Data Assimilation in Operator Algebras

TL;DR

This paper introduces an algebraic framework for data assimilation using operator algebras, enabling positivity-preserving computations and potential quantum implementation, with promising results in climate models.

Contribution

It develops a novel algebraic approach embedding Bayesian data assimilation into non-abelian operator algebras, linking quantum operations with classical data assimilation.

Findings

New computational schemes that are positivity-preserving.

Framework suitable for quantum computer implementation.

Promising results in climate model data assimilation.

Abstract

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a non-abelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to new computational data assimilation schemes that are (i) automatically positivity-preserving; and (ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to data assimilation of the Lorenz 96 multiscale system and the El Nino Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.