Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains
| Autor: | Molly Carton, Mete Yurtoglu, Duane W. Storti |
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| Rok vydání: | 2017 |
| Předmět: |
0209 industrial biotechnology
Finite volume method Stencil code Mathematical analysis Trigonometric integral 020207 software engineering 02 engineering and technology Grid Computer Graphics and Computer-Aided Design Stencil Research Papers Industrial and Manufacturing Engineering Computer Science Applications Regular grid Volume integral Order of integration (calculus) 020901 industrial engineering & automation 0202 electrical engineering electronic engineering information engineering Applied mathematics Software Computer Science::Distributed Parallel and Cluster Computing Mathematics |
| Zdroj: | Journal of computing and information science in engineering. 18(2) |
| ISSN: | 1530-9827 |
| Popis: | We present a unified method for numerical evaluation of volume, surface, and path integrals of smooth, bounded functions on implicitly defined bounded domains. The method avoids both the stochastic nature (and slow convergence) of Monte Carlo methods and problem-specific domain decompositions required by most traditional numerical integration techniques. Our approach operates on a uniform grid over an axis-aligned box containing the region of interest, so we refer to it as a grid-based method. All grid-based integrals are computed as a sum of contributions from a stencil computation on the grid points. Each class of integrals (path, surface, or volume) involves a different stencil formulation, but grid-based integrals of a given class can be evaluated by applying the same stencil on the same set of grid points; only the data on the grid points changes. When functions are defined over the continuous domain so that grid refinement is possible, grid-based integration is supported by a convergence proof based on wavelet analysis. Given the foundation of function values on a uniform grid, grid-based integration methods apply directly to data produced by volumetric imaging (including computed tomography and magnetic resonance), direct numerical simulation of fluid flow, or any other method that produces data corresponding to values of a function sampled on a regular grid. Every step of a grid-based integral computation (including evaluating a function on a grid, application of stencils on a grid, and reduction of the contributions from the grid points to a single sum) is well suited for parallelization. We present results from a parallelized CUDA implementation of grid-based integrals that faithfully reproduces the output of a serial implementation but with significant reductions in computing time. We also present example grid-based integral results to quantify convergence rates associated with grid refinement and dependence of the convergence rate on the specific choice of difference stencil (corresponding to a particular genus of Daubechies wavelet). |
| Databáze: | OpenAIRE |
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