Riemannian geometry provides a vast toolbox for the numerical treatment of nonlinear optimization problems. Alas, some infinite dimensional problems, e.g., those related to curvature energies of immersed submanifolds (such as curves and surfaces), can hardly be put into a Riemannian context. Still, many techniques from Riemannian optimization perform very well even in a Banach manifold setting—at least experimentally.
In this talk, we demonstrate the efficiency of the (projected) gradient method for the mini- mization of several curvature energies (Euler-Bernoulli, Willmore, and O’Hara energy). Instead of L2-bilinear forms (which lead to parabolic flows), we utilize H^s-bilinear forms with s > 0 to define gradients. This leads to an ordinary differential equation on the space of immersions which can be solved by standard methods. In case of constraints, we also provide conditions (constraint qualifications) for the existence of the projected gradient vector field.