On the Loewner Framework for Model Reduction of Burgers’ Equation

This paper addresses issues that originate in the extension of the Loewner framework to compute reduced order models (ROMs) of so-called quadratic-bilinear systems. The latter arise in semi-discretizations of fluid flow problems, such as Burgers’ equation or the Navier-Stokes equations. In the linear case, the Loewner framework is data-driven and constructs a ROM from measurements of the transfer function; it does not explicitly require access to the system matrices, which is attractive in many settings. Research on extending the Loewner framework to quadratic-bilinear systems is ongoing. This paper presents one extension and provides details of its implementation that allow application to large-scale problems. This extension is applied to Burgers’ equation. Numerical results show the potential of the Loewner framework, but also expose additional issues that need to be addressed to make it fully applicable. Possible approaches to deal with some of these issues are outlined.