Optimality of balanced proper orthogonal decomposition for data reconstruction II: Further approximation results

In our earlier paper Singler (2010) [7], 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.