GPU Integral Computations in Stochastic Geometry
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1 Outline GPU Integral Computations in Stochastic Geometry Department of Computer Science Western Michigan University June 25, 2013
2 Outline Outline 1 Introduction 2 3 4
3 Outline Outline 1 Introduction 2 3 4
4 Outline Outline 1 Introduction 2 3 4
5 Outline Outline 1 Introduction 2 3 4
6 Outline Introduction 1 Introduction 2 3 4
7 Problem Characteristics For given integrand f( x), over domain D, obtain integral approximation Q I = D f( x)d x It is useful to obtain an estimated error E Q I. Figure : 1 0 dx 1 0 dy 2εy (x+y 1) 2 +ε 2, ε = 0.1
8 Outline Introduction 1 Introduction 2 3 4
9 Adaptive methods Definition: Integration rule: Q = r j=1 w jf( x j ) D f( x)d x; also in composite rules Adaptive partitioning: concentrates subdivisions and function sampling in difficult areas. Competitive for: irregular function behavior, small to moderate dimensions Iterated adaptive method; e.g.: b d a c f(x, y) dx dy = [ b ] a dx d c dy f(x, y) Figure : 1 1 dx 1 1 dy εy 2 θ(1 x 2 y 2 ) with a = 0.8, ε = (x 2 +y 2 a 2 ) 2 +ε 2
10 Non-adaptive methods Sequence of rules, with increasing number of points; rules may be on a lattice. Monte Carlo (MC): Q = 1 N N j=1 f( x j) Competitive for: moderate to high dimensions, irregular function behavior, irratic integration domain Quasi-Monte Carlo (QMC): In a non-adaptive sequence, a number of randomized copies are generated for each rule, and the results over the randomized copies are averaged. Competitive: up to high dimensions, fairly smooth function behavior
11 Outline Introduction 1 Introduction 2 3 4
12 An adaptive region subdivision algorithm for distributed memory systems; layered over MPI (Message Passing Interface - see, e.g., [1]); uses task pool on each node, and load balancing among nodes to handle localized integrand difficulties such as peaks and singularities An iterated adaptive method, for shared memory/multi-core systems QMC method; load balancing is applied to deal with possible heterogeneous processors MC method: implemented using CUDA [2] on GPUs Goals: to extend ParInt to run efficiently on a hybrid architecture, with distributed nodes, each multicore, and possibly with GPUs (graphics processing units).
13 In computational geometry Example: cube tetrahedron picking yields the expected volume of a random tetrahedron in a cube. Other examples: ball tetrahedron picking, tetrahedron tetrahedron picking, also sphere tetrahedron picking Expected d-dimensional (dd) volume E[V n (K)] of the polyhedron formed by n points X 1,..., X n, uniformly distributed in the interior of a convex body K (cf., [3]): K E[V n (K)] = 1 K K Hull(X 1,...,X n ) dx 1 K K denotes the d-dimensional volume of K and dxn K Hull(X 1,...,X n ) is the volume of the convex hull generated by the n points.
