Combined Reduction

Real-life models of inverse problems often have high-dimensional state and parameter spaces.
For example, a network with many nodes and unknown connectivity. The underlying large-scale systems impede the simulation and parameter estimation of such models. Unlike in typical model order reduction scenarios, not only the state space dimension but also the parameter space dimension poses a source for computational cost. Combined state and parameter reduction tackles this obstacle, of which two pathways with distinct scopes of application are pursued. First, a gramian-based method, that employs balanced or direct truncation of extended empirical gramians. Due to the flexibility of this ansatz, also the combined reduction of nonlinear models is enabled. Second, an optimization-based method, which iteratively assembles a reduced basis using greedy optimization. Recent additions to this approach allow the reduction of Bayesian inverse problems even on extreme scales. Fields of application of both methods are presented along with implementations and numerical experiments.