Seminar Schedule
Professor Carsten Hartmann
Structure-preserving model reduction of partially-observed differential equations
We study model reduction of dissipatively perturbed linear Hamiltonian systems. Large-scale systems of this kind arise in a variety of physical contexts, e.g., in molecular dynamics or structural mechanics. Common spatial decomposition methods such as Proper Orthogonal Decomposition aim at identifying a subspace of ``high-energy modes onto which the dynamics are projected. These modes, however, may not be relevant for the dynamics. Moreover the methods tacitly assume that all degrees of freedom can actually be observed or measured. Here we adopt ideas from control theory and ask to which extend certain states are "controllable" by some external perturbation and give observable output. Model reduction then consists in (1) transforming the system such that those degrees of freedom that are least sensitive to the perturbation also give the least output and (2) neglecting the respective unobservable/uncontrollable modes. We explain how this idea which is known by the name of Balanced Truncation can be extended to stochastic differential equations (e.g. second-order Langevin processes) and discuss aspects of structure-preservation (e.g., when the equations admit a Hamiltonian structure).