this limitation for linear and nonlinear diffusion, including m > 1.Source: §1 · p. 3

Method

  • The method regularizes the internal energy rather than the continuity-equation flux, so particles remain particles while the underlying Wasserstein gradient-flow structure is preserved.The approach applies in all dimensions and covers equations with linear and nonlinear diffusion, including heat, porous medium, Fokker–Planck, and Keller–Segel equations.Source: §1 · pp. 3–4

Analysis

  • The regularized energies Γ-converge to the original energies, minimizers converge when confining or interaction potentials ensure their existence, and gradient flows converge for m ≥ 2 under sufficient regularity.For m = 2 with bounded-entropy initial data, the required regularity conditions hold automatically, extending the corresponding convergence result to the full equation on R^d.Source: §1 · p. 4

Numerical convergence

  • Under sufficient regularity, numerical blob particle solutions converge to exact solutions, with convergence theory requiring particle spacing h = o(ε).The numerical method uses the particle representation of the regularized equation, whose convolutions reduce to finite sums and combine naturally with drift and interaction potentials.Source: §1 · pp. 4–5

Numerical results

  • For heat and porous medium equations, the 2-Wasserstein error scales linearly with h ∼ N−1/d, while L1 errors are quadratic for m = 1, 2 and superlinear for m = 3.Numerical tests also capture Fokker–Planck convergence to its steady state and recover critical masses 1 and 8π for one- and two-dimensional Keller–Segel variants, with critical-mass convergence under refinement.Source: §1 · pp. 5–6

2. Preliminaries

This section establishes notation for measures, functions, and growth conditions, then develops mollifier-based convolution tools and the Wasserstein framework used to define gradient flows.

2.1. Basic notation.

  • The paper fixes notation for Euclidean balls, indicator functions, at-most-quadratic growth, probability measures with finite moments, densities, Lebesgue spaces, push-forwards, and weak-* convergence.These conventions support subsequent measure-theoretic and variational arguments.Source: §2.1 · p. 6

2.2. Convolution of measures.

  • Mollifiers regularize measures and integrands: the paper assumes ϕ = ζ ∗ ζ with smooth, even, normalized ζ, defines scaled kernels ϕε and ζε, and establishes exchange and convergence lemmas.The mollifier exchange lemma is used in the proofs of Γ-convergence of regularized energies and convergence of their gradient flows.Source: §2.2 · pp. 6–7
  • The convolution results include conditions for moving mollifiers through integrals and show that weak-* convergent probability measures retain the same limit after mollification.The assumptions cover Gaussians and smooth compactly supported functions, while the exchange lemma applies to Lipschitz integrands and finite signed measures.Source: §2.2 · p. 7

2.3. Optimal transport, Wasserstein metric, and gradient flows.

  • The optimal-transport framework defines transport plans and the W2 metric, identifies unique optimal maps for absolutely continuous sources, and introduces geodesics and generalized geodesics.Wasserstein convergence is stronger than weak-* convergence, and semiconvexity along generalized geodesics underpins uniqueness and stability of gradient flows.Source: §2.3 · pp. 7–8
  • Gradient flows are defined through proper lower-semicontinuous energies, subdifferentials, tangent velocity fields, and the continuity equation, with local slopes characterizing flows under semiconvexity.The framework also defines domains, reduced subdifferentials, tangent spaces, and curves of maximal slope.Source: §2.3 · pp. 8–9

3. Regularized internal energies

The section regularizes internal energies by mollifier convolution, preserving the gradient-flow structure while keeping particles atomic. Under structural assumptions on F, it establishes finiteness, lower semicontinuity, differentiability, convexity, and subdifferential properties for the regularized and full energies.

