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A blob method for diffusion

José Antonio Carrillo, Katy Craig, Francesco S. Patacchini

arXiv:1709.09195v4math.APmath.NA

TL;DR

Deterministic particle methods fail to retain particles under diffusion, while stochastic alternatives require averaging. This paper regularizes the internal energy within the Wasserstein gradient-flow framework, yielding a blob method whose regularized energies, flows, and particle solutions converge under stated assumptions, with numerical examples illustrating convergence and qualitative behavior.

  • Problem

    Diffusive equations instantly smooth Dirac-mass initial data, so basic deterministic particle methods no longer preserve particles and stochastic methods require averaging.

  • Method

    The paper regularizes the internal energy with a mollifier, preserving the Wasserstein gradient-flow structure while producing equations whose solutions remain particle-based.

  • Results

    Regularized energies Γ-converge for all m ≥1, gradient flows converge for m ≥2 under sufficient regularity, and the resulting blob-method particle solutions converge to exact solutions.

  • Takeaways & Limitations

    The method provides a deterministic blob framework for diffusive Wasserstein gradient flows and numerically illustrates convergence rates and preserved Fokker–Planck and Keller–Segel properties.

  • Takeaways & Limitations

    General convergence requires additional regularity assumptions, and convergence rates in particle spacing h and regularization ε are not quantified.

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 develops a deterministic particle method for diffusive Wasserstein gradient flows by regularizing the internal energy so particles remain particles. It proves variational and solution convergence under stated assumptions and demonstrates numerical convergence and preserved qualitative behavior.

  • Motivation: Diffusion destroys the Dirac-mass structure required by basic deterministic particle methods, motivating stochastic methods and deterministic regularizations.Stochastic approaches require averaging over many realizations, while prior deterministic approaches regularized the continuity-equation flux.
  • Approach: The paper introduces a deterministic method for linear and nonlinear diffusion that preserves the Wasserstein gradient-flow structure by regularizing the internal energy.The method uses a mollifier and extends the blob-method idea to equations including heat, porous medium, Fokker–Planck, and Keller–Segel models.
  • Variational results: Regularized energies Γ-converge to the unregularized energies for all m ≥1, and their minimizers converge when a confining drift or interaction potential ensures minimizer existence.The regularized energy has a novel form combining aspects of interaction and internal energies.
  • Gradient-flow convergence: For m ≥2 and semiconvex C2 potentials, regularized gradient flows are well-posed; under sufficient regularity, they converge to solutions of the original equation.For m = 2 with bounded-entropy initial data, the required regularity conditions hold automatically, extending the stated porous-medium result to all of R^d.
  • Numerical method: Because particles remain particles for the regularized equation, particle approximations define a blob method whose solutions converge to exact solutions under sufficient regularity.Numerical examples examine convergence rates and qualitative properties, including Fokker–Planck asymptotics and Keller–Segel critical mass.
  • Limitations: The convergence theory requires additional regularity assumptions in general, and the paper does not quantify convergence rates in the particle spacing h and regularization ε.For m > 2 or more general initial data, controlling relevant nonlocal norms remains open; the case 1 ≤m < 2 also remains unresolved.

2. Preliminaries

This section establishes notation for measures, convolutions, mollifiers, and Wasserstein transport, then introduces the geometric and variational definitions used for gradient-flow analysis.

  • Basic notation: The paper works with Borel probability measures on R^d, their moments, densities relative to Lebesgue measure, push-forwards, signed-measure variation, and weak-* convergence.The notation also uses Lp spaces relative to a measure and at-most-quadratic growth for functions.
  • Mollifiers and convolution: Convolution with a mollifier regularizes probability measures and is central to the paper’s regularization of internal energies.The section develops convolution identities, mollifier exchange, and convergence of mollified measures under weak-* convergence.
  • Mollifier assumptions: The mollifier is constructed as ϕ = ζ ∗ζ under smoothness, positivity, symmetry, integrability, and decay assumptions satisfied by Gaussians and smooth compactly supported functions.The scaled kernels are ϕε = ε^-dϕ(·/ε) and ζε = ε^-dζ(·/ε).
  • Optimal transport: Wasserstein distance is defined through transport plans, with optimal plans and transport maps providing the geometry for probability-measure evolution.The section introduces Γ(µ,ν), W2, optimal transport plans, and generalized geodesics.
  • Gradient-flow framework: Generalized geodesics, semiconvexity, subdifferentials, tangent spaces, and local slopes provide the variational framework for Wasserstein gradient flows.A gradient flow is represented by a continuity-equation solution whose velocity is related to the negative subdifferential of the energy.
  • Gradient-flow framework: When an energy is semiconvex along geodesics, its local slope is a strong upper gradient and gradient flows can be characterized as curves of maximal slope.The definitions specify the relevant regularity in time, metric derivative, tangent space, and continuity equation.

