On the connection between balanced proper orthogonal decomposition, balanced truncation, and metric complexity theory for infinite dimensional systems

In this paper, the connection between two important model reduction techniques, namely balanced proper orthogonal decomposition (POD) and balanced truncation is investigated for infinite dimensional systems. In particular, balanced POD is shown to be optimal in the sense of distance minimization in a space of integral operators under the Hilbert-Schmidt norm. Whereas balanced truncation is shown to be a particular case of balanced POD for infinite dimensional systems for which the impulse response satisfies certain finite energy constraints. POD and balanced truncation are related to certain notions of metric complexity theory. In particular both are shown to minimize different n-widths of partial differential equation solutions including the Kolmogorov, Gelfand, linear and Bernstein n-widths. The n-widths quantify inherent and representation errors due to lack of data and loss of information.