A parallel multiscale FIM approach in solving the Eikonal equation on GPU

Abstract

Signed Distance Fields (SDFs) are essential in various applications, particularly in level set problems, where computing the SDF is equivalent to solving the Eikonal equation. Common approaches to solving these equations include the Fast Marching Method (FMM), the Fast Sweeping Method (FSM), and the Fast Iterative Method (FIM). However, FMM and FSM face significant challenges in parallelization, increasing interest in developing FIM for GPU architectures. In this paper, we extend the innovative FIM algorithm (Huang, 2021), which is GPU-friendly but relies on a single uniform grid, by incorporating multiscale techniques to accelerate wavefront propagation from source points to infinity. Unlike the traditional Fast Iterative Method, which operates on a single uniform grid and propagates the wavefront at a constant speed of one grid spacing per iteration, our multiscale approach applies a hierarchy of varying propagation speeds to accelerate the convergence. Once all source and infinite points are properly initialized, only a few FIM iterations are required to refine the values of points near the source. A coarser-grained scale, with twice the spacing of the finer grid, is then used to propagate values from accepted and tentative points to the outer regions. This process is repeated until the top-level scale is reached. Subsequently, we reverse this process by performing FIM calculations from the coarsest scale until reaching the finest grid, thereby completing a V-cycle. With multiscale V-cycles, the solution progressively converges across the entire computational domain. Comparative experimental results show that our algorithm improves computational efficiency by approximately 128% over the GPU-based Fast Marching Method and by a factor of 23 compared to the improved FIM algorithm (Huang, 2021) at scale of $N=2E8$. This optimized approach applies to numerical simulations of multi-body systems, including fluid–structure interactions, as well as numerical analyses of flooding and earthquake scenarios.