Abstract
Stochastic geometry has applications in areas such as robotics, tomographic reconstruction with uncertainties, spatial Poisson-Vonoroi tessellations, and imaging from medical data using tetrahedral meshes. We examine numerical approaches for the computation of multivariate integrals for a family of problems where uniformly distributed points are picked as polyhedron vertices for tessellations, for example, tetrahedron vertices in a cube, tetrahedron or on a spherical surface. The classical cube tetrahedron picking problem yields the expected volume of a random tetrahedron in a cube, and helps furthermore assessing unsolved extremal problems (cf., A. Zinani, 2003). We demonstrate feasible numerical approaches including adaptive integration through region partitioning, quasi-Monte Carlo (based on a randomized Korobov lattice), and Monte Carlo techniques, which are the basic methods of our parallel integration package ParInt. We then describe our implementation of the Monte Carlo approach on GPUs (Graphics Processing Units) in CUDA C, and demonstrate its parallel performance for various stochastic geometry integrals.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Zinani, A.: The expected volume of a tetrahedron whose vertices are chosen at random in the interior of a cube. Monatsh. Math. 139, 341–348 (2003), doi:10.1007/s00605-002-0531-y
Philip, J.: The expected volume of a random tetrahedron in a cube (2007), http://www.math.kth.se/~johanph/ETC.pdf
Heinrich, L., Körner, Mehlhorn, N., Muche, L.: Numerical and analytical computation of some second-order characteristics of spacial poisson-vonoroi tesselations. Statistics 31, 235–259 (1998)
Pach, J., Sharir, M.: Combinatorial Geometry and Its Algorithmic Applications: The Alcala Lectures. Amer. Mathematical Society (2009) ISBN-10: 0821846914, ISBN-13: 978-0821846919
Dardenne, J., Siauve, N., Valette, S., Prost, R., Burais, N.: Impact of tetrahedral mesh quality for electromagnetic and thermal simulations. In: compumag, Florianopolis, Brazil, pp. 1044–1045 (2009)
Sitek, A., Huesman, R.H., Gullberg, G.T.: Tomographic reconstruction using an adaptive tetrahedral mesh defined by a point cloud. IEEE Transactions on Medical Imaging 25(9), 1172–1179 (2006)
Fedorov, A., Chrisochoides, N.: Tetrahedral mesh generation for non-rigid registration of brain mri: Analysis of the requirements and evaluation of solutions. In: Garimella, R.V. (ed.) Proceedings of the 17th International Meshing Roundtable. Springer, Heidelberg (2008), doi:10.1007/978-3-540-87921-3
Zhang, Y., Wang, W., Liang, X., Bazilevs, Y., Hsu, M.C., Kvamsdal, T., Brekken, R., Isaksen, J.: High-fidelity tetrahedral mesh generation from medical imaging data for fluid-structure interaction analysis of cerebral aneurysms. Computer Modeling in Engineering & Sciences (CMES) (2), 131–148 (2009)
Efron, B.: The convex hull of a random set of points. Biometrica 52, 331–343 (1965)
Buchta, C., Müller, J.: Random polytopes in a ball. J. Appl. Prob. 21, 753–762 (1984)
Affentranger, F.: The expected volume of a random polytope in a ball. J. Microscopy 151, 277–287 (1988)
Miles, R.E.: Isotropic random simplices. Adv. Appl. Probability 3, 353–382 (1971)
Buchta, C., Reitzneri, M.: The convex hull of random points in a tetrahedron: Solution of Blaschke’s problem and more general results. J. Reine Angew. Math. 536, 1–29 (2001)
Philip, J.: The average volume of a random tetrahedron in a tetrahedron. TRITA MAT 06 MA 02 (2006), http://www.math.kth.se/~johanph/ev.pdf
Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved problems in geometry, pp. 54–57. Springer, New York (1991), B5. Random Polygons and Polyhedra
Weisstein, E.W.: Sphere tetrahedron picking. In: MathWorld – A Wolfram Web Resource (March 2013), http://mathworld.wolfram.com/topics/RandomPointPicking.html
Buchta, C.: Distribution-independent properties of the convex hull of random points. J. Theor. Prob. 3, 387–393 (1990)
Berntsen, J., Espelid, T.O., Genz, A.: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Softw. 17, 437–451 (1991)
Genz, A.: MVNDST (1998), http://www.math.wsu.edu/faculty/genz/software/fort77/mvndstpack.f
de Doncker, E., Kaugars, K., Cucos, L., Zanny, R.: Current status of the ParInt package for parallel multivariate integration. In: Proc. of Computational Particle Physics Symposium (CPP 2001), pp. 110–119 (2001)
Li, S., Kaugars, K., de Doncker, E.: Distributed adaptive multivariate function visualization. International Journal of Computational Intelligence and Applications (IJCIA) 6(2), 273–288 (2006)
Genz, A.: MVNPACK (2010), http://www.math.wsu.edu/faculty/genz/software/fort77/mvnpack.f
L’Equyer, P.: Combined multiple recursive random number generators. Operations Research 44, 816–822 (1996)
Cranley, R., Patterson, T.N.L.: Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal. 13, 904–914 (1976)
OpenMP website, http://www.openmp.org
CUDA NVIDIA Developer Zone, https://developer.nvidia.com/category/zone/cuda-zone
OpenACC website, http://www.openacc-standard.org
Open MPI website, http://www.open-mpi.org
MPI website, http://www-unix.mcs.anl.gov/mpi/index.html
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
de Doncker, E., Assaf, R. (2013). GPU Integral Computations in Stochastic Geometry. In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2013. ICCSA 2013. Lecture Notes in Computer Science, vol 7972. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39643-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-39643-4_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39642-7
Online ISBN: 978-3-642-39643-4
eBook Packages: Computer ScienceComputer Science (R0)