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Research Papers

Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains

[+] Author and Article Information
Mete Yurtoglu

747 6th Street South,
Kirkland, WA 98033
e-mail: myurtoglu@me.com

Molly Carton

Mechanical Engineering,
University of Washington,
Seattle, WA 98195-2600
e-mail: mcarton@uw.edu

Duane Storti

Mechanical Engineering,
University of Washington,
P.O. Box: 352600,
Seattle, WA 98195-2600
e-mail: storti@uw.edu

Manuscript received September 8, 2017; final manuscript received March 9, 2018; published online April 26, 2018. Assoc. Editor: Yong Chen.

J. Comput. Inf. Sci. Eng 18(2), 021013 (Apr 26, 2018) (9 pages) Paper No: JCISE-17-1179; doi: 10.1115/1.4039639 History: Received September 08, 2017; Revised March 09, 2018

Abstract

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).

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Figures

Fig. 1

Data from convergence study for grid-based evaluation of surface area of a torus. (a) Raw data and best fit log–log plots of relative error versus grid refinement are shown for genus 1 and 2. The genus 2 stencil (with radius 2) produces a significant reduction in relative error across the full range of refinement. (b) Best-fit lines for log–log plots of relative error versus refinement for genus 1–7. Increasing genus beyond G = 2 does not consistently provide decreases in relative error to make up for increased computational cost associated longer stencils.

Fig. 2

Results of convergence study for genus 1 grid-based evaluation of moment of inertia of a toroidal shell

Fig. 3

Results of convergence study for genus 1 grid-based evaluation of the volume of a solid torus. Log–log plot of relative error versus grid refinement shows raw data and best-fit line.

Fig. 4

Results of convergence study for genus 1 grid-based evaluation of the moment of inertia of a solid torus. Log–log plot of relative error versus grid refinement shows raw data and best-fit line.

Fig. 5

Results of convergence study for genus 1 grid-based evaluation of the path length of a sphere-sphere intersection curve. Log–log plot of relative error versus grid refinement shows raw data and best-fit line.

Fig. 6

Visualization of stoichiometric surface implicitly defined by a 512 × 512 grid of direct numerical simulation data. Area and area density were successfully computed using the grid-based surface integral method [30].

Fig. 7

Computed volume/actual volume versus number of points for a the calculation of the volume of a torus, comparing a Monte Carlo method with direct integration. Direct integration results are within ±0.0005 in under 1 million points, where some Monte Carlo runs remain outside that range at 7 million points. (a) Volume error versus number of points and (b) detail of (a).

Fig. 8

Timing comparison of calculation of torus volume with wavelet genus 1 in number of points evaluated versus execution time (ms). Indicated are timings for direct integration on the CPU and GPU (including full execution time and kernel time) as well as Monte Carlo timing.

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