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15th International Conference on High Performance Computing & Simulation (HPCS)

作者： Ganellari, Daniel Haase, Gundolf Karl Franzens Univ Graz Inst Math & Sci Comp Graz Austria

ISBN: (纸本)9781538632505

The numerical solution of the Eikonal equation follows the fast iterative method with its application for tetrahedral meshes. Therein the main operations in each discretization element T contain various inner products in the M-metric as <(e) over right arrow (k,s), (e) over right arrow (s,l)> M-T equivalent to (e) over right arrow (T)(k,s) center dot M-T center dot (e) over right arrow (s,l) with (e) over right arrow (s,l) as connecting edge between vertices s and l in element T. Instead of passing all coordinates of the tetrahedron together with the 6 entries of M-T we precompute these inner products and use only them in the wave front computation. This first change requires less memory transfers for each tetrahedron. The second change is caused by the fact that <(e)over right arrow>(k,s), (e) over right arrow (s,l)> M-T (k not equal l) represents an angle of a surface triangle whereas <(e)over right arrow>(k,s), (e)over right arrow>(k,s)> M-T represents the length of an edge in the M-metric. Basic geometry as well as vector arithmetics yield to the conclusion that the angle information can be expressed by the combination of three edge lengths. Therefore we only have to precompute the 6 edge lengths of a tetrahedron and compute the remaining 12 angle data on-the-fly which reduces the memory footprint per tetrahedron to 6 numbers. The efficient implementation of the two changes requires a local Gray-code numbering of edges in the tetrahedron and a bunch of bit shifts to assign the appropriate data. First numerical experiments on CPUs show that the reduced memory footprint approach is faster than the original implementation. Detailed investigations as well as a CUDA implementation are ongoing work.

关键词： Eikonal equation tetrahedral mesh Gray-code reduced memory footprint parallel algorithm shared memory multiple-processor computer system CUDA GPU

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

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2013年第5期35卷 C473-C494页

作者： Fu, Zhisong Kirby, Robert M. Whitaker, Ross T. Univ Utah Sci Comp & Imaging Inst Salt Lake City UT 84112 USA

Generating numerical solutions to the eikonal equation and its many variations has a broad range of applications in both the natural and computational sciences. Efficient solvers on cutting-edge, parallel architectures require new algorithms that may not be theoretically optimal, but that are designed to allow asynchronous solution updates and have limited memory access patterns. This paper presents a parallel algorithm for solving the eikonal equation on fully unstructured tetrahedral meshes. The method is appropriate for the type of fine-grained parallelism found on modern massively-SIMD architectures such as graphics processors and takes into account the particular constraints and capabilities of these computing platforms. This work builds on previous work for solving these equations on triangle meshes;in this paper we adapt and extend previous two-dimensional strategies to accommodate three-dimensional, unstructured, tetrahedralized domains. These new developments include a local update strategy with data compaction for tetrahedral meshes that provides solutions on both serial and parallel architectures, with a generalization to inhomogeneous, anisotropic speed functions. We also propose two new update schemes, specialized to mitigate the natural data increase observed when moving to three dimensions, and the data structures necessary for efficiently mapping data to parallel SIMD processors in a way that maintains computational density. Finally, we present descriptions of the implementations for a single CPU, as well as multicore CPUs with shared memory and SIMD architectures, with comparative results against state-of-the-art eikonal solvers.

关键词： Hamilton-Jacobi equation eikonal equation tetrahedral mesh parallel algorithm shared memory multiple-processor computer system graphics processing unit

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A FAST ITERATIVE METHOD FOR SOLVING THE EIKONAL EQUATION ON TRIANGULATED SURFACES

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2011年第5期33卷 2468-2488页

作者： Fu, Zhisong Jeong, Won-Ki Pan, Yongsheng Kirby, Robert M. Whitaker, Ross T. Univ Utah Sci Comp & Imaging Inst Salt Lake City UT 84112 USA UNIST Ulsan 689798 South Korea

This paper presents an efficient, fine-grained parallel algorithm for solving the Eikonal equation on triangular meshes. The Eikonal equation, and the broader class of Hamilton-Jacobi equations to which it belongs, have a wide range of applications from geometric optics and seismology to biological modeling and analysis of geometry and images. The ability to solve such equations accurately and efficiently provides new capabilities for exploring and visualizing parameter spaces and for solving inverse problems that rely on such equations in the forward model. Efficient solvers on state-of-the-art, parallel architectures require new algorithms that are not, in many cases, optimal, but are better suited to synchronous updates of the solution. In previous work [W. K. Jeong and R. T. Whitaker, SIAM J. Sci. Comput., 30 (2008), pp. 2512-2534], the authors proposed the fast iterative method (FIM) to efficiently solve the Eikonal equation on regular grids. In this paper we extend the fast iterative method to solve Eikonal equations efficiently on triangulated domains on the CPU and on parallel architectures, including graphics processors. We propose a new local update scheme that provides solutions of first-order accuracy for both architectures. We also propose a novel triangle-based update scheme and its corresponding data structure for efficient irregular data mapping to parallel single-instruction multiple-data (SIMD) processors. We provide detailed descriptions of the implementations on a single CPU, a multicore CPU with shared memory, and SIMD architectures with comparative results against state-of-the-art Eikonal solvers.

关键词： Hamilton-Jacobi equation Eikonal equation triangular mesh parallel algorithm shared memory multiple-processor computer system graphics processing unit

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