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Влияние аппаратно-программных средств на скорость вычисления алгоритмов решения уравнения переноса излучения для плоской геометрии среды.

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69
?????????????? ??????????
??????? ?????????-??????????? ??????? ??
???????? ?????????? ?????????? ???????
????????? ???????? ????????? ??? ???????
????????? ?????
????????? ?. ?., ????? ?. ?., ????????? ?. ?., ?????????? ?. ?.
?????????? ?????????????? ???????? (??????????? ???????????), 111250 ??????, ??????
????????? ? ???????? 17.05.2011, ????? ????????? 20.05.2011.
???????????? ??.-????. ???? ?.?. ???????? 23.05.2011
????????, ??? ? ??????? ????????? ???????????????? ????????? ???????? ????????? (???) ?????
???????????? ????????????? ???????. ????????? ??????? ???????????? ????? ???????? ?????????
?????????, ??? ???????? ?????????? ???????????????? ??????????, ?? ?????????? ???????????
????????? ???????? ????? ???????. ????????????? ??? ???????? ?????? ????? ?????????
???????????? ????? ???????, ??????????? ??? ??? ???????????. ????????? ?????????? ??????????
??????? ??? ?????????? ???????? ????????? ???????????? ?????, ????? ??????? ????????
??????????? ???????? ??????????? ??????????? ?????? ??????????? ????????. ????????????????
??????? ?????????-??????????? ??????? ?? ????????????? ?????????? ?????????? ??????? ??? ?
??????? ?????.
???????? ?????: ????????? ???????? ?????????, ??????? ?????????, ???????????? ?????????
PACS 42.68.Ay
??????????
1. ???????? (69).
2. ???????????? ???? (70).
3. ????????????? ???? (71).
4. ??????? ??? ??????? ????? (72).
5. ????????????? ????????????? ????????????
????? (72).
6. ?????? ??????? ??????????? ? ??????????
???????? ????????? (73).
7. ?????????? ? ?????????? (74).
8. ?????????? (76).
?????????? (76).
1. ????????
??????? ????? ?????????????? ?????????? ??????? ????????? ?? ?????? ?????? ???????????? ???????????? ???????? ???????? ???????
?????? ???????? ?????????. ??? ???????, ???????? ?????? ????????????? ??????? ?????? ??????. ??-?? ?????? ??????????????? ??????????? ??????, ?????????? ?? ???????????? ??????, ????? ???????? ? ????????, ???????? ??????????? ?????? [1, 2]. ??????? ????? ???????????? ?????? ???????, ?????????? ?? ????????????
??????? ?????? ??????. ????? ????, ? ????????? ????????????????? ??????? ?????????? ??????
??????, ?????????? ?? ?????????. ? ???? ????? ?
??????? ??????? ????????? ???????? ????????????? ??????? ?????????? ? ???????? ??????????,
? ?????? ????????? ?????????????? ???????? ???????? ??????????.
? ?????? ???????, ?????????? ?????????? ?
???????? ??????????. ??? ? ??????? GOSAT (The
Greenhouse gases Observing SATellite) ?????????
???????????? ????????????? ??????????? ????????? ? ???????????? ?? ????? 1% [3]. ??? ???????? ????? ????? ?????????? ????????? ??????????? ?? ????????? ?????. ??? ????? ??????????
?????? ?????? ?????????????? ???? ????? TMS
[4]. ??? ??????? ???? ????? ??????????? ?? ?????
???????????. ??? ??????????? ??????? ???????? ????? ????????? ?????? ??????? ?????????
????????, ??????? ????? ????????? ??????????.
???????? ????????? ??????? ??????? ????? ??
???? ??????????-?????????? ???????. ?????????
?????? ????????? ??????????? ?????? ??????? ?????, ?? ? ????????? ????? ? ??????? ?????????????? ???????????? ??????? ?????? ???????
????????? ?????. ?????????????? ?? ???????????? ???? ???????? ?? ???????, ?????????? ??????? ?????????? ????????? ???????? ?????????
(????) ? ???????????? ?? ?????????? ??? ???????? ???? ?????? ?????.
?????? | 2011 | ??? 3 | ????? 1
70 ????????? ?.?., ????? ?.?.,
????????? ?.?., ?????????? ?.?.