Structural assumptions

  • Assumption 3.1 requires F∈C2, coercivity at infinity, controlled lower behavior near zero, and convexity, boundedness, and vanishing of U(s)=sF(s).These conditions imply that F is nondecreasing, but they do not generally ensure Wasserstein geodesic convexity unless F itself is convex.Source: §3 · pp. 9–10

Construction and scope

  • Mollifier convolution produces finite regularized internal energies for every probability measure and yields a novel functional blending internal and interaction effects.The regularized energies are combined with potential and interaction energies to approximate the original diffusion equation while preserving particle structure.Source: §3 · p. 10

Bounds and lower semicontinuity

  • For entropy and Rényi cases, regularized energies approximate the unregularized energies with errors vanishing as ε→0 and satisfy the appropriate weak-* or Wasserstein lower semicontinuity.Lower semicontinuity holds for m>1 under weak-* convergence, while the entropy case m=1 requires quadratic Wasserstein convergence and a Gaussian mollifier.Source: §3 · p. 11

Differentiability and convexity

  • When F is convex, regularized internal energies are differentiable and λF-convex along generalized geodesics.Differentiability follows from regularity of F and the mollifier, while convexity is obtained through the above-the-tangent inequality.Source: §3 · p. 11

Subdifferential characterization

  • The subdifferential of the regularized internal energy is characterized explicitly, and this extends to the full energy under semiconvex, even, at-most-quadratic potential and interaction terms.The full-energy result assumes V and W are C1 and semiconvex, with W even and both having at most quadratic growth.Source: §3 · p. 12

4. Γ-convergence of regularized internal energies

The regularized internal energies Γ-converge to their unregularized counterparts as ε → 0. With suitable confinement, minimizers exist and converge weakly-* to minimizers of the unregularized energy, although stronger convergence is obstructed by missing L^m bounds.

Γ-convergence

  • F_ε Γ-converges to F as ε → 0 with respect to the weak-* topology.The theorem treats m = 1 on P_2(R^d) and m > 1 on P(R^d).Source: §4 · p. 13

Existence of minimizers

  • Under confining assumptions on V or W, the regularized energies are lower semicontinuous and admit minimizers.For m > 1, existence holds under (CV) or (CW); for m = 1, it holds under (CV′) with Gaussian ϕ.Source: §4 · p. 15

Convergence of minimizers

  • As ε → 0, minimizers of the regularized energies converge, up to subsequences and, under (CW), translations, to minimizers of the unregularized energy.The result applies for m > 1 under (CV) or (CW), and for m = 1 under (CV′) with Gaussian ϕ.Source: §4 · pp. 15–16

Limitations

  • Stronger convergence is unavailable because the regularized energy does not control L^m norms of minimizing sequences.This limitation also obstructs improving the Γ-convergence results for gradient flows.Source: §4 · p. 16

5. Γ-convergence of gradient flows

This section establishes well-posed, particle-preserving regularized gradient flows for m ≥ 2 and gives conditions for their convergence to unregularized gradient flows as ε → 0. In the m = 2 case with C² potentials having bounded Hessians, finite initial second moments and internal energies ensure convergence of the full sequence to the unique limiting gradient flow.

Well-posedness

  • Under Assumption 5.1, each regularized energy admits a unique gradient flow for every initial datum in its domain.The assumptions require semiconvex, at-most-quadratic-growth potentials V and W, with W even.Source: §5 · p. 16

Particle dynamics

  • For regularized energies, a gradient flow starting from a finite sum of Dirac masses remains a finite sum of Dirac masses, with particle trajectories governed by an ordinary-differential-equation system.The associated ODE system is well-posed forward in time.Source: §5 · pp. 17–18

Γ-convergence framework

  • A Serfaty-type criterion shows that weak convergence plus suitable energy and slope conditions identifies the limit as a gradient flow of the unregularized energy.The criterion assumes regularized gradient flows, almost-everywhere weak convergence, and the stated convergence conditions; the limit lies in AC2([0,T]; P2(Rd)).Source: §5 · pp. 18–19

Gradient-flow convergence

  • Under Assumption 5.1 and Theorem 5.8’s hypotheses, µε converges weakly almost everywhere to µ, which is a gradient flow of Em with initial data µ(0).The hypotheses include uniform control of the nonlocal BV-type quantity and convergence of ζε ∗µε to µ in L1([0,T]; Lm_loc(Rd)).Source: §5 · pp. 20–21

Corollary 5.9

  • For m = 2, bounded Hessians of V and W make the additional assumptions unnecessary when initial data have finite second moments and internal energies.This establishes convergence without separately imposing Assumptions (A5.8)–(A2).Source: §5 · p. 24

