A methodological framework for ensemble-based estimation and simulation of high dimensional dynamical systems such as the oceanic or atmospheric flows is proposed. To that end, the dynamical system is embedded in a family of reproducing kernel Hilbert spaces (RKHS) with kernel functions driven by the dynamics. In the RKHS family, the Koopman and Perron-Frobenius operators are unitary and uniformly continuous. This property warrants they can be expressed in exponential series of diagonalizable bounded evolution operators defined from their infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. The RKHS family enables us the devise of strikingly simple ensemble data assimilation methods for trajectory reconstructions in terms of constant-in-time linear combinations of trajectory samples. Such an embarrassingly simple strategy is made possible through a fully justified superposition principle ensuing from several fundamental theorems.

翻译：提出了一种用于高维动力系统（如海洋或大气流动）集合估计与模拟的方法论框架。为此，将动力系统嵌入由动力学驱动的核函数所生成的再生核希尔伯特空间（RKHS）族中。在该RKHS族中，Koopman算子和Perron-Frobenius算子是酉算子且一致连续。这一性质保证了它们可以表示为从其无穷小生成元定义的可对角化有界演化算子的指数级数。李雅普诺夫指数以及正切线性动力学的精确集合表达式也可直接获取。该RKHS族使我们能够设计出异常简洁的集合数据同化方法，通过轨迹样本的常系数线性组合实现轨迹重构。这种极为简单的策略得益于若干基本定理推导出的完全合理的叠加原理。