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On the hidden beauty of the proper orthogonal decomposition

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Abstract

The proper orthogonal decomposition theorem (Loève, 1955) of probability theory has been proposed by Lumley (1967, 1972, 1981) for detection of spatial coherent patterns in turbulent flows. More specifically, the decomposition extracts deterministic functions from second-order statistics of a random field and converges optimally fast in quadratic mean (i.e., in energy). The technique can be made completely deterministic in the sense that it can be applied to spatially and temporally evolving flows. The remarkable property of this deterministic decomposition is not only in its optimal convergence (as emphasized before) but also in its space/time symmetry which permits access to the spatiotemporal dynamics. The flow is decomposed into both spatial and temporal orthogonal modes which are coupled: each space component is associated with a time component partner. The latter is the time evolution of the former and the former is the spatial configuration of the latter. This generalizes the notion of spatial and temporal structures which can be followed through the various instabilities that the flow undergoes as Reynolds number increases. It also provides a nonlinear dynamics tool for spatiotemporal dynamical systems and can be used for bifurcation detection and analysis as well as dimension and degree of complexity estimates.

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Dedicated to Professor J.L. Lumley on the occasion of his 60th birthday.

This work was supported by an NSF/PYI award MSS89-57462, and partially by a NATO Grant No. 900265 which are gratefully acknowledged.

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Aubry, N. On the hidden beauty of the proper orthogonal decomposition. Theoret. Comput. Fluid Dynamics 2, 339–352 (1991). https://doi.org/10.1007/BF00271473

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