Ensemble forecasts in reproducing kernel Hilbert space family

TL;DR

This paper introduces a novel ensemble forecasting framework using reproducing kernel Hilbert spaces (RKHS) that simplifies high-dimensional dynamical system analysis and data assimilation, leveraging operator properties in RKHS.

Contribution

It develops a new RKHS-based methodology for ensemble estimation and simulation of complex dynamical systems, enabling straightforward data assimilation techniques.

Findings

Operators are unitary and uniformly continuous in RKHS

Exact ensemble-based tangent linear dynamics are accessible

Simple linear combination methods for trajectory reconstruction

Abstract

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.