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Convergence Analysis of a Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Systems

Boyi Hu, Zhongjian Wang, Jack Xin, Zhiwen Zhang

TL;DR

The paper tackles the computational challenge of simulating the 3D fully parabolic Keller-Segel system by proposing a stochastic interacting particle-field method with a random batch approximation (SIPF-$r$). The approach couples particle-based density with a spectrally represented chemical concentration and uses RBM to bypass the mean-field limit, achieving scalable computations in 3D. The authors prove high-probability convergence to the McKean-Vlasov solution in the $\mathcal{W}_1$ distance and deriving explicit convergence rates in terms of the discretization parameters $\delta t$, Fourier mode cutoff $H$, particle count $P$, and batch size $R$, together with detailed lemmas and a discrete Gronwall argument. Numerical experiments validate the predicted convergence rates, confirm the crucial role of the CFL-like conditions, and illustrate the method’s robustness and accuracy for capturing complex chemotactic dynamics in 3D.

Abstract

Chemotaxis models describe the movement of organisms in response to chemical gradients. In this paper, we present a stochastic interacting particle-field algorithm with random batch approximation (SIPF-$r$) for the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) system, also known as the fully parabolic KS system. The SIPF-$r$ method approximates the KS system by coupling particle-based representations of density with a smooth field variable computed using spectral methods. By incorporating the random batch method (RBM), we bypass the mean-field limit and significantly reduce computational complexity. Under mild assumptions on the regularity of the original KS system and the boundedness of numerical approximations, we prove that, with high probability, the empirical measure of the SIPF-$r$ particle system converges to the exact measure of the limiting McKean-Vlasov process in the $1$-Wasserstein distance. Numerical experiments validate the theoretical convergence rates and demonstrate the robustness and accuracy of the SIPF-$r$ method.

Convergence Analysis of a Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Systems

TL;DR

The paper tackles the computational challenge of simulating the 3D fully parabolic Keller-Segel system by proposing a stochastic interacting particle-field method with a random batch approximation (SIPF-). The approach couples particle-based density with a spectrally represented chemical concentration and uses RBM to bypass the mean-field limit, achieving scalable computations in 3D. The authors prove high-probability convergence to the McKean-Vlasov solution in the distance and deriving explicit convergence rates in terms of the discretization parameters , Fourier mode cutoff , particle count , and batch size , together with detailed lemmas and a discrete Gronwall argument. Numerical experiments validate the predicted convergence rates, confirm the crucial role of the CFL-like conditions, and illustrate the method’s robustness and accuracy for capturing complex chemotactic dynamics in 3D.

Abstract

Chemotaxis models describe the movement of organisms in response to chemical gradients. In this paper, we present a stochastic interacting particle-field algorithm with random batch approximation (SIPF-) for the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) system, also known as the fully parabolic KS system. The SIPF- method approximates the KS system by coupling particle-based representations of density with a smooth field variable computed using spectral methods. By incorporating the random batch method (RBM), we bypass the mean-field limit and significantly reduce computational complexity. Under mild assumptions on the regularity of the original KS system and the boundedness of numerical approximations, we prove that, with high probability, the empirical measure of the SIPF- particle system converges to the exact measure of the limiting McKean-Vlasov process in the -Wasserstein distance. Numerical experiments validate the theoretical convergence rates and demonstrate the robustness and accuracy of the SIPF- method.

Paper Structure

This paper contains 14 sections, 7 theorems, 102 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Derivation of SIPF-$r$ Method
  3. Updating chemical concentration $\widetilde{c}$
  4. Updating density of active particles $\widetilde{\rho}$
  5. Particle-wise Independence due to RBM
  6. $L^2$ Convergence of SIPF-$r$ method to smooth solutions
  7. Numerical Experiments
  8. Validation of Convergence Rate
  9. Accuracy of SIPF-$r$ Method
  10. Convergence of the SIPF-$r$ Method
  11. Validation of Theoretical Assumptions
  12. Spatial Lipschitz Continuity
  13. CFL-like Condition
  14. Conclusions

Key Result

Theorem 3.3

Suppose that the exact solutions and the solutions of the SIPF-$r$ method satisfy Assumptions assump:1, assump:2, assump:3 in $\mathbb{R}^3$, consider the SIPF-$r$ method with $H$ being the Fourier mode, $P$ being the number of particles, $R$ being the batch size, and $\delta t$ being the uniform ti where $S_i$, $i = 0, \dots, 10$, are constants specified in Eqs.S_specified-S_specified-for-alpha,

Figures (4)

  • Figure 1: Scattering plot of particles with $M_0$ = 20.
  • Figure 2: Cumlative distribution of $\rho$ computed by SIPF-$r$ and radial FDM
  • Figure 3: $L^2$ error of $\widetilde{c}$ in SIPF-$r$
  • Figure 4: Relative Error

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Lemma 3.5
  • Proof 1
  • Lemma 3.6
  • Proof 2
  • Lemma 3.7
  • ...and 5 more