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.
翻译:我们利用Stochacist Koopman操作员与Kolmogorov后方方程式之间的关系,为随机差异方程式构建重要取样计划。 具体地说,我们建议使用Stochacist Koopman操作员的天分,以近似 Doob 变形, 以达到可见的兴趣( 例如, 与稀有事件相关), 从而得出相应的零差分重要取样估计仪的近似值。 我们的方法广泛适用且系统化, 处理非正常系统、 非梯度系统, 以及带有骨质动态或级级低声的系统, 在一个共同框架中。 在非线性环境中, 随机科普曼操作员无法通过分析推导出, 我们使用动态模式分解法进行数字化, 但这个框架对所使用的特定数字方法是不可知性。 数值实验显示, 即使是微小的精度近, 也能够产生罕见事件的有效重要取样计划。