Convergence of the random batch method for interacting particles with disparate species and weights
We consider in this work the convergence of the random batch method proposed in our previous work [Jin et al., J. Comput. Phys., 400(2020), 108877] for interacting particles to the case of disparate species and weights. We show that the strong error is of O(√ τ) while the weak error is of O(τ) where τ is the time step between two random divisions of batches. Both types of convergence are uniform in N, the number of particles. The proof of strong convergence follows closely the proof in [Jin et al., J. Comput. Phys., 400(2020), 108877] for indistinguishable particles, but there are still some differences: Since there is no exchangeability now, we have to use a certain weighted average of the errors; some refined auxiliary lemmas have to be proved compared with our previous work. To show that the weak convergence of empirical measure is uniform in N, certain sharp estimates for the derivatives of the backward equations have been used. The weak convergence analysis is also illustrating for the convergence of the Random Batch Method for N-body Liouville equations.
Duke Scholars
Published In
DOI
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- Numerical & Computational Mathematics
- 4903 Numerical and computational mathematics
- 4901 Applied mathematics
- 0103 Numerical and Computational Mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics
Citation
Published In
DOI
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- Numerical & Computational Mathematics
- 4903 Numerical and computational mathematics
- 4901 Applied mathematics
- 0103 Numerical and Computational Mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics