Multidimensional Integration Package: DICE and its parallelization

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Multidimensional Integration Package
DICEandits parallelization

• F.Yuasa / KEK
• K.Tobimatsu / Kogakuin Univ.
• S.Kawabata / KEK
• ACAT2002
• 24-28, Jun. 2002 at MSU, Moscow

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BASES

• Multidimensional Integration Package
• Stratified and Importance sampling method
• Singular function can be integrated
• Up to 100 dimensions
• Heavily used in GRACE framework

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When singularities go along a diagonal line, we
need appropriate variable transformation.
y
Y
x
X
(x , y)
(X , Y)
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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 2Ndim hypercubes
• Two kinds of sampling method
• DICE Input
• Ndim, Expected Error, of Sampling points,
  Maximum division level, Maximum of iteration

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How to divide Hypercube
Ndim2
regular
divide
divide
Level 3
Level 2
Regular sampling and random sampling
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Example 1
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Example 2
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Example 3
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Example 4
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Results of I4
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Results of I4 (part2)
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Example 5

• More complicated Integrand
• of dimensions 4
• of lines in FORTRAN about 300 lines

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Results of Example5
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Results of Example5 (part2)
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Results of Example5 (part3)
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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

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Speed Up
Example5
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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.