Nicola Guglielmi, Università dell'Aquila and Gran Sasso Science Institute (GSSI), Italy
Low-rank ODEs for matrix nearness problems.
The topic of this talk is a methodology based on differential equations on low-rank matrix manifolds for the efficient solution of structured matrix nearness problems such as the distance to instability or to passivity of a matrix with a certain structure, or the distance of a given connected weighted graph to a nearest non-connected weighted graph.
In all cases we obtain a characterization of extremal perturbations, which turn out to be of low rank and are attractive stationary points of the low-rank differential equations that we derive.
We use a two-level approach; in the inner level we determine extremizers over the set of perturbations of a given norm, say epsilon, by following the low-rank differential equations up to a stationary point, and in the outer level we optimize with respect to epsilon.
This permits us to obtain fast algorithms - exhibiting quadratic convergence - for solving the considered matrix nearness problems.
The talk is inspired by recent and current works with Paolo Butta', Daniel Kressner, Christian Lubich, Manuela Manetta, Wim Michiels, Silvia Noschese, and Michael Overton.