A Parallel Two-Scale Method for Eikonal Equations

Adam Chacon, Alexander Vladimirsky
2015 SIAM Journal on Scientific Computing  
Numerous applications of Eikonal equations prompted the development of many efficient numerical algorithms. The Heap-Cell Method (HCM) is a recent serial two-scale technique that has been shown to have advantages over other serial state-of-the-art solvers for a wide range of problems [A. Chacon and A. Vladimirsky, SIAM J. Sci. Comput., 34 (2012), pp. A547-A578]. This paper presents a parallelization of HCM for a shared memory architecture. The numerical experiments in R 3 show that the parallel
more » ... w that the parallel HCM exhibits good algorithmic behavior and scales well, resulting in a very fast and practical solver. Introduction. The Eikonal equation is a nonlinear first-order static PDE used in a range of applications, including robotic navigation, wavefront propagation, seismic imaging, optimal control, and shape-from-shading calculations. The computational efficiency on a fixed grid is an important practical consideration in many of these applications. Several competing "fast" serial algorithms have been introduced in the last two decades to solve the grid-discretized Eikonal equation. The two most widely used are the Fast Marching Method (FMM) and the Fast Sweeping Method (FSM). The Heap-Cell Method (HCM), introduced in the authors' previous work [8] , is a two-scale technique based on combining the ideas of FMM and FSM. The current paper focuses on the parallelization of HCM for a shared memory architecture. We will start by briefly describing the relevant discretization of the Eikonal PDE (section 1.1) and the prior algorithms for solving it (sections 2 and 3). HCM is reviewed in section 4, and the new parallel HCM (pHCM) is described in detail in section 5. The numerical experiments in section 6 demonstrate that pHCM is efficient and achieves good parallel scalability for a wide range of grid resolutions and domain decomposition parameters. Additional experimental results are included in an extended version of this paper [10] . We conclude in section 7 by discussing the limitations of our approach and the directions of future work. Eikonal PDE and its upwind discretization. An important subclass of Hamilton-Jacobi equations is formed by static Eikonal PDEs: Classical (smooth) solutions of equation (1) generally do not exist, and weak solutions are not unique [2] . However, existence and uniqueness can be shown for the viscosity *
doi:10.1137/12088197x fatcat:nnnql6dsgbbzxhla6gafzrhfhq