On the mean-field limit of the Cucker-Smale model with Random Batch Method
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929444173643776 |
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| author | Wang, Yuelin Lin, Yiwen |
| author_facet | Wang, Yuelin Lin, Yiwen |
| contents | In this work, we focus on the mean-field limit of the Random Batch Method (RBM) for the Cucker-Smale model. Different from the classical mean-field limit analysis, the chaos in this model is imposed at discrete time and is propagated to discrete time flux. We approach separately the limits of the number of particles $N\to\infty$ and the discrete time interval $τ\to 0$ with respect to the RBM, by using the flocking property of the Cucker-Smale model and the observation in combinatorics. The Wasserstein distance is used to quantify the difference between the approximation limit and the original mean-field limit. Also, we combine the RBM with generalized Polynomial Chaos (gPC) expansion and proposed the RBM-gPC method to approximate stochastic mean-field equations, which conserves positivity and momentum of the mean-field limit with random inputs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21297 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the mean-field limit of the Cucker-Smale model with Random Batch Method Wang, Yuelin Lin, Yiwen Numerical Analysis Analysis of PDEs Classical Analysis and ODEs Probability In this work, we focus on the mean-field limit of the Random Batch Method (RBM) for the Cucker-Smale model. Different from the classical mean-field limit analysis, the chaos in this model is imposed at discrete time and is propagated to discrete time flux. We approach separately the limits of the number of particles $N\to\infty$ and the discrete time interval $τ\to 0$ with respect to the RBM, by using the flocking property of the Cucker-Smale model and the observation in combinatorics. The Wasserstein distance is used to quantify the difference between the approximation limit and the original mean-field limit. Also, we combine the RBM with generalized Polynomial Chaos (gPC) expansion and proposed the RBM-gPC method to approximate stochastic mean-field equations, which conserves positivity and momentum of the mean-field limit with random inputs. |
| title | On the mean-field limit of the Cucker-Smale model with Random Batch Method |
| topic | Numerical Analysis Analysis of PDEs Classical Analysis and ODEs Probability |
| url | https://arxiv.org/abs/2407.21297 |