Treat All Integrals as Volume Integrals: A uni?ed, parallel, grid-based method for evaluation of volume, surface, and path integrals on implicitly de?ned domains

[+] Author and Article Information
Mete Yurtoglu

Google, 747 6th St S., Kirkland, WA

Molly Carton

Student Mem. ASME, University of Washington - Seattle, Mechanical Engineering, Seattle, WA 98195-2600

Duane W. Storti

University of Washington - Seattle, Mechanical Engineering, Box 352600, Seattle, WA 98195-2600

1Corresponding author.

ASME doi:10.1115/1.4039639 History: Received September 08, 2017; Revised March 09, 2018


Wepresentauni?edmethodfornumericalevaluationofvolume,surface,andpathintegralsofsmooth,bounded functions on implicitly de?ned bounded domains. The method avoids both the stochastic nature (and slow convergence) of Monte Carlo methods and problem-speci?c domain decompositions required by most traditional numericalintegrationtechniques. Ourapproachoperatesonauniformgridoveranaxis-alignedboxcontainingtheregion ofinterest,sowerefertoitasagrid-basedmethod. Allgrid-basedintegralsarecomputedasasumofcontributions fromastencilcomputationonthegridpoints. Eachclassofintegrals(path,surface,orvolume)involvesadifferent 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 de?ned over the continuous domain so that grid re?nement 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 numericalsimulation(DNS)of?uid?ow,oranyothermethodthatproducesdatacorrespondingtovaluesofafunctionsampledonaregulargrid. Everystepofagrid-basedintegralcomputation(includingevaluatingafunctionon a grid, application of stencils on a grid, and reduction of the contributions from the grid points to a single sum) is well-suitedforparallelization. WepresentresultsfromaparallelizedCUDAimplementationofgrid-basedintegrals that faithfully reproduces the output of a serial implementation but with signi?cant reductions in computing time. We also present example grid-based integral results to quantify convergence rates associated with grid re?nement and dependence of the convergence rate on the speci?c choice of difference stencil (corresponding to a particular genus of Daubechies wavelet).

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