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Published August 1, 2000 | public
Journal Article

Reconstruction Equations and the Karhunen-Loève Expansion for Systems with Symmetry


We present a method for applying the Karhunen-Lo`eve decomposition to systems with continuous symmetry. The techniques in this paper contribute to the general procedure of removing variables associated with the symmetry of a problem, and related ideas have been used in previous works both to identify coherent structures in solutions of PDEs, and to derive low-order models via Galerkin projection. The main result of this paper is to derive a simple and easily implementable set of reconstruction equations which close the system of ODEs produced by Galerkin projection. The geometric interpretation of the method closely parallels techniques used in geometric phases and reconstruction techniques in geometric mechanics. We apply the method to the Kuramoto- Sivashinsky equation and are able to derive accurate models of considerably lower dimension than are possible with the traditional Karhunen-Loève expansion.

Additional Information

© 2000 Elsevier Science B.V. Received 10 August 1999; revised 3 February 2000; accepted 10 February 2000. Communicated by F.H. Busse. Available online 25 May 2000. The authors would like to thank Richard M. Murray for many helpful comments and discussions. We also thank Mic hael Dellnitz, Dieter Armbruster, Larry Sirovich, and Emily Stone for their interest and insightful comments on the problem considered in this work. This work was partially supported by NSF grant KDI/ATM-9873133 and NSF/DARPA/Opaal grant DMS-9874082.

Additional details

August 21, 2023
October 20, 2023