???????????? ?????? ????????? ?????????? ? ????????????????? ????????? ?? ????????? ???????. ?????????? ??????????? ?????????? ??? ???????? ?????? ????? ?????????? (mpich
? ?.?.), ??????????? ???????????? ?? ?????? ???
???? ??????????, ?? ? ?????????????? ????????.
?????? ? ???? ?????????? ?????????????? ??????????, ? ??????? ??????????? ??? ??????? ??????? ???????? ???????, ???????????????? ???
??????????? ???? ???????????. ???????, ???????? ????? ????????????? ??????????? ??????????? ??? ?????????? ?????????? (CUDA). ?????
??????? ??????? ??????? ??????? ?????????? ???????.
2. ???????????? ????
????? ??????? ?????? ???? ??? ???? ?????? ????? ???????? ?0, ?? ??????? ?????? ??????? ????? ??????????? ??????????????? ????? ?
??????????? н0 [5]:
?
? ? ??? ?
? ? ? ? ? ?
R(?) x ( l , l ?) R(? ?)L( ?, ?l ?) d?l ?,
╣ L(?, l ) ? L(?, l ) ?
?? ??
4? ??
??
?
?
?L(?, ?l )
? L0 ?(?l ? ?l 0 ), L(?, ?l )
? 0;
?? 0, ( z? , ?l ) ? 0
??
???0 , ( z? , ?l ) ? 0
(1)
?
??? L (?,н) - ?????? ?????????? ?????? ????????? ???? ? ????? ?? ?????????? ??????? ? ?? ??????????? н. ??????? ????????? ??????? ????????? OXYZ, ??? OZ ?????????? ???? ??????????????? ??????? ????, z? - ????????? ???
????? OZ, н = { 1 ? ╣ 2 cos?, 1 ? ╣ 2 sin?, ?}, н0
?
= { 1 ? ╣ 2 0, ?0}, ?(н, z? ), ?0(н0, z? ). x (н,н') - ??????? ?????????,
? ? ??????? ???????????? ??????
????. R (?)- ??????? ????????? (???????) ?????????? ?????? ??? ???????? ????????? ??????????. ???? ? - ??????????? ????? ??????????? ( z? вн) ? (нвн'), ? ?? ? ????? ??????????? (нвн') ? (н'в z? ). ????????? ??????? ???????? ???????? ???????? ??????, ??????? ???????? ??????, ?????? ? ?????, ??????? ? ???????
???????? ??? ????????.
??? ?????????? ??????? ???? ??????????
????????? ???????? ????????? ???????, ??? ???????? ????? ?? ???? ???????: ??????????? ???????? (??) ? ?????????? ??????? (???), ??????????????? ??????? ???????? ? [6]. ??? ?????????? ????????? ???????? ???: ???? ??????????? ?????????? ??????? ?????????, ??? ??????????? ???????? ???????????? ??????? ????????? ???????.
1 ????? | ??? 3 | 2011 | ??????
?????????????? ??????????
??????? ???????????? ? ???????, ???????????? ???????? ??????????? (?????? ??????? ╚????-????╩, ???????????? ?????????? ?
???????), ?????? ??????????? ?????????????
????????? ? ???? ???????? ????????? ?????.
??? ????? ???????????? ? ??????? ??????????
??????? ?????? ? ???? ????? ???? ?????? ? ????????????, ?????????? ??? ???????????, ? ?????????? ? ??????? ??????? ?? ??????? ??????????:
?
?
?
(2)
L (?,н) = L ?(?,н) + L r(?,н).
??????????????, ??? ???????????? ????? ????? ????? ????????????, ??? ???????? ????????
???????? ?? ??? ????? ???????????? - ????? ????????? ????????????? [7].
??????? ????? ????????????? ??????????
Eddington [8] ? Milne [9], ??????? ?? ??????? ?????? ???????????? ?????????. ? ?????? ??????????????? ????????? ????????? ??? ??????? ????????? ????? ???
?
?
? ?
L ?(?,?,?) ? L 0(?,?,?) = e-?/?0 R (?) L 0?(н ? н0),
? 1 ?
? p cos ? ?
0?
L0 = E0 ?
,
? p sin ?0 ?
?
?
? q ?
?
??? L 0 - ?????? ?????????? ?????? ????????? ?????????, E0- ?????????? ? ????? ????????????, p,
q ? ??????? ???????? ? ??????????? ??????????? ?????, ?0 ? ?????? ???????????.
