Multi-index importance sampling for McKean--Vlasov stochastic differential equations

TL;DR

The paper addresses rare-event estimation for solutions to McKean–Vlasov stochastic differential equations by marrying a decoupled IS strategy with a multi-index Monte Carlo framework. The authors derive a two-layer estimator where the IS control is computed offline via a PDE and applied uniformly across multiple discretization indices, leading to substantial variance reduction and a theoretical complexity improvement to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$ for smooth observables. The multi-index approach leverages the propagation of chaos to ensure mixed-difference variances decay faster, enabling efficient coupling of particle and time discretizations; an adaptive algorithm constructs the optimal index set and sample allocations. Numerical experiments on the Kuramoto model demonstrate orders-of-magnitude computational savings over standard MC and outperform the prior multilevel DLMC approach, while verifying the necessary regularity assumptions for the mixed-difference analysis. The work provides a practical framework for robust uncertainty quantification in MV-SDEs and outlines clear directions for rigorous analysis and extension to higher dimensions and less regular observables.

Abstract

This work addresses the estimation of rare-event quantities expressed as expectations of smooth observables of solutions to a broad class of McKean--Vlasov stochastic differential equations (MV-SDEs). Building on the double loop Monte Carlo (DLMC) method with stochastic optimal control-based importance sampling (IS) introduced by Ben Rached et al. (2024a), this work extends this framework to the multi-index Monte Carlo (MIMC) setting. The resulting multi-index DLMC estimator mitigates the explosion of the coefficient of variation for rare event quantities. Moreover, it exploits the sampling efficiency of MIMC by leveraging the propagation of chaos to ensure mixed-difference variances vanish in the mean-field limit. The complexity analysis relies on assumptions on mixed-difference bias and variance decay, similar to standard MIMC assumptions. Although not rigorously proved, this work presents strong numerical evidence in support of these assumptions. The primary contribution of this work is the novel numerical integration of the MIMC method with IS for MV-SDEs. This approach reduces the computational complexity from $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the DLMC estimator to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$, enabling an accurate estimation of rare-event quantities within a prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$. Numerical experiments on the Kuramoto model from statistical physics demonstrate computational savings of several orders of magnitude for the multi-index DLMC estimator with IS, compared with the standard Monte Carlo (MC) method.

Figures (8)

• Figure 1: Rare-event observables $G$ plotted for $\epsilon = 0.5$ and $K=0$, including the indicator function and corresponding mollifiers of varying regularity.
• Figure 2: Variance reduction due to the IS scheme introduced in Section \ref{['sec:dlmcis']} for the Kuramoto model \ref{['eqn:kuramoto_model']} with the $C^\infty$-mollifier observable with $\varepsilon = \frac{1}{3}$ and $K = 3.5$.
• Figure 3: Kuramoto example, rate verification ($C^\infty$-mollifier observable with $K = 3.5$ and $\varepsilon = \frac{1}{3}$): Sample means and variances of mixed-differences in the multi-index DLMC estimator \ref{['eqn:midlmc_estimator']}.
• Figure 4: Kuramoto example, numerical rate verification ($C^\infty$-mollifier observable with $K = 3.5$ and $\varepsilon = \frac{1}{3}$): Contour plots of the sample mean and variances in the multi-index DLMC estimator \ref{['eqn:midlmc_estimator']}. All plots numerically and asymptotically verify Assumptions \ref{['ass:midlmc_bias']} and \ref{['ass:midlmc_var']}.
• Figure 5: Kuramoto example, adaptive DLMC algorithms with IS ($C^\infty$-mollifier observable with $K = 3.5$ and $\varepsilon = \frac{1}{3}$): Comparing multilevel and multi-index estimators. Here, MLDLMC+IS denotes multilevel DLMC with IS and MIDLMC+IS denotes multi-index DLMC with IS.

Theorems & Definitions (5)

• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3