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A FAST ITERATIVE METHOD FOR SOLVING THE EIKONAL EQUATION ON TETRAHEDRAL DOMAINS.

Zhisong Fu1, Robert M Kirby1, Ross T Whitaker1

  • 1The Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112.

SIAM Journal on Scientific Computing : a Publication of the Society for Industrial and Applied Mathematics
|September 16, 2014
PubMed
Summary
This summary is machine-generated.

This study introduces a novel parallel algorithm for solving the eikonal equation on 3D tetrahedral meshes, optimizing performance for modern parallel architectures like GPUs. The method enhances computational density and efficiency for scientific computing applications.

Keywords:
Hamilton–Jacobi equationeikonal equationgraphics processing unitparallel algorithmshared memory multiple-processor computer systemtetrahedral mesh

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Area of Science:

  • Computational Science
  • Applied Mathematics
  • Scientific Computing

Background:

  • The eikonal equation is crucial for numerous applications in natural and computational sciences.
  • Existing solvers often struggle with the demands of modern parallel architectures, particularly regarding asynchronous updates and memory access patterns.
  • Adapting 2D methods to 3D unstructured tetrahedral meshes presents significant algorithmic and data structure challenges.

Purpose of the Study:

  • To develop a parallel algorithm for solving the eikonal equation on unstructured tetrahedral meshes suitable for massively-SIMD architectures.
  • To extend previous 2D strategies to efficiently handle 3D domains, addressing data increase and maintaining computational density.
  • To provide a generalized solution for inhomogeneous and anisotropic speed functions on both serial and parallel platforms.

Main Methods:

  • A parallel algorithm is presented for solving the eikonal equation on fully unstructured tetrahedral meshes.
  • The method incorporates a local update strategy with data compaction, adapted from 2D triangle mesh solvers.
  • Two new update schemes and specialized data structures are proposed to manage 3D data complexity and optimize for parallel SIMD processors.

Main Results:

  • The algorithm demonstrates efficient solutions on both serial and parallel architectures, including multicore CPUs and SIMD platforms.
  • New strategies effectively mitigate data increase in 3D and maintain high computational density.
  • Implementations are described for various architectures, with comparative results against state-of-the-art solvers.

Conclusions:

  • The developed parallel algorithm offers an efficient approach to solving the eikonal equation on 3D unstructured tetrahedral meshes for modern parallel computing platforms.
  • The method's adaptability to inhomogeneous, anisotropic speed functions and its performance on various architectures make it a valuable tool for scientific applications.
  • This work advances the state-of-the-art in numerical solutions for the eikonal equation on complex geometries and parallel hardware.