ALTERNATIVE APPROACHES TO THE KARHUNEN-LOEVE DECOMPOSITION FOR MODEL-REDUCTION AND DATA-ANALYSIS

The Karhunen-Loeve (KL) decomposition is a statistical pattern analysi s technique for finding the dominant structures in an ensemble of spat ially distributed data. Recently the technique has been used to analyz e and perform model reduction on both experimental and simulated spati otemporal patterns from reactive and fluid-dynamical systems. We propo se alternative ensembles for the KL decomposition that address some of the shortcomings of the usual procedure. Two examples are presented. In the first, the question of optimal low-dimensional bases for a reac tion-diffusion model is addressed. We consider an ensemble constructed from short time integrations of a large set of initial conditions. Th is ensemble contains information about the global dynamics that is not contained in an ensemble comprised only of snapshots close to a parti cular attractor. A low-dimensional KL basis for this alternative ensem ble is found to represent the dynamics better than a KL basis obtained only from points on the attractor. The second example shows how diffe rent ensemble averages give different results for the representation o f ''intermittent'' attractors. An average based on arclength in phase space stresses the intermittent components of an attractor, features t hat are de-emphasized in the usual time-average based procedure.