Reconstruction equations and the Karhunen-Loève expansion for systems with symmetry

We present a method for applying the Karhunen-Loève 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.