Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916394637983744 |
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| author | Rached, Nadhir Ben Haji-Ali, Abdul-Lateef Tempone, Raúl Wilkosz, Leon |
| author_facet | Rached, Nadhir Ben Haji-Ali, Abdul-Lateef Tempone, Raúl Wilkosz, Leon |
| contents | This work develops a particle system addressing the approximation of McKean-Vlasov stochastic differential equations (SDEs). The novelty of the approach lies in involving low discrepancy sequences nontrivially in the construction of a particle system with coupled noise and initial conditions. Weak convergence for SDEs with additive noise is proven. A numerical study demonstrates that the novel approach presented here doubles the respective convergence rates for weak and strong approximation of the mean-field limit, compared with the standard particle system. These rates are proven in the simplified setting of a mean-field ordinary differential equation in terms of appropriate bounds involving the star discrepancy for low discrepancy sequences with a group structure, such as Rank-1 lattice points. This construction nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An asymptotic error analysis reveals that the proposed approach outperforms methods based on the classic particle system with independent initial conditions and noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09821 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC Rached, Nadhir Ben Haji-Ali, Abdul-Lateef Tempone, Raúl Wilkosz, Leon Numerical Analysis This work develops a particle system addressing the approximation of McKean-Vlasov stochastic differential equations (SDEs). The novelty of the approach lies in involving low discrepancy sequences nontrivially in the construction of a particle system with coupled noise and initial conditions. Weak convergence for SDEs with additive noise is proven. A numerical study demonstrates that the novel approach presented here doubles the respective convergence rates for weak and strong approximation of the mean-field limit, compared with the standard particle system. These rates are proven in the simplified setting of a mean-field ordinary differential equation in terms of appropriate bounds involving the star discrepancy for low discrepancy sequences with a group structure, such as Rank-1 lattice points. This construction nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An asymptotic error analysis reveals that the proposed approach outperforms methods based on the classic particle system with independent initial conditions and noise. |
| title | Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2409.09821 |