Proof of Corollary 5.9

  • Uniform moment, energy, and Sobolev estimates establish the compactness and convergence conditions required to apply the gradient-flow convergence theorem.The proof derives uniform bounds using bounded Hessians, entropy estimates, cutoff tests, Gronwall’s inequality, and local Rellich–Kondrachov compactness.Source: §5 · pp. 25, 27

Corollary 5.9

  • The regularized flows µε converge, along the full sequence, to the unique gradient flow µ of E2 with the same initial data.Compactness and convergence of the regularized densities verify Theorem 5.8; uniqueness of the limiting flow upgrades subsequential convergence to full convergence.Source: §5 · p. 27

6. Numerical results

The numerical results assess the blob method across diffusion, Fokker–Planck, and Keller–Segel equations, testing convergence and qualitative behavior. They examine density evolution, second moments, steady states, and blow-up dynamics, including mass-dependent aggregation and agreement with theoretical slopes.

6.1. Numerical method and convergence.

  • Theorem 6.1 establishes weak-* convergence of particle solutions to the unique unregularized gradient flow for h = o(ε), under the stated regularity assumptions.The result applies for almost every t ∈ [0,T] when m ≥ 2 and the required assumptions hold along the particle solutions.Source: §6.1 · pp. 27–28
  • The method provides numerical evidence for its convergence assumptions: the relevant nonlocal Sobolev quantity decreases or remains bounded, while blob solutions converge in L1 and L∞ on bounded time intervals.For purely diffusive problems the quantity decreases in time; for nonlinear Fokker–Planck it converges toward the steady-state norm.Source: §6.1 · p. 30, Heat and Porous Medium Equations:… · p. 32

6.2. Numerical implementation.

  • Gaussian mollifiers provide the best observed balance between computational speed and convergence speed, while ODE-solver choices produce nearly identical results and lower-order time-discretization error.The implementation uses ε = h^0.99, consistent with the theoretical requirement h = o(ε), and evaluates errors in L1, L∞, and Wasserstein metrics.Source: §6.2 · p. 30

6.3. Simulations.

  • For heat and porous-medium equations, numerical and exact solutions are visually indistinguishable for m = 1 and 2, while m = 3 tails converge as the grid is refined.The simulations also reproduce infinite propagation for m = 1 and finite propagation with emerging Barenblatt profiles for m = 2 and 3.Source: §6.3 · p. 31, Figure 1 · p. 32

Heat and Porous Medium Equations: Evolution of Nonlocal Sobolev Norm

  • The nonlocal Sobolev norm converges as h → 0 and decreases in time for heat and porous-medium solutions, supporting its interpretation as an approximation of ||∇μ^m||_L1.For nonlinear Fokker–Planck data, it decreases for Barenblatt initial data and remains bounded while approaching the steady-state norm for double-bump data.Source: Heat and Porous Medium Equations:… · pp. 32, 34, Figure 3 · p. 33

Heat and Porous Medium Equations: Fundamental Solution

  • The Wasserstein error scales linearly with h, while L1 and L∞ rates depend on m; two-dimensional experiments show similar convergence rates to the one-dimensional case.For m = 1,2, L1 convergence is quadratic or better, whereas L∞ convergence deteriorates for m = 3 because of the exact solution’s sharp boundary transition.Source: Heat and Porous Medium Equations:… · p. 32, Figure 4 · p. 34, Figure 5 · p. 35

Fokker–Planck: Two Dimensions

  • The one-dimensional Keller–Segel variants reproduce blow-up or steady-state behavior through second moments and particle trajectories, with modest improvement over the prior DGF particle method after grid refinement.For the nonlinear variant, both the second moment and trajectories converge to a steady state; for the blow-up case, trajectories merge as the second moment approaches zero.Source: Heat and Porous Medium Equations:… · p. 35, Fokker–Planck: Two Dimensions · p. 37, Figure 7 · p. 37
  • For the two-dimensional Keller–Segel equation, the method reproduces mass-dependent aggregation and second-moment evolution, with performance similar across dimensions and trajectories shaped by diffusion and attraction.Supercritical data aggregate more rapidly, the second moment is approximately linear in time, and the numerical solution reaches roughly half the exact zero-time before peeling away.Source: Fokker–Planck: Two Dimensions · p. 38, Figure 9 · p. 38
  • For double-bump initial data with masses 7π, 8π, and 9π, the second-moment slopes agree well with theoretically predicted slopes.The slopes are estimated from the particle-solution evolutions and compared with the predictions shown in Figure 10.Source: Fokker–Planck: Two Dimensions · p. 38, Figure 10 · p. 39

