Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation

TL;DR

The paper addresses efficient numerical simulation of the McKean–Vlasov equation by proposing and analyzing three discretization schemes: a standard particle method with convergence in Wasserstein distance and two quantization-based schemes (recursive quantization for the Vlasov form and a hybrid particle–quantization approach). It provides rigorous convergence and error analyses, deriving Wasserstein-rate bounds for the particle method and $L^2$-error bounds for the quantization-based schemes, while clarifying practical trade-offs between determinism, randomness, and computational cost. The work also demonstrates practical applicability through Burgers’ equation and a 3D FitzHugh–Nagumo neuronal network, highlighting the potential to reduce data volume and maintain accuracy with quantization, especially via Lloyd’s algorithm. Overall, the results offer a toolbox of scalable, implementable strategies for mean-field-type dynamics with quantifiable accuracy, suitable for high-dimensional and computationally intensive settings.

Abstract

In this paper, we study three numerical schemes for the McKean-Vlasov equation \[\begin{cases} \;dX_t=b(t, X_t, μ_t) \, dt+σ(t, X_t, μ_t) \, dB_t,\: \\ \;\forall\, t\in[0,T],\;μ_t \text{ is the probability distribution of }X_t, \end{cases}\] where $X_0$ is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients $b$ and $σ$, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends a previous work [BT97] established in one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme, inspired by the $K$-means clustering). Two examples are simulated at the end of this paper: Burger's equation and the network of FitzHugh-Nagumo neurons in dimension 3.

Figures (11)

• Figure 1: A Voronoï partition.
• Figure 2: The quadratic optimal quantization for the standard normal distribution $\mathcal{N}(0, \mathbf{I}_2)$.
• Figure 3: Log-error of the particle method \ref{['Deq']} with respect to $\log_{2}(N)$.
• Figure 4: Standard deviation of the error of the particle method. The horizontal axis is $\log_{2}(N)$.
• Figure 5: Log-error of the recursive quantization scheme \ref{['Eeq']}-\ref{['Geq']}-\ref{['Geq2']} with respect to $\log_{2}(K)$.

Theorems & Definitions (23)

• Proposition 2.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6: "À la Gronwall" Lemma
• Theorem 3.7
• Lemma 3.8
• proof : Proof of Theorem \ref{['thm1']}