Reconstruction equations and the Karhunen-Loeve expansion for systems withsymmetry

We present a method for applying the Karhunen-Loeve decomposition to system s with continuous symmetry. The techniques in this paper contribute to the general procedure of removing variables associated with the symmetry of a p roblem, and related ideas have been used in previous works both to identify coherent structures in solutions of PDEs, and to derive low-order models v ia Galerkin projection. The main result of this paper is to derive a simple and easily implementable set of reconstruction equations which close the s ystem of ODEs produced by Galerkin projection. The geometric interpretation of the method closely parallels techniques used in geometric phases and re construction techniques in geometric mechanics. We apply the method to the Kuramoto-Sivashinsky equation and an able to derive accurate models of cons iderably lower dimension than are possible with the traditional Karhunen-Lo eve expansion. (C) 2000 Elsevier Science B.V. All rights reserved.