Symmetry and the Karhunen-Loeve analysis

The Karhunen-Loeve (K-L) analysis is widely used to generate low-dimensional dynamical systems, which have the same row-dimensional attractors as some large-scale simulations of PDEs, If the PDE is symmetric with respect to a symmetry group G, the dynamical system has to be equivariant under G to capture the full phase space. It is shown that symmetrizing the K-L, eigenmodes instead of symmetrizing the data leads to considerable computational savings if the K-L analysis is done in the snapshot method. The feasibility of the approach is demonstrated with an analysis of Kolmogorov flow.