ACAT-2002

The Evolutionary Model of Physics Large-Scale Simulation on Parallel Dataflow Architecture

Dr. Andrey V. NikitinDr. Ludmila I. NikitinaLomonosov Moscow State University

ACAT-2002

Physics Large-Scale Simulation

Nonlinear 3D simulation of experimentally observed magnetohydrodynamic (MHD) burst in the DIII-D tokamak

Fast (~10-6 sec) processes Slow (~10-2 sec) processes 3D calculations Different scales for time and space

ACAT-2002

Physics Large-Scale Simulation

Model:

V – velocity

B – magnetic fieldP - pressure

VReBBPVVt

V

1),(

)(1bsm jBVReBV

t

B

OHQPPVPPVt

P

||||||||)1()(

0

P

dt

d

PB

BP

,||

PPP ||

,Vtdt

d

ACAT-2002

Physics Large-Scale Simulation

Numerical model:

straight field line coordinate system

)(

)( 1

max

min

max

)sin(),()cos(),(),,,(nM

nMm

N

n

smn

cmn nmtUnmtUtU

),,(

ACAT-2002

Physics Large-Scale Simulation

Requirements to the numerical simulation process:

Inversion of matrixes 106

Near real time Evolution of simulation process

ACAT-2002

1(P)

4(R)

2(P)

5(W)

3(W)

1

2

h = 3

d = (a + b) * c

<a,4,-> <b,4,->

<c,5,->

<+,5,->

<*,-,->

Dataflow model of calculations

Dataflow graph decomposition to layers

аГ

ACAT-2002

Dataflow model of calculations

buffer

SwitchS2

SwitchS1

PE

tokens

packets

tokens

tokens

Associativememorymodule

jM

ACAT-2002

Dataflow model of calculations

jij

jijjаji BM

FpMXГ

,

,, ),,,(

n

l

n

m

n

kjmilmklkаji xxssXГ

1 1 1, 2

1),(

),(),,(1

1 1

XГpXГ а

p

i

p

ijijа

pаp

ээа MpMXГT

MTpMXГE

),,,(),,,(

),,(max),,,(1

,,1

pXГpMXГT а

h

iji

pjа

Write time

Search time

Total time

Effectiveness

ACAT-2002

Dataflow model of calculations

Goal:

max),,,( pMXГE а

ACAT-2002

Genetic algorithm

1 2

9

7 8

12

5 6

11

3 4

10

14

15

13

1

2

3

Граф алгоритмаAlgorithm

1

0

0

1

0

0

1

0

0

0

1

0

0

1

0

0

0

1

0

0

1

1

0

0

1

0

0

0

1

0

0

1

0

0

1

0

0

1

0

0

0

1

0

0

1

Partition matrix

аГ

Chromosome x’ = 

1 1 2 2 2 2 3 3 1 1 2 2 1 1 1

ACAT-2002

MHD dataflow graph

*

+-

C0B0 F0 C1 B1 F1A1

()-1

* *

()-1

*

*

*

CN-1 FN-1AN-1

<2> <2>

*

+-

()-1

*

*

*

*<N-1> *<N-1>

BN-1

<N> <N> UN-1

*

- UN-2

<1> <1>

к U0

t+1

1,1,1ˆ t

U

2,1,1ˆ t

U

Pt

U,1,1ˆ

...

1,1

1

t

U1,1

max

t

NU...

2,1

1

t

U2,1

max

t

NU...

1,2,1ˆ t

U

2,2,1ˆ t

U

Pt

U,2,1ˆ

...

...

r = 1

r = 1,p = 1,...,P

r = 2

r = 2,p = 1,...,P

Rt

U,1

1

Rt

NU,1

max

...

1,,1ˆ Rt

U

2,,1ˆ Rt

U

PRt

U,,1ˆ

...

r = R

r = 2,p = 1,...,P

t

Ut

Ut1,1

2

t

U

2,1

2

t

U

Rt

U,1

2

ACAT-2002

Genetic algorithm

0% 0.01% 1% 2% 5% 10%

Сходимость относительной ошибки при различных вероятностях мутации

1501401301201101009080706050403020100

110

105

100

95

90

85

80

75

70

65

60

55

50

45

40

35

30

25

20

15

10

5

0

Convergence of GA (mutation)

ACAT-2002

Genetic algorithm vs. MK

ГА МК (0.01) МК (0.5)

Сравнение работы ГА и метода Монте-Карло

5045403530252015105

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Convergence of GA and MK

ACAT-2002

Conclusion

GA was used to organize evolutionary computations of physics large-scale simulation on parallel dataflow architecture

GA allows to find optimal distribution of data by modules with high effectiveness

Code NFTC (1 iteration): SUN Ultra 400 Mhz/2 FPU ~ 50s, for dataflow ~ 4·10-2s.

Total time of typical computations:

Current (SUN Ultra Enterprise) – 30 hrs Dataflow (modules - 102,module capacity - 106,PE –

104/10 GFlops) – real time

ACAT-2002

Questions?