An O(N) level set method for eikonal equations

A propagating interface can develop corners and discontinuities as it advances. Level set algorithms have been extensively applied for the problems in which the solution has advancing fronts. One of the most popular level set algorithms is the so-called fast marching method (FMM), which requires total O(N log(2)N) operations, where N is the number of grid points. The article is concerned with the development of an O(N) level set algorithm called the group marching method (GMM). The new method is based on the narrow band approach as in the FMM. However, it is incorporating a correction-by-iteration strategy to advance a group of grid points at a time, rather than sorting the solution in the narrow band to march forward a single grid point. After selecting a group of grid points appropriately, the GMM advances the group in tw iterations for the cost of slightly larger than one iteration. Numerical results are presented to show the efficiency of the method, applied to the eikonal equation in tw and three dimensions.