Ben Rached, N., Haji-Ali, A.-L., Subbiah Pillai, S.M. et al. (1 more author) (2024) Multilevel importance sampling for rare events associated with the McKean–Vlasov equation. Statistics and Computing, 35 (1). ISSN 0960-3174
Abstract
This work combines multilevel Monte Carlo with importance sampling to estimate rare-event quantities that can be expressed as the expectation of a Lipschitz observable of the solution to a broad class of McKean–Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator introduced in this context in Ben Rached et al. (Stat Comput, 2024. https://doi.org/10.1007/s11222-024-10497-3) to the multilevel setting. We formulate a novel multilevel DLMC estimator and perform a comprehensive cost-error analysis yielding new and improved complexity results. Crucially, we devise an antithetic sampler to estimate level differences guaranteeing reduced computational complexity for the multilevel DLMC estimator compared with the single-level DLMC estimator. To address rare events, we apply the importance sampling scheme, obtained via stochastic optimal control in Ben Rached et al. (2024), over all levels of the multilevel DLMC estimator. Combining importance sampling and multilevel DLMC reduces computational complexity by one order and drastically reduces the associated constant compared to the single-level DLMC estimator without importance sampling. We illustrate the effectiveness of the proposed multilevel DLMC estimator on the Kuramoto model from statistical physics with Lipschitz observables, confirming the reduced complexity from O(TOL−4 r ) for the single-level DLMC estimator to O(TOL−3 r ) while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance TOLr.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2024. This is an open access article under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. |
Keywords: | McKean–Vlasov stochastic differential equation, Importance sampling, Multilevel Monte Carlo, Decoupling approach, Double loop Monte Carlo |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 19 Nov 2024 09:32 |
Last Modified: | 19 Nov 2024 09:32 |
Status: | Published online |
Publisher: | Springer |
Identification Number: | 10.1007/s11222-024-10508-3 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:219806 |