??????????? (2) ? (1) ???????? ??????? ??????
?? ?????????????
? ? L r (?, ?l ) + L r (?, ?l ) =
?╣
??
?
? ? ?? ?l ? ) d?l ? + ?(?, ?l );
?
?
x
R(
)
(
l
,
l
)
R(
)
L
(
,
?
?
?
?=
r
?
? 4?
? L (?, ?l )
= 0; L r (?, ?l )
= 0,
r
?= 0, ╣> 0
?=?0 , ╣< 0
?
?
? ???????? ?????????? ? ?????? ?????
? ?? R(?) x (l , l ?) R(? ?) L a ( ?, ?l ? ) d?l ? ?
? (?, ?l ) =
?
4?
? L a (?, ?l ) ?╣
? L a (?, ?l ),
??
(3)
(4)
??????? ? ?????? ????????? ??????? ?????????
????????? ?????
? ?? / ╣0 e
R(?) x (?l , ?l 0 ).
?(?, ?l ) =
4?
(5)
?????????????? ??????????
??????? ?????????-???????????
??????? ?? ???????? ??????????
????????????? ?????????? ??? ? ??????
? Lmc (?, ╣) m
+ L c (?, ╣) =
???? ????? ?????? ???????, ????????? ? ??????- ╣
??
??? ???????? ??? ???????????? ???????????? ????K
m
m 1 m
????????? ??????????? ????????? ? ??? ?? ??????- = ? (2k + 1)?
?
?
?
(
)
(
)
L
╣
?
?
╣
?
k
k ?
k
c ( ?, ╣ ) d ╣ +
2 k =0
??? ????????, ??? ?? ?????? ??????? ???????? ???1
??????? ?????? ?????????? ???????? ? ????????????. +?
(?, ╣), c = 1, 2,
(9)
? ?????? ??????????? ??????? ????????? ?????????
??? M ? ????? ?????? ???? ????? ?? ???????,
?? ????????? ????????? R , ??? ???????? ?????????m ? 0, M ; ???????? ??????? ????????? ??????????? ????????? ? ?? ????????? ???????????? ??????????? ? ???? ?????????? ?? ?????????? ????? ???????? ??? ??????????? ???????.
???????? ????????
3. ????????????? ????
? k = TSC xk TCS , xk ? ?? xrsk ?? , [ xCP (cos ? ) ]rs =
Ku??er-Ribari? [10] ?????????? ????????K
???? ??? ??????????? ??????????? ??????????? = ? (2k + 1) xrsk Prk, s (cos ? ),
k = max( r , s )
(10)
?????
? L+2 ?
?Q ? iU ?
r ? s ????????? ?????? ???????? ╠0 ? ╠2, K
?L ?
?
?
? ?????????? ???????? ? ?????????? ???????
1 ? I ?V ?
+0 ?
?
=
=
LCP =
????????? ?? ?????????? ??????????? ????? L?0 ? 2 ? I + V ?
????;
? ?
?
?
?Q + iU ?
? L?2 ?
?Q mk (╣)
0
0
0 ?
?
i
I
0
1
0
?
?
?
?? ?
k
0
R mk (╣) ? Tkm (╣) 0 ?
?
?1 0 0 ?1? ? Q ? ? m (╣) =
,
1
? 0 ? Tkm (╣) R mk (╣)
? ? ? = TCS L SP ,
0 ?
= ?
?
?
2 ?1 0 0 1 ? ?U ?
0
0
Q mk (╣) ??
?? 0
?
?? ?
?0 1 i 0 ? ?V ?
?1 (?) = diag ( cos ?, cos ?, sin ?, sin ? ) ,
?1
TSC ? TCS
,
(6)
?2 (?) = diag ( ? sin ?, ? sin ?, cos ?, cos ? ) , (11)
??? ???????? ??? ???????? ? ????????? ???????????? ??? ????????????? ??????? ????????? ????- Qln (╣) = (k ? m)! Pkm (╣),
k
(k + m)!
?????? ??????????? ??????? Pmn
(cos ?), ??? ??????? ??????????? ??????????? ??????? ??????? R n (╣) = 0.5i m ( P k (?) + P k (?) ) ,
l
m ,2
m , ?2
????????
k ? ?
