Stochastic particle method with birth-death dynamics

Abstract

In order to numerically solve high-dimensional nonlinear PDEs and alleviate the curse of dimensionality, a stochastic particle method (SPM) has been proposed to capture the relevant feature of the solution through the adaptive evolution of particles [J. Comput. Phys. 527 (2025) 113818]. In this paper, we introduce an active birth-death dynamics of particles to improve the efficiency of SPM. The resulting method, dubbed SPM-birth-death, sample new particles according to the nonlinear term and execute the annihilation strategy when the number of particles exceeds a given threshold. A rigorous error estimation for SPM-birth-death is established, elucidating the first-order convergence in time and space, as well as half-order accuracy in the initial sample size with explicit variance estimates. We also extend the analysis framework to SPM and provide theoretical justification for the existing numerical convergence study. Our theoretical results reveal that the introduced active birth-death dynamics of particles results into less frequent resampling and SPM-birth-death is thus able to achieve higher efficiency than SPM. Validating benchmarks are provided. In particular, preliminary numerical experiments on the Allen-Cahn equation demonstrate that SPM-birth-death can achieve smaller errors at the same computational cost compared with the original SPM.

Figures (7)

• Figure 1: Flowchart for proving Theorem \ref{['thm:error_SPM_birth_death']}, an error estimator of SPM-birth-death.
• Figure 2: 1-D benchmark: Results with parameters $h = 0.2$ and $\tau = 0.01$, $N=1\times 10^6$ for SPM, $N(0) = 1\times 10^5$, $n_A=9$ for SPM-birth-death. (Top) The absolute $L^2$ error of SPM and SPM-birth-death($R=4$) over time $t\in [0,10]$. The maximal particle number and memory usage of SPM-birth-death are guaranteed to be less than those of SPM since $n_AN(0)< N$. The comparison highlights the superior performance of SPM-birth-death. (Bottom) Zoomed-in view of SPM-birth-death error. The error jumps when death mechanism occurs and oscillates more thereafter, since the resampling procedure introduces the spatial bias and statistical variance.
• Figure 3: 1-D benchmark: The numerical effects of SPM-birth-death with different particle numbers show that its variance decreases as the initial sampling number $N(0)$ increases.
• Figure 4: 1-D benchmark: The variation of particle number over time for SPM-birth-death. When the particle number surpasses $n_A\times N(0)$, the particle annihilation mechanism is activated, resulting in a reduction of the particle number.
• Figure 5: 1-D benchmark: The correlation between relative $L^2$ error $\mathcal{E}_2[u]$ and computational cost for SPM and SPM-birth-death. SPM-birth-death demonstrates superior computational efficiency: achieving lower errors at equivalent computational times (CPU time) and requiring shorter times to reach equivalent error levels. We set in SPM: $N = 1\times 10^5, 2\times 10^5, 4\times 10^5, 8\times 10^5, 1\times 10^6, 1\times 10^7$ with the corresponding running time respectively being 1.7502, 3.4546, 6.8315, 13.42, 16.8924, 168.81 seconds, and in SPM-birth-death: $N(0) = 2\times 10^4, 4\times 10^4, 8\times 10^4, 2\times 10^5, 4\times 10^5, 8\times 10^5$ with the corresponding running time respectively being 2.0102, 3.2941, 6.3855, 13.9954, 28.31, 60.82 seconds. The maximal particle number and memory usage of SPM-birth-death are guaranteed to be less than those of SPM since $n_AN(0)\le N$ for $n_A=3$. Here sample sizes were strategically selected to ensure comparable runtimes, enabling direct comparison within the same figure.

Theorems & Definitions (11)

• Theorem 1
• Corollary 1
• Theorem 2
• Corollary 2
• Lemma 3
• proof : Proof of Lemma \ref{['lem:Linfupper']}
• Lemma 4
• Lemma 5
• Lemma 6
• proof : Proof of Lemma \ref{['lem:gronwall']}