1. F.Yuasa at ACAT2002 Multidimensional Integration Package: DICE and its parallelization F.Yuasa / KEK K.Tobimatsu / Kogakuin Univ. S.Kawabata / KEK ACAT2002 24-28, Jun. 2002 at MSU, Moscow

2. 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

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

4. 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

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

6. F.Yuasa at ACAT2002 Example 1

7. F.Yuasa at ACAT2002

8. Example 2

9. F.Yuasa at ACAT2002

10. Example 3

11. F.Yuasa at ACAT2002

12. Example 4 4

13. F.Yuasa at ACAT2002

14. Results of I4 PackageEps =10**(-1)Eps =10**(-2) DICE-mpi(3.1408 +-0.0029)E-02 (-1.0662 +-0.0011)E-02 ParInt1.10.031216955 +-0.0016138 -0.010710149 +-0.00055466 BASES(3.174411 +-0.047391)E-02 (-1.056945 +-0.035239)E-02 Analytical results 0.0314313-0.0106773

15. F.Yuasa at ACAT2002 Results of I4 (part2) PackageEps =10**(-3)Eps =10**(-4) DICE-mpi(-1.6070 +-0.0023)E-02 (-1.6676 +-0.0024)E-02 ParInt1.1-0.01593609 0.01293747 -0.0114787661 0.0206822789 BASES(-1.614249 +-0.009806)E-02 (-1.639091 +-0.033053)E-02 Analytical results -0.0160761-0.0166246

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

17. F.Yuasa at ACAT2002 Results of Example5 PackageResult# of Sample points DICE-mpi 1 processor (1.0638+-0.0011)E-1324798768 ParInt1.1 1 processor (1.0622+-0.0529)E-132100000110 BASES(1.064086+-0.000337)E- 13 9996350

18. F.Yuasa at ACAT2002 Results of Example5 (part2) PackageResult# of Sample points DICE-mpi 1 processor ( -1.1529+-0.0019) E-13 24798768 ParInt1.1 1 processor (-1.1452+-0.0652)E-132100000110 BASES(-1.154076+-0.000681)E- 13 9996350

19. F.Yuasa at ACAT2002 Results of Example5 (part3) PackageResult# of Sample points DICE-mpi 1 processor ( -8.8675+-0.0872)E-15 0.98% 33956096 ParInt1.1 1 processor We did not try BASESWe did not try

20. 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

21. F.Yuasa at ACAT2002 Speed Up # of CPUs1248 CPU time [sec] 3232.801648.40839.00431.68 Speed Up1.001.963.857.49 Example5

22. 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.