Tln (╣) = 0.5i m ( Pmk ,2 (?) ? Pmk ,?2 (?) ) .
e ? im? Pmn
(l ? l ?) e ? in?? =
k
??? ??????? ??????????? ????????? ?
k ?
(l ? z? ) Pqnk (z? ? ?l ?) eiq?? ,
= ? (?1) q Pmq
??????? ??? ?????????? ?????????, ???q =? k
????? ? ????????? (9), ? ?????? ??????? ????? = ? ? ??.
(7) ?????? ?????? ? ???? ??????? ????????? ????????? ? ???? ?????? ??? ???????????? ? ??????? [14]:
???? ?????????? ????????????, ??? ????????- 1 m
m
?? ????????????? ??????????? ????????? ??- ? ? k (╣ ?) L c (?, ╣ ?)d ╣ ? ?
????? ??????? ?????? ?????????, ??????? ??- ?1
N 2
??? ?????????? ??????? ???????? ??? ????? 0.5? w j ? mk (╣ +j ) Lmc (?, ╣ +j ) +
?????? ??????????? ??????? ????? ???????j =1
?? ? c??????? (SP ? Stokes presentation) ????N 2
?????????. ?????????? ??????? ?????? (3) ? +0.5? w j ? mk (╣ ?j ) Lmc (?, ╣ ?j ),
j =1
????? [11-13]
(12)
m
╠
M
??? ╣ j = 0.5(?j ╠ 1), ?j, wj ? ???? ? ???? ?????????
??1 (m?)L1 (?, ╣) + ?
L r (?, ?l ) = ? (2 ? ?0,m ) ?
m
?,
?????????? ??????? N/2. ???????, ??? ╣ ╠j ????m=0
?? +?2 (m?)L 2 (?, ╣) ??
(8) ????????? ?????????? ╚+╩ ? ?????????? ╚-╩ ????? ???????? ???? ? ????
?????.
?????? | 2011 | ??? 3 | ????? 1
71
72 ????????? ?.?., ????? ?.?.,
?????????????? ??????????
????????? ?.?., ?????????? ?.?.
?????????????? ??? ????????????? m ? c (??????? ?? ???????? ? ?????? ?????????) ????? ??????? ???????????? ???????????????? ?????????
d ╠
Li (?) + L╠i (?) =
d?
╣i╠
m + +
K
m ╠ ? ? k (╣ j ) Li (?) + ?
?N2
?+
= ? w j ? (2k + 1)? k (╣i )? k ? m ? ?
?
?
4 j =1 k =0
? +? k (╣ j ) Li (?) ?
╠
+? i (?),
??? L╠i (?) ? Lmc (?, ╣i╠ ), ? i╠ (?) ? ? cm (?, ╣i╠ ).
?????? ????????? ??????? ? ???????
(13)
A?
m + m +
?K
? ? (2k + 1)? k (╣i )? k ? k (╣ j )
k =0
??K
m ? m +
?
? ? (2k + 1)? k (╣i )? k ? k (╣ j )
? k =0
???
?
(╣i+ )? k ? km (╣ ?j ) ?
k =0
?,
K
m ? m ? ?
(2k + 1)? k (╣i )? k ? k (╣ j ) ?
?
k =0
? (14)
K
m
? (2k + 1)?
k
? I (?, ╣1╠ ), Q(?, ╣1╠ ), U (?, ╣1╠ ), V (?, ╣1╠ ), ? , ?
L ╠ (?) = ?
?
╠
╠
╠
╠
?? I (?, ╣ N /2 ), Q(?, ╣ N /2 ), U (?, ╣ N /2 ), V (?, ╣ N /2 ) ??
T
? ?????????? ??????????? ????????? ???????.
????????? ?????????? ????? ???????? ??????? ???????? ????, ?? ??? ??????? ???????,
??? ????????? ???????? ??????? (13) ? ?????????
????
d L(?)
= ? BL(?) + M ?1 ?(?),
d?
?1 B ? M (1 ? AW).
(15)
4. ??????? ??? ??????? ?????
??????? ?????????? ??????? ????????? ????? ?????????????
???
?1 ? ????????? ????? [15]
??0
?1
? U L(0) + e U L(?0 ) =
= ? e?? U ?1 M ?1 ?(?, ╣ 0 )d ?,
?0
0
(16)
?
??? ?U - ???????
? ? ??????????? ???????? ???????
