TY - JOUR
T1 - Multilevel Importance Sampling for Rare Events Associated With the McKean--Vlasov Equation
AU - Ben Rached, Nadhir
AU - Haji-Ali, Abdul-Lateef
AU - Pillai, Shyam Mohan Subbiah
AU - Tempone, Raúl
N1 - Follow-up to arXiv:2207.06926
PY - 2024/9/23
Y1 - 2024/9/23
N2 - This work combines multilevel Monte Carlo (MLMC) 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., 2023) 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., 2023), 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 $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the single-level DLMC estimator to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-3})$ while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$.
AB - This work combines multilevel Monte Carlo (MLMC) 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., 2023) 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., 2023), 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 $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the single-level DLMC estimator to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-3})$ while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$.
KW - math.NA
KW - cs.NA
KW - stat.CO
KW - 60H35. 65C30. 65C05. 65C35
M3 - Article
SN - 0960-3174
JO - Statistics and Computing
JF - Statistics and Computing
ER -