Efficient level set methods for constructing wavefronts in three spatial dimensions

Wavefront construction in geometrical optics has long faced the twin difficulties of dealing with multi-valued forms and resolution of wavefront surfaces. A recent change in viewpoint, however, has demonstrated that working in phase space on bicharacteristic strips using eulerian methods can bypass both difficulties. The level set method for interface dynamics makes a suitable choice for the eulerian method. Unfortunately, in three-dimensional space, the setting of interest for most practical applications, the advantages of this method are largely offset by a new problem: the high dimension of phase space. In this work, we present new types of level set algorithms that remove this obstacle and demonstrate their abilities to accurately construct wavefronts under high resolution. These results propel the level set method forward significantly as a competitive approach in geometrical optics under realistic conditions.