A generalized POD space-time Galerkin scheme for parameter dependent dynamical systems

In a recent work [Baumann et al. 'Discrete Input/Output Maps and their Relation to Proper Orthogonal Decomposition' (2015)], a generalized version of Proper Orthogonal Decomposition (POD) has been formulated based on generalized measurements replacing the snapshot matrix of the standard POD. We extend this approach in two directions. Firstly, we add a parameter dependence as a third dimension of the measurement matrix. Secondly, we use a higher-order SVD to obtain optimized low-dimensional bases for both the space and the time discretization. Then, applying a space-time Galerkin scheme, a time-dependent PDE can be transformed into a small system of algebraic equations -- the reduced model. We illustrate the properties and benefits of this approach for an example of nonlinear Burgers equation with varying diffusion parameters.