Optimality of Balanced Proper Orthogonal Decomposition for Data Reconstruction II: Further Approximation Results

In our earlier paper [Numer. Funct. Anal. Optim. 31 (2010), no. 7-9, 852-869], we showed two separate data sets can be optimally approximated using balanced proper orthogonal decomposition (POD) modes derived from the data. In this work, we prove new results concerning the approximation capability of the balanced POD modes. We give exact computable expressions for the errors between the individual data sets and the low order balanced POD data reconstructions. We also consider approximating elements of the Hilbert space using various projections onto the balanced POD modes. We discuss the relevance of these results to balanced POD model reduction of nonlinear partial differential equations. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Mathematical Analysis and Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, [VOL#421, ISSUE#2, (15 January 2015)] http://dx.doi.org/10.1016/j.jmaa.2014.07.059