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Parallel Computation of Multivariate Normal Probabilities
"... We present methods for the computation of multivariate normal probabilities on parallel/ distributed systems. After a transformation of the initial integral, an approximation can be obtained using MonteCarlo or quasirandom methods. We propose a metaalgorithm for asynchronous sampling methods and d ..."
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Cited by 207 (9 self)
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We present methods for the computation of multivariate normal probabilities on parallel/ distributed systems. After a transformation of the initial integral, an approximation can be obtained using MonteCarlo or quasirandom methods. We propose a metaalgorithm for asynchronous sampling methods and derive efficient parallel algorithms for the computation of MVN distribution functions, including a method based on randomized Korobov and Richtmyer sequences. Timing results of the implementations using the MPI parallel environment are given. 1 Introduction The computation of the multivariate normal distribution function F (a; b) = j\Sigmaj \Gamma 1 2 (2) \Gamma n 2 Z b a e \Gamma 1 2 x \Sigma \Gamma1 x dx: (1) often leads to computationalintensive integration problems. Here \Sigma is an n \Theta n symmetric positive definite covariance matrix; furthermore one of the limits in each integration variable may be infinite. Genz [5] performs a sequence of transformations resu...
Methods for the Computation of Multivariate tProbabilities
 Computing Sciences and Statistics
, 2000
"... This paper compares methods for the numerical computation of multivariate tprobabilities for hyperrectangular integration regions. Methods based on acceptancerejection, sphericalradial transformations and separationofvariables transformations are considered. Tests using randomly chosen problems ..."
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Cited by 83 (11 self)
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This paper compares methods for the numerical computation of multivariate tprobabilities for hyperrectangular integration regions. Methods based on acceptancerejection, sphericalradial transformations and separationofvariables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz (1992) for multivariate normal probabilities. These methods allow moderately accurate multivariate tprobabilities to be quickly computed for problems with as many as twenty variables. Methods for the noncentral multivariate tdistribution are also described. Key Words: multivariate tdistribution, noncentral distribution, numerical integration, statistical computation. 1 Introduction A common problem in many statistics applications is the numerical computation of the multivariate t (MVT) distribution function (see Tong, 1990) defined by T(a; b; \Sigma; ) = \Gamma( +m 2 ) \Gamma( 2 ) p j\Sigma...
Use of ParInt for Parallel Computation of Statistics Integrals
 Computing Science and Statistics
, 1996
"... We present applications of ParInt, a package of Parallel and distributed multivariate Integration algorithms. The integrals arise in Bayesian statistical computation. The integrals are transformed according to the methods of Genz and Kass [10]. The parallel integration algorithms are based on subreg ..."
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Cited by 4 (4 self)
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We present applications of ParInt, a package of Parallel and distributed multivariate Integration algorithms. The integrals arise in Bayesian statistical computation. The integrals are transformed according to the methods of Genz and Kass [10]. The parallel integration algorithms are based on subregion partitioning. An overview of ParInt and its graphical user interface is given and the parallel speedups are discussed for the problems under consideration. 1 Introduction An investigation of fast techniques for numerical integration is motivated by the need to compute computationally intensive multiple integrals arising in various areas of science and engineering, for example in statistics and in finite element applications. The work reported in this paper is part of a project (ParInt) involving the design, analysis and development of a set of coarse grain parallel and distributed algorithms for multivariate numerical integration. We discuss an automatic integration algorithm which uses...
Numerical computation of a nonplanar twoloop vertex diagram
 In LoopFest V
, 2006
"... Overview.} The twoloop crossed vertex diagram gives rise to a sixdimensional integral, where the outer integration is over the simplex z1+z2+z3 = 1 and the inner integration over the hyperrectangle [1; +1]3: The factor 1=D23 in the integrand has a nonintegrable singularity interior to the integr ..."
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Cited by 4 (3 self)
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Overview.} The twoloop crossed vertex diagram gives rise to a sixdimensional integral, where the outer integration is over the simplex z1+z2+z3 = 1 and the inner integration over the hyperrectangle [1; +1]3: The factor 1=D23 in the integrand has a nonintegrable singularity interior to the integration domain and a singularity on the boundary.} The integral can be evaluated by iterated numerical integration.
Error Distribution for Iterated Integrals
"... Abstract: In earlier work we demonstrated that iterated numerical integration outperforms ”standard ” multivariate integration techniques for various function classes with singularities inside the domain of integration. We focus on the accuracy requirements to be applied in different directions or c ..."
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Abstract: In earlier work we demonstrated that iterated numerical integration outperforms ”standard ” multivariate integration techniques for various function classes with singularities inside the domain of integration. We focus on the accuracy requirements to be applied in different directions or combinations of directions of iterated integrals. The integrals are approximated numerically in an ”automatic ” (blackbox) manner, where the user poses an accuracy requirement for the multivariate integral and expects a result within that tolerance. For an iterated integration it is generally accepted that an inner integral should be evaluated more accurately than its outer integrals in order to avoid the impression of roundoff error. We propose techniques for setting the error tolerances in different directionsi and give numerical results. Key–Words: Automatic, numerical, iterated integration, error distribution
Joint Statistical Meetings Statistical Computing Section PARALLEL MULTIVARIATE INTEGRATION: PARADIGMS AND APPLICATIONS
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"... (2) Numerical extrapolation and 1loop 3point vertex (3) Extrapolation by the algorithm (4) 4point function ..."
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(2) Numerical extrapolation and 1loop 3point vertex (3) Extrapolation by the algorithm (4) 4point function