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Quantum mechanics and data assimilation.

Authors :
Giannakis D
Source :
Physical review. E [Phys Rev E] 2019 Sep; Vol. 100 (3-1), pp. 032207.
Publication Year :
2019

Abstract

A framework for data assimilation combining aspects of operator-theoretic ergodic theory and quantum mechanics is developed. This framework adapts the Dirac-von Neumann formalism of quantum dynamics and measurement to perform sequential data assimilation (filtering) of a partially observed, measure-preserving dynamical system, using the Koopman operator on the L^{2} space associated with the invariant measure as an analog of the Heisenberg evolution operator in quantum mechanics. In addition, the state of the data assimilation system is represented by a trace-class operator analogous to the quantum mechanical density operator, and the assimilated observables by self-adjoint multiplication operators. An averaging approach is also introduced, rendering the spectrum of the assimilated observables discrete and thus amenable to numerical approximation. We present a data-driven formulation of the quantum mechanical data assimilation approach, utilizing kernel methods from machine learning and delay-coordinate maps of dynamical systems to represent the evolution and measurement operators via matrices in a data-driven basis. The data-driven formulation is structurally similar to its infinite-dimensional counterpart and shown to converge in a limit of large data under mild assumptions. Applications to periodic oscillators and the Lorenz 63 system demonstrate that the framework is able to naturally handle highly non-Gaussian statistics, complex state space geometries, and chaotic dynamics.

Details

Language :
English
ISSN :
2470-0053
Volume :
100
Issue :
3-1
Database :
MEDLINE
Journal :
Physical review. E
Publication Type :
Academic Journal
Accession number :
31639900
Full Text :
https://doi.org/10.1103/PhysRevE.100.032207