3. Regularized internal energies

The paper regularizes internal energies by mollifier convolution, creating a novel Wasserstein-gradient-flow functional that blends internal and interaction effects while retaining useful analytical properties.

  • Analytical properties: The assumptions imply that F is nondecreasing, but they do not ensure convexity along Wasserstein geodesics unless F itself is convex.This separates the baseline assumptions from the stronger convexity condition used for the geodesic analysis.
  • Definition and assumptions: The regularization convolves the density with a mollifier before applying the internal-energy function, producing finite regularized energies under the stated assumptions.The construction is well-defined on P(Rd) for m > 1 and on P2(Rd) for m = 1 under the specified lower-growth conditions.
  • Definition and assumptions: The regularized internal energy is novel because it blends interaction-like convolution with internal-energy composition.The authors distinguish this form from earlier particle methods through the choice of mollifier.
  • Analytical properties: The regularized entropies satisfy lower-semicontinuity properties under weak-* or Wasserstein convergence, depending on m and the mollifier.For m > 1, weak-* convergence is used; for m = 1, Wasserstein convergence is obtained when the mollifier is Gaussian.
  • Analytical properties: For convex F, the regularized internal energies are differentiable and semiconvex along generalized Wasserstein geodesics.The paper also characterizes their subdifferentials and extends the characterization to full energies containing potential and interaction terms.

4. Γ-convergence of regularized internal energies

The regularized energies Γ-converge to the original energies as ε tends to zero, and under confining assumptions their minimizers converge to minimizers of the unregularized problem.

  • Γ-convergence: The regularized internal energies Γ-converge to the unregularized energies with respect to the weak-* topology.This result applies to the entropy and Rényi-entropy cases considered in the section.
  • Minimizers: Confining drift or interaction potentials provide the compactness needed for existence of minimizers of the regularized energies.The stated conditions include compact sublevel sets, quadratic confinement, or a radially growing interaction potential.
  • Minimizers: Under the corresponding confining assumptions, minimizers of the regularized energies converge, up to subsequences and possibly translations, to minimizers of the unregularized energies.The m = 1 result additionally assumes a Gaussian mollifier and quadratic confinement.
  • Limitations: The convergence of minimizers is only established in the weak-* topology because the regularized formulation does not control the required L^m norms.This lack of control is also identified as an obstacle to stronger Γ-convergence results for gradient flows.

5. Γ-convergence of gradient flows

The paper establishes well-posed regularized Wasserstein gradient flows whose particle solutions remain discrete, then proves their convergence to unregularized gradient flows under sufficient assumptions. For m = 2, bounded Hessians of V and W make the additional convergence assumptions automatic.

  • Convergence of gradient flows: Under sufficient regularity and uniform bounds along the regularized flows, the limit curve is an absolutely continuous gradient flow of the unregularized energy.The convergence argument uses a Γ-convergence scheme with assumptions controlling convergence and a nonlocal gradient quantity.
  • Regularized gradient flows: The regularized energies are designed for semiconvex, at-most-quadratic-growth potentials V and W, with W even, and admit unique gradient flows.The proof verifies properness, coercivity, lower semicontinuity, and semiconvexity along generalized geodesics.
  • Regularized gradient flows: Regularized gradient flows can be characterized as continuity equations with an explicitly defined regularized velocity field.This connects the Wasserstein gradient-flow formulation to a partial differential equation.
  • Particle representation: Finite sums of Dirac masses remain sums of Dirac masses under the regularized flow, with particle trajectories governed by ordinary differential equations.The resulting particle system is well-posed forward in time under the stated regularity assumptions.
  • Limitations: For m > 2 or more general initial data, controlling the nonlocal norms required by the convergence theorem remains an open problem.The authors expect these norms to approximate the BV-norm of ρ^m, but do not establish the needed bound generally.
  • Convergence of gradient flows: For m = 2 with V, W ∈ C2(Rd) and bounded Hessians, finite second moments and internal energies suffice to obtain convergence without additional assumptions.The limiting gradient flow is unique, so convergence holds for the full sequence rather than only subsequences.

6. Numerical results

The paper develops and analyzes a deterministic blob method for Wasserstein gradient-flow diffusion, proving convergence under regularity assumptions and testing numerical accuracy and qualitative behavior.

