18. Dezember 2019
Henrik Schumacher (RWTH Aachen)
Gradient Flows for the Möbius Energy
Abstract:
Aiming at optimizing the shape of closed embedded curves within prescribed
isotopy classes, we use a gradient-based approach to approximate
stationary points of the Möbius energy. The gradients are computed with
respect to certain fractional-order Sobolev scalar products that are
adapted to the Möbius energy. In contrast to $L^2$-gradient flows, the
resulting flows are ordinary differential equations on an
infinite-dimensional manifold of embedded curves. In the fully discrete
setting, this allows us to completely decouple the time step size from the
spatial discretization, resulting in a very robust optimization algorithm
that is orders of magnitude faster than following the discrete
$L^2$-gradient flow.