CAIMS Prize Award

Prize Award year: 
1996
Prize Winner: 
citation: 

Institute of Applied Mathematics, University of British Columbia

Supervisors:  U. Ascher and B. Wetton  

Thesis title:  Efficient algorithms for diffusion-generated motion by mean curvature

Abstract:

This thesis considers the problems of simulating the motion of evolving surfaces with a normal velocity equal to mean curvature plus a constant. Such motions arise in a variety of applications. A general method for this purpose was proposed by Merriman, Bence and Osher, and consists of alternately diffusing and sharpening the front in a certain manner. This method (referred to as the MBO-method) naturally handles complicated topological changes with junctions in several dimensions. However, the usual finite difference discretization of the method is often exceedingly slow when accurate results are sought, especially in three spatial dimensions.

We propose a new, spectral discretization of the MBO-method which obtains greatly improved efficiency over the usual finite difference approach. These efficiency gains are obtained, in part, through the use of a quadrature-based refinement technique, by integrating Fourier modes exactly, and by neglecting the contribution of rapidly decaying solution transients. The resulting method provides a practical tool, not available hitherto, for accurately treating the motion by mean curvature of complicated surfaces with junctions. Indeed, we present numerical studies which demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization.

New analytic and experimental results are also developed to explain important properties of the MBO-method such as the order of the approximation error. Extrapolated algorithms, not possible when using the usual finite difference discretization, are proposed and demonstrated to achieve more accurate results.

We apply our new, spectral method to simulate the motion of a number of three dimensional surfaces with junctions, and we visualize the results. We also propose and study a simple extension of our method to a nonlocal curvature model which is impractical to treat using the previously available finite difference discretization.