Alternative approaches to the Karhunen-Loeve decomposition for model reduction and data analysis

Michael D. Graham, Ioannis G. Kevrekidis

Research output: Contribution to journalArticlepeer-review

148 Scopus citations


The Karhunen-Loève (KL) decomposition is a statistical pattern analysis technique for finding the dominant structures in an ensemble of spatially distributed data. Recently the technique has been used to analyze and perform model reduction on both experimental and simulated spatiotemporal patterns from reactive and fluid-dynamical systems. We propose 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 reaction-diffusion model is addressed. We consider an ensemble constructed from short time integrations of a large set of initial conditions. This ensemble contains information about the global dynamics that is not contained in an ensemble comprised only of snapshots close to a particular attractor. A low-dimensional KL basis for this alternative ensemble is found to represent the dynamics better than a KL basis obtained only from points on the attractor. The second example shows how different ensemble averages give different results for the representation of "intermittent" attractors. An average based on arclength in phase space stresses the intermittent components of an attractor, features that are de-emphasized in the usual time-average based procedure.

Original languageEnglish (US)
Pages (from-to)495-506
Number of pages12
JournalComputers and Chemical Engineering
Issue number5
StatePublished - 1996

All Science Journal Classification (ASJC) codes

  • General Chemical Engineering
  • Computer Science Applications


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