Data Assimilation in Operator Algebras
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 nonabelian 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 finitedimensional matrix algebras leads to new computational data assimilation schemes that are (i) automatically positivitypreserving; and (ii) amenable to consistent datadriven 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.13659
 Bibcode:
 2022arXiv220613659F
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Chaotic Dynamics;
 Physics  Data Analysis;
 Statistics and Probability;
 Quantum Physics
 EPrint:
 46 pages, 4 figures