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Multilevel Importance Sampling for Rare Events Associated With the McKean--Vlasov Equation

Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Shyam Mohan Subbiah Pillai, Raúl Tempone

Abstract

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 $\mathrm{TOL}_{\mathrm{r}}$.

Multilevel Importance Sampling for Rare Events Associated With the McKean--Vlasov Equation

Abstract

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 for the single-level DLMC estimator to while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance .
Paper Structure (26 sections, 2 theorems, 49 equations, 5 figures, 5 algorithms)

This paper contains 26 sections, 2 theorems, 49 equations, 5 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. McKean--Vlasov Stochastic Differential Equation
  3. Example: fully connected Kuramoto model for synchronized oscillators
  4. Problem setting
  5. Double Loop Monte Carlo Estimator Using a Decoupling Approach
  6. Decoupling Approach for McKean--Vlasov Stochastic Differential Equation
  7. Importance Sampling for the Decoupled McKean--Vlasov Stochastic Differential Equation
  8. Optimal Importance Sampling Control for Decoupled McKean--Vlasov Stochastic Differential Equation
  9. Double Loop Monte Carlo Estimator With Importance Sampling
  10. Multilevel Double Loop Monte Carlo Estimator With Importance Sampling
  11. Multilevel Double Loop Monte Carlo Estimator for Decoupled McKean--Vlasov Stochastic Differential Equation
  12. Error analysis
  13. Complexity analysis
  14. Importance Sampling Scheme for the multilevel DLMC Estimator for the Decoupled MV-SDE
  15. Adaptive Multilevel Double Loop Monte Carlo Algorithm With Importance Sampling
  16. ...and 11 more sections

Key Result

Proposition 1

Let the process $\Bar{X}^P$ follow the dynamics eqn:decoupled_mvsde. We consider the following Itô SDE for the controlled process $\Bar{X}^P_\zeta: [0,T] \cross \Omega \rightarrow \mathbb{R}^d$ with control $\zeta: [0,T] \cross \mathbb{R}^d \rightarrow \mathbb{R}^d$: where eqn:strong_approx_mvsde is used to compute $\{\mu^P_t:t \in [0,T]\}$ in eqn:decoupled_mvsde and eqn:dmvsde_sde_is. The value

Figures (5)

  • Figure 1: Convergence rates of level differences using antithetic ($\hat{\mathcal{G}}$) and naïve ($\bar{\mathcal{G}}$) samplers for the Kuramoto model with $G(x)=\cos{x}$.
  • Figure 2: Algorithm \ref{['alg:mldlmc_is_adaptive']} applied to the Kuramoto model for $G(x) = \cos{x}$. (MLDLMC: multilevel double loop Monte Carlo) .
  • Figure 3: Numerical experiments verifying variance reduction in the double loop Monte Carlo estimator for level differences using importance sampling on the Kuramoto model for $G(x)=\Psi(x-K)$ with $K=2.5$.
  • Figure 4: Convergence rates of level differences using the antithetic estimator ($\hat{\mathcal{G}}$) for the Kuramoto model with $G(x)=\Psi(x-K)$.
  • Figure 5: Algorithm \ref{['alg:mldlmc_is_adaptive']} applied to the Kuramoto model for $G(x) = \Psi(x-K)$.

Theorems & Definitions (13)

  • Proposition 1: Hamilton--Jacobi--Bellman PDE for decoupled MV-SDE my_paper
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1: Optimal multilevel DLMC complexity
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 3 more