Two-Dimensional Keller–Segel Equation: Evolution of Density

  • For M = 7π, 8π, and 9π, larger initial mass produces faster aggregation at the origin in the two-dimensional Keller–Segel equation.Figure 12 evolves blob solutions initialized from constant multiples of a linear combination of Barenblatts.Source: Figure 12 · p. 40

Appendix A. Proofs of preliminary results

Appendix A proves the preliminary analytical results underlying the regularized energies and their Wasserstein gradient-flow analysis. It establishes mollifier convergence, energy bounds and lower semicontinuity, then characterizes derivatives and subdifferentials.

Mollifier convergence

  • The appendix proves the mollifier exchange lemma and shows that ϕε * µε converges to µ in distribution.The distributional convergence is tested against smooth, compactly supported functions.Source: Appendix A · p. 41

Internal-energy inequalities

  • Proposition 3.8 establishes inequalities relating regularized and unregularized internal energies, including entropy, power-law, and lower-bound estimates.The proof uses Jensen’s inequality, Hölder’s inequality, Carleman-type estimates, and second-moment bounds.Source: Appendix A · pp. 41–42

Lower semicontinuity

  • Proposition 3.9 proves lower semicontinuity of regularized energies under weak-* convergence for m > 1 and Wasserstein convergence for Gaussian kernels when m = 1.The argument combines liminf estimates with Gaussian growth control and continuity of the logarithm.Source: Appendix A · pp. 42–43

Geodesic differentiability

  • Proposition 3.10 proves differentiability of regularized energies along generalized geodesics using Hessian bounds and dominated convergence.The proof constructs an integrable dominating function for the relevant derivative terms.Source: Appendix A · pp. 43–44

Subdifferential characterization

  • Proposition 3.12 identifies the Wasserstein subdifferential and its minimal-norm element for the regularized internal energies.The characterization follows from first- and second-derivative estimates along interpolating measures and a tangent-space orthogonality argument.Source: Appendix A · pp. 44–46

Full regularized energies

  • Corollary 3.13 extends the subdifferential characterization to the full regularized energies by incorporating the semiconvexity constants of V and W.Its proof follows the construction for the internal energy while modifying the auxiliary functions to include potential and interaction terms.Source: Appendix A · pp. 46–47

Appendix B. Weak convergence of measures

Appendix B recalls weak convergence tools for sequences of functions and measures, including convergence under varying probability measures and a corresponding Fatou lemma. These results support the paper’s Γ-convergence proofs for energies and gradient flows.

Role in the paper

  • The recalled convergence theorem for varying measures is used in the proofs of both Γ-convergence of the energies and Γ-convergence of the gradient flows.The appendix identifies this theorem as a key ingredient in those proofs.Source: Appendix B · p. 47

Definitions

  • The appendix defines weak convergence of functions when the underlying probability measures converge weakly, together with strong Lp convergence for p > 1.The functions belong to L1(µn; Rd), while the limit belongs to L1(µ; Rd).Source: Appendix B · p. 47

Compactness and convergence properties

  • Under weak convergence of measures and uniformly bounded Lp norms, a subsequence converges weakly, weak limits satisfy additional convergence properties, and strong Lp convergence yields further conclusions under bounded moments.These properties are stated as Proposition B.2 and are attributed to [3, Theorem 5.4.4].Source: Appendix B · p. 47

Lower-semicontinuity tool

  • A Fatou lemma generalized to varying measures is recalled for nonnegative functions when µn converges weakly to µ.The result is cited to [46, Theorem 1.1] and [4, Lemma 3.3].Source: Appendix B · p. 47

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