?
B , ? = diag(? ? , ? + ) - ???????????? ??????? ???-
???????? ????????
?
? ????????????? ?? ??????????? ???, ??? ? ? = ?? + .
???????? ???????
(16) ??????? ? ????????
???
?
?
????????????? ? ? , ??? ????????????? ? + ????????? ? ???????, ??? ???????? ? ???????? ????????? ??????????????? ????????? ???????.
1 ????? | ??? 3 | 2011 | ??????
??? ?????????? ????? ???????????? ??????????
?????????????? [15] ? ??????? ????????? (16)
?????
?? ???????
S = diag(1, e ??+ ?0 ),
(17)
??? ???????? (16) ? ?????????? ?????????? ?????????
?1 ? L + (0) ? ?1 ? L + (?0 ) ? ? J ? ?
? SU ? ? = ? ? ,
? + HU ? ? L ? (0) ?
? L ? ( ?0 ) ? ? J + ?
(18)
??
? ? ?0
0
??? H? Se = diag(e , 1),
? J ? ? ?0 ??
J ? ? ? = S ? e U ?1 M ?1 ?(?, ╣ 0 ) d ?.
0
? J+ ?
? ????????? (18) ???????
? ????????
?
????????
????????? ?? ???? ??L + (0), L ? (?0 )
?????? ? ???????????? ?????????? ?????????,
?
?
? L ? (0), L + (?0 ) ???????? ??????? ???????? ?
?????????? ????????? ?? ???? ??????: ?????????? ? ?????????. ????? ??????? (18) ????????????
?????????
??
????
???????
? L ? (0) ? ? F? ? ? R ? T? ? ? L + (0) ?
?
? = ? ? + ? ? ? ?,
L
(
)
L
(
)
?
?
F
T
R
? + 0 ? ? +? ? + +?? ? 0 ?
(19)
???
? F? ? ? J ? ?
? ? = A ? ? ,
? F+ ?
? J+ ?
? R ? T? ? ? u11
? e?? ?0 u12 ?
?,
? ? = A ? ?? + ?0 T
R
e
u
u
?
?
??
+
+
?
?
21
22
?
?u u ?
U ?1 ? ? 11 12 ? ,
? u 21 u 22 ?
?1
? ? u12
e?? ?0 u11 ?
A??
? .
?? ? ?? ? e + 0 u 22 u 21 ??
??????????? ????????? ??? ?????????? ?????? ????????? ? ?????? [5].
5. ????????????? ?????????????
???????????? ?????
????????????? ????? ???????????? ? ????
?????? ???????????? ?????????? ????????? [8,
9], ????????? ???? ????????? [16], ? ??? ?????????
? ??????????? ???????????
[4]. ???????? ???
??? ????? ?????? ? L ?(?,н) ??? ????????? ?????????
? ??????????? ??????????? [12, 17, 18]. ?????
?????? ???????? ?????? ? ??????? [5, 11, 12] ??
?????? ??????????? ??????????? ?????? ??????????? ???????? (???). ? ???? ?????? ???????????? ????? ????? ???
?????????????? ??????????
L a (?, ?l , ?l 0 ) =
=
M
? ? (2 ? ?
c =1,2 m = 0
m ,0
K
2k + 1 m
)?c (m?)?
? k (╣) Zk (?)? km (╣ 0 ) D c L0 ,
4?
k =0
(20)
?????
Zk (?) = exp ?? ? (1 ? ?? k )? ╣ 0 ?? ,
D1 =diag(1 1 0 0), D 2 =diag(0 0 1 1).
?????????? ????????? ????????? ? (4) ? ????????? ??????????? ?????????????? [6, 14], ????? ???????? ????????? ??? ??????? ??????????
╣ 0 ? cm (?, ╣i ) =
K
= ? (2k + 1)(╣i ? ╣ 0 )? km (╣i ) d k Zk (?)? mk (╣ 0 ) D c L 0 +
k =m
m ?? mK (╣i ) d K +1 Z K +1 (?)? mK +1 (╣ 0 ) ? ? + A K +1 ? ? Dc L0 ,
K
K +1
?? ?? m (╣i ) d K Z K (?)? m (╣ 0 ) ??
(21)
k
1
(k 2 ? m 2 )(k 2 ? s 2 )?rs .
