We also discuss work in progress with N. Le (Columbia) and Peletier which exploits the Gamma-limit structure of the energy
to prove convergence of the associated gradient flows. In particular, we connect to the well-known LSW theory for Ostwald ripening.
Joint work with Almut Burchard and Benjamin Stephens
This is a joint work with S. Keraani (University of Lille 1, France)
In general these equations are considered too difficult to solve, which is why linearized models or other approximations are commonly used. Progress has recently been made in building solvers for a class of Geometric PDEs. These solvers naturally give better geometric results and, in some cases, are competitive in terms of cost with the simplified models.
In this talk I'll give examples of a few important geometric PDEs which can be solved using a numerical method called monotone finite difference schemes: Monge-Ampere, Convex Envelope, Infinity Laplace, and Mean Curvature.
These methods have been implemented for registration of Brain Images. For Surface Registration, the Infinity Laplace equation is used to match surfaces using geodesic lengths [Sapiro]. For Volume Registration, the Monge-Ampere equation is used to minimize distortion of volumes [Tannenbaum-Haker-Haber]. Convergent numerical schemes are important in these applications: bad discretizations lead to artificial singularities in the mappings.
Focussing in on the Monge-Ampere equation, I'll show how naive schemes can work well for smooth solutions, but break down in the singular case. This makes having a convergent scheme even more important. I'll present a convergent solver which which is fast: comparable to solving the Laplace equation a few times.