Christian Grussler

A symmetry approach for balanced truncation of positive linear systems

We consider model order reduction of positive linear systems and show how a symmetry characterization can be used in order to preserve positivity in balanced truncation. The reduced model has the additional feature of being symmetric.

On optimal low-rank approximation of non-negative matrices

For low-rank Frobenius-norm approximations of matrices with non-negative entries, it is shown that the Lagrange dual is computable by semi-definite programming. Under certain assumptions the duality gap is zero. Even when the duality gap is non-zero, several new insights are provided.

Modified balanced truncation preserving ellipsoidal cone-invariance

We consider model order reduction of stable linear systems which leave ellipsoidal cones invariant. We show how balanced truncation can be modified to preserve cone-invariance. Additionally, this implies a method to perform external positivity preserving model reduction for a large class of systems.

The use of the r* heuristic in covariance completion problems

We consider a class of structured covariance completion problems which aim to complete partially known sample statistics in a way that is consistent with the underlying linear dynamics. The statistics of stochastic inputs are unknown and sought to explain the given correlations. Such inverse problems admit many solutions for the forcing correlations, but can be interpreted as an optimal low-rank approximation problem for identifying forcing models of low complexity. On the other hand, the quality of completion can be improved by utilizing information regarding the magnitude of unknown entries. We generalize theoretical results regarding the r* norm approximation and demonstrate the performance of this heuristic in completing partially available statistics using stochastically-driven linear models.

Local convergence of proximal splitting methods for rank constrained problems

We analyze the local convergence of proximal splitting algorithms to solve optimization problems that are convex besides a rank constraint. For this, we show conditions under which the proximal operator of a function involving the rank constraint is locally identical to the proximal operator of its convex envelope, hence implying local convergence. The conditions imply that the non-convex algorithms locally converge to a solution whenever a convex relaxation involving the convex envelope can be expected to solve the non-convex problem.