A Koopman framework for rare event simulation in stochastic differential equations

Abstract

We exploit the relationship between the stochastic Koopman operator and the Kolmogorov backward equation to construct importance sampling schemes for stochastic differential equations. Specifically, we propose using eigenfunctions of the stochastic Koopman operator to approximate the Doob transform for an observable of interest (e.g., associated with a rare event) which in turn yields an approximation of the corresponding zero-variance importance sampling estimator. Our approach is broadly applicable and systematic, treating non-normal systems, non-gradient systems, and systems with oscillatory dynamics or rank-deficient noise in a common framework. In nonlinear settings where the stochastic Koopman eigenfunctions cannot be derived analytically, we use dynamic mode decomposition (DMD) methods to compute them numerically, but the framework is agnostic to the particular numerical method employed. Numerical experiments demonstrate that even coarse approximations of a few eigenfunctions, where the latter are built from non-rare trajectories, can produce effective importance sampling schemes for rare events.

Added minor revisions to the description of the methodology in Section 2, explanation of numerical examples in Section 3, and additional discussion in Section 6. One additional numerical example is provided in Section 3. Results are unchanged. Journal of Computational Physics, 2022

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Cited by in corpus (4)

• Scalable Methods for Computing Sharp Extreme Event Probabilities in Infinite-Dimensional Stochastic Systems
• Learning Koopman eigenfunctions of stochastic diffusions with optimal importance sampling and ISOKANN
• Computing eigenfunctions of the multidimensional Ornstein-Uhlenbeck operator
• Rare event sampling for moving targets: extremes of temperature and daily precipitation in a general circulation model