A Koopman framework for rare event simulation in stochastic differential equations

TL;DR

This paper introduces a Koopman operator-based method for importance sampling in stochastic differential equations, enabling efficient rare event simulation through eigenfunction approximations, applicable to complex systems with various dynamics.

Contribution

It develops a systematic framework linking the stochastic Koopman operator with importance sampling, using eigenfunctions and DMD methods for broad applicability in rare event simulation.

Findings

Eigenfunction-based importance sampling improves rare event estimation.

Coarse eigenfunction approximations from non-rare trajectories are effective.

The framework applies to non-normal, non-gradient, and oscillatory systems.

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.