The Evolutionary Model of Physics
Large-Scale Simulation on Parallel
Dataflow Architecture
Dr. Andrey V. Nikitin
Dr. Ludmila I. Nikitina
Lomonosov Moscow State University
ACAT-2002
Physics Large-Scale Simulation
Nonlinear 3D simulation of experimentally
observed magnetohydrodynamic (MHD)
burst in the DIII-D tokamak
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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 , )V  P    B   B   Re 1V
V

t
B
   V  B   Re m1   (V  B  jbs )
t
P
 ( PV )  (   1) P  V         P   ||   ||||P||   QOH
t
d  P 
    0
V – velocity
dt   
B – magnetic field
B

P - pressure
||P   ,   P
B

d

  V ,     P  P  || P
dt t
ACAT-2002
Physics Large-Scale Simulation
Numerical model:
U (  , ,  , t ) 
  U
M max ( n ) N max
m  M min ( n ) n 1
c
mn
s
(  , t ) cos( m  n )  U mn
(  , t ) sin( m  n )
straight field line coordinate system (  ,  ,  )
ACAT-2002
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Physics Large-Scale Simulation
Requirements to the numerical simulation
process:
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Inversion of matrixes 106
Near real time
Evolution of simulation process
ACAT-2002
Dataflow model of calculations
Dataflow graph decomposition to layers
1(P)
<a,4,->
2(P)
<b,4,->
3(W)
d = (a + b) * c
<c,5,->
1
4(R)
<+,5,->
2
Га
5(W)
ACAT-2002
<*,-,->
h=3
Dataflow model of calculations
packets
tokens
PE
Switch
S1
Switch
S2
buffer
tokens
Associative
memory
M j module
ACAT-2002
tokens
Dataflow model of calculations
Write time
Search time
 Fi , j 
i , j ( Г а , X , M j , p)   j 

 M j  Bi , j 
1 n n n
 i , j ( Г а , X )   s lk s mk x il x jm
2 l 1 m 1 k 1
p 1
p
( Г а , X , p)    ij ( Г а , X )
i 1 j i 1
h
Total time
T ( Г а , X , M , p )   max  i , j  ( Г а , X , p )
i 1
Effectiveness
j 1, p
Tэ M э
E ( Г а , X , M , p) 
Tp ( Г а , X , M , p) M p
ACAT-2002
Dataflow model of calculations
Goal:
E ( Г а , X , M , p)  max
ACAT-2002
Genetic algorithm
1
2
3
4
5
6
7
8
3
9
10
2
11
13
14
15
1
Partition matrix
Chromosome
x’ =
12
Га
Algorithm
Граф алгоритма
1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 
  0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 


0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 
1 1 2 2 2 2 3 3 1 1 2 2 1 1 1
ACAT-2002
MHD dataflow graph
U
t
t 1,1
r=1
U1
t
Û
t 1,1
t
B0
...
U2
C0
t 1,1
U N max
F0
B1
F1
<N-1>
*
AN-1
CN-1
<1>
*
...
r = 1,
p = 1,...,P
C1
()-1
<1>
t 1,1,1
Uˆ
A1
*
*
*
t 1,1, 2
Uˆ
-
+
t 1,1, P
Uˆ
к U0
r=2
t 1, 2
U1
t 1, 2
...
U2
()-1
t 1, 2
U N max
<2>
t+1
Uˆ
*
<2>
<N-1>
*
*
t 1, 2 ,1
...
r = 2,
p = 1,...,P
t 1, 2 , 2
Uˆ
t 1, 2 , P
Uˆ
*
*
...
t 1, R
r=R
U1
t 1, R
...
U2
-
t 1, R
+
U N max
()-1
t 1, R ,1
Uˆ
<N>
...
r = 2,
p = 1,...,P
*
<N>
*
t 1, R , 2
Uˆ
*
t 1, R , P
Uˆ
ACAT-2002
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UN-2
UN-1
BN-1
FN-1
Genetic algorithm
Convergence of GA (mutation)
Сходимость относительной ошибки при различных вероятностях мутации
0%
0.01%
1%
2%
5%
10%
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
0
10
20
30
40
50
60
70
80
ACAT-2002
90
100
110
120
130
140
150
Genetic algorithm vs. MK
Convergence of GA and MK
Сравнение работы ГА и метода Монте-Карло
ГА
МК (0.01)
МК (0.5)
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
5
10
15
20
25
30
ACAT-2002
35
40
45
50
Conclusion
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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:
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Current (SUN Ultra Enterprise) – 30 hrs
Dataflow (modules - 102,module capacity - 106,PE – 104/10
GFlops) – real time
ACAT-2002
Questions?
ACAT-2002