Parametric representation and sensitivity analysis of proper-orthogonal-decomposition modes on matrix manifolds

We propose a framework for the parametric analysis of proper-orthogonal-decomposition (POD) modes, which form a locally optimal orthonormal basis at a specific flow parameter (e.g., Reynolds number). It is well known that POD modes vary depending on the flow parameters and control inputs. Our approach based on matrix manifolds represents the set of POD modes for different parameters, or a set of the subspaces they span. This enables the modeling of the parameter dependence of POD modes and allows for a quantitative analysis of their sensitivity to parameter variations. As a result, it provides a strategy for constructing parametric reduced-order models (ROMs) and offers potential applications to ROMs that account for uncertainties in flow parameters and control inputs. In this study, we focus on evaluating the sensitivity of POD modes and the subspaces they span with respect to variations in the Reynolds number, and on developing a parametric ROM for flow field around a circular cylinder.