[Paper Review] A parallel Heap-Cell Method for Eikonal equations

Adam Chacon, Alexander Vladimirsky|arXiv (Cornell University)|Jun 20, 2013

Advanced Numerical Methods in Computational Mathematics24 references4 citations

TL;DR

This paper presents a parallel Heap-Cell Method (pHCM) for solving Eikonal equations on shared memory architectures, combining the efficiency of the serial Heap-Cell Method (HCM) with parallelization across cells. pHCM achieves high scalability and speedup, particularly on problems with high work per cell, outperforming serial solvers and parallelized Fast Sweeping Methods in numerical experiments on 3D grids.

ABSTRACT

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. This paper presents a parallelization of HCM for a shared memory architecture. The numerical experiments in $R^3$ show that the parallel HCM exhibits good algorithmic behavior and scales well, resulting in a very fast and practical solver. We further explore the influence on performance and scaling of data precision, early termination criteria, and the hardware architecture. A shorter version of this manuscript (omitting these more detailed tests) has been submitted to SIAM Journal on Scientific Computing in 2012.

Motivation & Objective

• To develop a scalable parallel algorithm for solving Eikonal equations on shared memory architectures.
• To improve upon the performance of serial solvers like HCM and parallelized Fast Sweeping Methods (FSM/LSM) for large-scale 3D problems.
• To investigate the impact of data precision, early termination, and hardware architecture on pHCM's performance and scaling.
• To enable efficient solution of Eikonal equations in applications requiring high computational throughput, such as robotic navigation and seismic imaging.
• To explore the feasibility of extending pHCM to anisotropic Hamilton-Jacobi equations and hybrid CPU-GPU architectures.

Proposed method

• pHCM partitions the computational domain into spatial cells and processes each cell independently using a heap-based label-setting strategy, similar to HCM.
• Each thread manages a subset of cells, performing sequential sweeping within each cell while maintaining a shared global heap for active nodes.
• The method uses a two-scale approach: local sweeping within cells and global propagation via a heap data structure to ensure causality and convergence.
• A novel cell-value heuristic (equation 4) is introduced to reduce redundant work, especially in problems with misaligned discontinuities in the speed function.
• The algorithm supports early termination within cells based on convergence criteria, reducing unnecessary sweeps.
• Memory access is optimized through spatial and temporal locality heuristics, and the method is designed for shared memory systems with cache-aware access patterns.

Experimental results

Research questions

• RQ1 How does pHCM scale across multiple cores on shared memory systems for 3D Eikonal problems?
• RQ2 What is the impact of data precision (e.g., 32-bit vs. 64-bit) on pHCM’s performance and accuracy?
• RQ3 How do early termination criteria affect the total work and runtime of pHCM?
• RQ4 What cell decomposition strategy (e.g., cell size, number of cells) optimizes performance for a given problem size and hardware?
• RQ5 Can pHCM outperform existing parallel Eikonal solvers, such as parallelized FSM/LSM, in terms of speedup and work efficiency?

Key findings

• pHCM achieves strong scalability on 3D grids up to 64³, with speedup increasing with problem size and work per cell.
• For M=64³, pHCM32 using the new cell heuristic (eq. 4) reduced average sweeps per cell from 8366 (with heuristic 3) to 20.4, demonstrating significant work reduction.
• pHCM32 outperformed HCM32 in terms of work efficiency (AvS), though raw time increased slightly due to overhead, indicating better algorithmic behavior.
• The new cell heuristic (eq. 4) improved scaling for larger cells, especially in problems with non-aligned discontinuities in the speed function.
• Performance gains were most pronounced when cell sizes were large or sweeping within cells required multiple iterations, indicating high work-per-cell scenarios benefit most.
• The method shows promise for extension to anisotropic Hamilton-Jacobi equations and hybrid CPU-GPU architectures, though distributed memory scaling remains challenging.