F.Yuasa at ACAT2002

Multidimensional Integration Package: DICE

andits parallelization

F.Yuasa / KEKK.Tobimatsu / Kogakuin Univ.

S.Kawabata / KEKACAT2002

24-28, Jun. 2002 at MSU, Moscow

F.Yuasa at ACAT2002

BASES

• Multidimensional Integration Package• Stratified and Importance sampling metho

d• 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 2Ndim 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 HypercubeNdim=2

Regular sampling and random sampling

regular

F.Yuasa at ACAT2002

Example 1

I1 dx1dx2

2x2

(x1 x2 1)2 20

1

0

1

F.Yuasa at ACAT2002

F.Yuasa at ACAT2002

Example 2

I2 dx1dx2

x22(1 x1

2 x22)

(x12 x2

2 a2)2 2 1

1

1

1

F.Yuasa at ACAT2002

F.Yuasa at ACAT2002

Example 3

I3 dx1dx2dx3

x12 x2

2 x32(1 x1

2 x22 x3

2)

(x12 x2

2 x32 a2)2 2

1

1

1

1

1

1

F.Yuasa at ACAT2002

F.Yuasa at ACAT2002

Example 4

I3 dx1dx2dx3dx4 f (1 x1

2 x22 x3

2 x42)

(x12 x2

2 x32 x4

2 a2)2 2 1

1

1

1

1

1

1

1

,

Rx12 x2

2 x32 x4

2,

and

f (R) (R 1/4)(R 1/2)2(R 3/4)

4

F.Yuasa at ACAT2002

f (R) (R 1/4)(R 1/2)2(R 3/4)

10 1

a0.8

10 2

a0.8

F.Yuasa at ACAT2002

Results of I4Package Eps =10**(-1) Eps =10**(-2)

DICE-mpi (3.1408

+-0.0029)E-02

(-1.0662

+-0.0011)E-02

ParInt1.1 0.031216955

+-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

F.Yuasa at ACAT2002

Results of I4 (part2)Package Eps =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

F.Yuasa at ACAT2002

• More complicated Integrand

• # of dimensions = 4

• # of lines in FORTRAN = about 300 lines

Example 5

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Results of Example5Package Result # of Sample

points

DICE-mpi1 processor

(1.0638+-0.0011)E-13 24798768

ParInt1.1

1 processor

(1.0622+-0.0529)E-13 2100000110

BASES (1.064086+-0.000337)E-13

9996350

F.Yuasa at ACAT2002

Results of Example5 (part2) Package Result # of Sample

points

DICE-mpi1 processor

( -1.1529+-0.0019)

E-13

24798768

ParInt1.1

1 processor

(-1.1452+-0.0652)E-13 2100000110

BASES (-1.154076+-0.000681)E-13

9996350

F.Yuasa at ACAT2002

Results of Example5 (part3) Package Result # of Sample

points

DICE-mpi1 processor

( -8.8675+-0.0872)E-15 0.98%

33956096

ParInt1.1

1 processor

We did not try

BASES We 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 CPUs 1 2 4 8

CPU time

[sec]

3232.80 1648.40 839.00 431.68

Speed Up 1.00 1.96 3.85 7.49

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.