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 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.

46 pages, 4 figures

References in corpus (5)

• Quantum advantage in learning from experiments
• Stable machine-learning parameterization of subgrid processes for climate modeling at a range of resolutions
• Efficient quantum algorithm for dissipative nonlinear differential equations
• Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics
• Nonparametric forecasting of low-dimensional dynamical systems

Cited by in corpus (2)

• Consistent spectral approximation of Koopman operators using resolvent compactification
• Koopman and transfer operator techniques from the perspective of quantum theory