Optimization on manifolds

Many of the mathematical problems emerging from science and engineering involve matrix manipulations. It is quite common that applications are linked to particular matrix structures. Preserving this structure in the numerical solution is of crucial importance, since together with the structure also the physical interpretation might get lost.

Traditionally, numerical linear algebra attempts to preserve the structure by projection and data restriction methods. These methods can result in the loss of accuracy and valuable computing time. Pure mathematics furnishes us elegant differential geometric tools to replace these projection techniques. Several classes of matrices have a natural geometric manifold structure, so that one can apply ``optimization on matrix manifolds'' techniques to effectively find the solutions without losing the structure.

The novelty of this promising and challenging research domain implies, unfortunately, that many important matrix structures have not been studied thoroughly yet. For many classes of matrices that are common in applications, in particular matrices with low displacement structure, the associated geometric spaces have singular points, and the existing optimization techniques can not be applied directly.

We plan to solve this problem using Hironaka's resolution of singularities, a powerful tool coming from algebraic and analytic geometry. This requires an intimate collaboration between pure and applied mathematics.