  • Numerical method and convergence: The numerical scheme approximates initial data on a grid, evolves the resulting regularized particle measures through an ODE system, and converges weakly to the exact solution when h = o(ε) under stated assumptions.The particle measures are gradient flows of the regularized energy, and the convergence theorem applies for almost every time.
  • Numerical method and convergence: The convergence proof assumes compactly supported initial data, although arbitrary finite-second-moment data can be approximated by compactly supported data in Wasserstein distance.The compact-support assumption is therefore presented as a technical condition rather than a restrictive practical boundary.
  • Numerical method and convergence: The proof requires h = o(ε), while verifying the additional assumptions along particle solutions can be analytically challenging and is supported numerically instead.The authors use numerical boundedness and convergence of relevant nonlocal Sobolev quantities as evidence for these conditions.
  • Numerical experiments: Gaussian mollifiers gave the best observed balance between computational speed and convergence speed among the tested compactly supported and oscillatory alternatives.The alternative mollifiers produced similar numerical results, but Gaussian mollifiers were preferred in practice.
  • Numerical experiments: The nonlocal Sobolev norm converged as h →0 and decreased over time for heat and porous medium simulations, supporting its interpretation as an approximation of the exact gradient norm.This behavior provides numerical evidence for a key assumption in the convergence theorem.
  • Numerical experiments: The Wasserstein error scaled linearly with h for m = 1, 2, 3 in one dimension, while L1 and L∞ rates varied with m and deteriorated for m = 3.The reported L∞ deterioration for m = 3 is attributed to the sharp boundary transition of the exact solution; two-dimensional experiments showed similar rates.
  • Numerical experiments: The simulations reproduced qualitative behavior across test equations, including convergence toward Fokker–Planck steady states, particle merging, and Keller–Segel critical-mass dynamics.For two-dimensional Keller–Segel, second-moment slopes agreed with theoretical predictions for masses 7π, 8π, and 9π, while larger mass aggregated faster.

Appendix A. Proofs of preliminary results

The appendix proves foundational properties of mollifier-regularized internal energies and establishes convergence of these regularizations as ε→0. It also develops the variational and measure-convergence tools used for the paper’s main energy and gradient-flow results.

  • Preliminary convolution tools: The mollifier exchange lemma supports moving functions through convolutions, a key ingredient in proving Γ-convergence of regularized energies and convergence of gradient flows.The regularization assumes ϕ=ζ∗ζ, hence ϕε=ζε∗ζε, enabling the required convolution manipulations.
  • Regularization inequalities: For entropy, Jensen’s inequality shows nonnegativity of the relative entropy, while integrability controls the regularization error as ε→0.The first term vanishes because µ∈D(Fm) implies µ∈L^m(Rd), while the second remains bounded.
  • Regularization inequalities: The regularized internal energies satisfy inequalities relating them to the unregularized energies for both 1≤m≤2 and m≥2, with convexity reversing the relevant bounds in the latter regime.The proof also derives lower bounds using Carleman-type estimates and second-moment control.
  • Lower semicontinuity: The regularized energies are lower semicontinuous under weak-* convergence for m>1 and Wasserstein convergence for m=1 when the mollifier is Gaussian.The proof combines varying-measure lower-semicontinuity arguments with Gaussian growth bounds and continuity of the logarithm.
  • Variational structure: The appendix characterizes the subdifferential of the regularized energies by proving differentiability along generalized geodesics and identifying the minimal-norm tangent element.The argument uses semiconvexity, mollifier Hessian bounds, dominated convergence, and tangent-space orthogonality.
  • Variational structure: The same variational characterization extends to the full regularized energies including confining and interaction potentials.The proof modifies the generalized-geodesic functions to incorporate V and W while retaining the semiconvexity argument.

Appendix B. Weak convergence of measures

Appendix B recalls convergence concepts for vector fields associated with varying probability measures and the corresponding compactness and lower-semicontinuity tools. These results underpin the paper’s convergence proofs.

  • Definitions: Weak convergence with varying measures defines convergence of vector fields whose underlying probability measures converge weakly.The framework applies to sequences vn∈L1(µn;Rd) converging toward v∈L1(µ;Rd).
  • Definitions: Strong convergence in Lp is introduced for p>1 as a refinement of weak convergence in the varying-measure setting.The definition is stated alongside weak convergence for vector fields carried by changing measures.
  • Compactness properties: Uniform Lp bounds yield a weakly convergent subsequence, while established weak convergence provides the associated lower-bound property.Additional moment bounds support strong-convergence conclusions in the varying-measure framework.
  • Lower semicontinuity: A Fatou-type lemma for varying measures supplies the lower-semicontinuity principle needed when integrating nonnegative functions against converging probability measures.The appendix presents this as the final preliminary result on weak convergence of measures.
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