14 Cube = C 3 tetrahedron picking (n = 4, d = 3): E[V 4 (C 3 )] = dx 1 dy 1 dz 1 dx 2 dy 2 dz x 1 y 1 z 1 1 with V 4 = 1 x 2 y 2 z x 3 y 3 z 3 1. x 4 y 4 z dx 3 dy 3 dz 3 dx 4 dy 4 dz 4 V
15 Tetrahedron = T 3 tetrahedron picking (cf., [4, 3, 5]) 1 1 x1 E[V 4 (T 3 )] = 6 4 dx x3 dx dy 1 1 x1 y 1 0 dy 3 1 x3 y x2 dz 1 dx x4 dz 3 dx x2 y 2 dy 2 dz x4 y 4 dy 4 dz 4 V 4 0 Ball = B 3 tetrahedron picking [6, 7]: We implemented as above but with integration over: [ x 2 ] 1 x y 1 2 dx i dy i dz i 1 1 x1 2i i=1 1 x 2 1 y 2 1
16 Sphere = S 2 tetrahedron picking: According to the interpretation of [8] (the vertices are varied over S 2, and using the parameter representation: S 2 = { s(θ, u) (sin(θ) 1 u 2, cos(θ) 1 u 2, u), the integral is: 1 du du du du 2π du du du 1 3 u [ 1, 1] and θ [0, 2π]} 1 1 du 4 2π 0 2π 2π 2π dθ 0 1 dθ 0 2 dθ 0 3 dθ 0 4 V 4 2π 2π 2π 2π dθ 0 1 dθ 0 2 dθ 0 3 dθ With some simplifications (cf., [9]: u 1 = 1 (first vertex); θ 2 = 0 (second vertex)): E[V 4 (S 2 )] = 1 du du du 2π du du 1 3 2π dθ 0 3 dθ 0 4 V 4 1 du 2π 2π 1 4 dθ 0 3 dθ 0 4 1
17 Outline Introduction 1 Introduction 2 3 4
18 Random number generation The MC estimate is obtained as the avarage of the function values at a set of N uniformly distributed random points; using the CUDA pseudo-random number generator library (curand) [10] to generate N d random floats for the coordinates of N d-dimensional points, directly on the GPU (device); and using a program section to allocate space for ndim = N d floats in array a on the device, and call the CURAND functions cudamalloc, curandcreategenerator, curandsetpseudorandomgeneratorseed and curandgenerateuniform.
19 Outline Introduction 1 Introduction 2 3 4
20 # EV. PROBLEM RESULT ABS. ERR. SEQ. TIME PAR. TIME SPEEDUP (M) (s) (s) 0.1 E[V 4 (S 2 )] e e e e e+00 E[V 4 (C 3 )] e e e e e+00 E[V 4 (T 3 )] e e e e e+00 1 E[V 4 (S 2 )] e e e e e+01 E[V 4 (C 3 )] e e e e e+01 E[V 4 (T 3 )] e e e e e E[V 4 (S 2 )] e e e e e+02 E[V 4 (C 3 )] e e e e e+01 E[V 4 (T 3 )] e e e e e E[V 4 (S 2 )] e e e e e+02 E[V 4 (C 3 )] e e e e e+02 E[V 4 (T 3 )] e e e e e E[V 4 (S 2 )] e e e e e+02 E[V 4 (C 3 )] e e e e e+02 E[V 4 (T 3 )] e e e e e E[V 4 (S 2 )] e e e e e+02 E[V 4 (C 3 )] e e e e e+02 E[V 4 (T 3 )] e e e e e+02 Table : GPU results for E[V 4 (S 2 )], E[V 4 (C 3 )] and E[V 4 (T 3 )].
21 We gave a brief overview of ParInt, some applications, and described a many-core parallelization of MC using an application to problems in stochastic geometry. Point-picking problems have applications, e.g., with tetrahedron vertices on a spherical cap, for analyzing the properties of the points in a Poisson-Vonoroi distribution [8]. With respect to the point-picking type of problem, extensions are possible as well (n > 4 points, d > 3). This work is part of efforts to port the to a hybrid parallel environment.
22 References Open-MPI: CUDA: NVIDIA Developer Zone, Zinani, A.: The expected volume of a tetrahedron whose vertices are chosen at random in the interior of a cube. Monatsh. Math. 139 (2003) DOI /s y. Buchta, C., Reitzner, M.: The convex hull of random points in a tetrahedron: Solution of Blaschke s problem and more general results.
23 J. Reine Angew. Math 536 (2001) 1 29 Philip, J.: The average volume of a random tetrahedron in a tetrahedron. TRITA MAT 06 MA 02 (2006) johanph/ev.pdf. Buchta, C., Müller, J.: Random polytopes in a ball. J. Appl. Prob. 21 (1984) Affentranger, F.: The expected volume of a random polytope in a ball. J. Microscopy 151 (1988) Heinrich, L., Körner, Mehlhorn, N., Muche, L.:
24 Numerical and analytical computation of some second-order characteristics of spacial Poisson-Vonoroi tesselations. Statistics 31 (1998) Weisstein, E.W.: Sphere tetrahedron picking. In: MathWorld A Wolfram Web Resource. (2013) NVIDIA: CUDA CURAND LIBRARY
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