F.Yuasa at ACAT2002 Multidimensional Integration Package: DICE and its parallelization F.Yuasa / KEK K.Tobimatsu / Kogakuin Univ. S.Kawabata / KEK ACAT , Jun at MSU, Moscow

F.Yuasa at ACAT2002 BASES Multidimensional Integration Package Stratified and Importance sampling method Singular function can be integrated Up to 100 dimensions Heavily used in GRACE framework

F.Yuasa at ACAT2002 x y (x, y) Y X (X, Y) When singularities go along a diagonal line, we need appropriate variable transformation.

F.Yuasa at ACAT2002 DICE Developed by K.Tobimatsu and S.Kawabata –First version of DICE in 1992 –Research Reports of Kogakuin Univ. No.72 (1992) Divide the integral region into 2 Ndim hypercubes Two kinds of sampling method DICE Input –Ndim, Expected Error, # of Sampling points, Maximum division level, Maximum # of iteration

F.Yuasa at ACAT2002 Level = 2 Level = 3 How to divide Hypercube Ndim=2 Regular sampling and random sampling regular

F.Yuasa at ACAT2002 Example 1

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Example 2

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Example 3

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Example 4 4

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Results of I4 PackageEps =10**(-1)Eps =10**(-2) DICE-mpi( )E-02 ( )E-02 ParInt BASES( )E-02 ( )E-02 Analytical results

F.Yuasa at ACAT2002 Results of I4 (part2) PackageEps =10**(-3)Eps =10**(-4) DICE-mpi( )E-02 ( )E-02 ParInt BASES( )E-02 ( )E-02 Analytical results

F.Yuasa at ACAT2002 More complicated Integrand # of dimensions = 4 # of lines in FORTRAN = about 300 lines Example 5

F.Yuasa at ACAT2002 Results of Example5 PackageResult# of Sample points DICE-mpi 1 processor ( )E ParInt1.1 1 processor ( )E BASES( )E

F.Yuasa at ACAT2002 Results of Example5 (part2) PackageResult# of Sample points DICE-mpi 1 processor ( ) E ParInt1.1 1 processor ( )E BASES( )E

F.Yuasa at ACAT2002 Results of Example5 (part3) PackageResult# of Sample points DICE-mpi 1 processor ( )E % ParInt1.1 1 processor We did not try BASESWe did not try

F.Yuasa at ACAT2002 Parallelization We use MPI for the parallelization. Parallelization is useful for higher dimensional integrand Parallelization is useful for complicated integrand Example 5 is calculated by the parallelized DICE

F.Yuasa at ACAT2002 Speed Up # of CPUs1248 CPU time [sec] Speed Up Example5

F.Yuasa at ACAT2002 Summary We have developed DICE. DICE is available to Vector Processor. DICE is available to Parallel Processor. We have used MPI for parallelization. For the complicated integrand, parallelization shows good scalability.