??? d k = 1 ? ?? k , ?? A m ?? =
rs
k
????? ?????? ?????? ??????? ?????????? ????
??????? ????? ??????????, ??? ????????? ????????? ????? ???????????? ???????? M.
???????, ??? ??? ??????? ????????? ???????????? ????? ?? ?????? ??? ?????????? ????????? (19). ???????? ?????? ?????????? ??????? ??????????. ????????? (19) ????????? ?????? ? ?????????? ?????, ?? ????????? ??????? ? ???????? ???????????? ?????. ???????, ??? (19) ???????? ??????? ????????????? ???????? ???????
?????? ??? ????????????????? ???? ?? ??????
?????????? ???????.
?????????? ??????? (19) ???????? ???????.
?? ?? ??????? ????? ?????????? ????????????? ??????? ??????????? ? ?????????? ?????????. ??????? ?? ??? (1) ? ??????? (19) ????????? ?????????? ???????????? ?? ??????? (2) ? ?????? ????????? ????????? (?????????????) ????????? ??????????? (12).
??? ?????????? ??????????? ??????? (19) ???????? ????????:
? ?????????? ????? ?????? ? ????? ????????????
??????? ??? ????????????? ???? (12);
? ?????????? ??????? ?????????? (4);
? ??????????? ??????????? ???????? ? ????????
??????? ??????? (16);
? ???????????? ?????? (19).
???????????? ?????????? ?????????? ????????? ? ???????????? ??????? ??????? ?? ??????? ??????, ???????? ? (19). ???????, ??? ?????????
??????? ?????????-???????????
??????? ?? ???????? ??????????
(19) ???????? ??? ?????????? ?????, ???????
?? ?????? ???????????? ?? ??????? ??????????? ??????????? ????? ? ?????????? ?????????, ??????? ???, ????????? ????? ?? ????? ???????? ???????????? ????? ???????. ? ????? ?????? M?N?K, ?? ??? ??????? ????????? ???????????? ????? ???????, ????? ??????? ??????????
????? ?????? ? ??????????? ????????????? ??
????, ????????, ??? M<<N<<K, ??? ???????????
??????? ????????????? ?????????. ? ???? ????? ?????? ????????, ??? ?????????? ?????????
???????????? ?????????? ????????? ??? ??????
????????????? ?????????.
6. ?????? ??????? ??????????? ?
?????????? ???????? ?????????
??? ??? ???? ????????, ????????? ?????? ????? ??????????? ???????????? ????????? ?? ???????? ????? ??? ?????????? ????????? (19). ??? ???
??????? ?????????? ?????? ??? ???????? ????
????? ????????? ???????: ???????????? ??????,
?????????? ??????????? ???????? ? ???????????
????????, ?????????? ???????? ???????.
??? ??????? ???? ????? ????? ??????????? ?????????? ?? ?????? ?? ? ??????? [19].
?????????? ????????? ?????????, ? ??????? ??????? ????????? ???????????? (LAPACK, IMSL,
NAG ? ??.). ??? ?????? ???????? ?????????? ???? ?????????? ?????????. ??? ?????????
Intel ?????????? Math Kernel Library (MKL) [http://
software.intel.com/en-us/intel-mkl],
???????????????? ?????????? ??? ?????????? Intel. ? ????????? ???????????? ?????????? ????? ??????? ???
????, ??? ??? ???????????? ??????????????? ? ????????????? ???????????????? ?????? ?? ?????????
????? ????, ??? ??????????? ??????????? ????????.
????? ? ?????? MKL ?????? ?????????? ?????????
???????? VML (????????, ?????? ????? ??????????? ?? ????????? ???????) ? ???????????? ??????.
????????????? MKL ?? ??????????? Intel ?????
??????, ??? ?????????? ?????????????? ??????, ????? ??? MATLAB ? MAPLE, ?????????? MKL, ???????? ??????? ???????? ??????????.
?????? ?? ???????? SCALAPACK ? ??????
LAPACK, ?????????????? ???????????? ??????????. ??? ?? ?????????? ?????????? ??????
MPICH. SCALAPACK ???????? ? ???????? ???????? ?????????? ?????????????????? ??? ?????????? ??????????? ??????? ?????????? ??????????????? ?????????? ????.
?????? | 2011 | ??? 3 | ????? 1
73
74 ????????? ?.?., ????? ?.?.,
????????? ?.?., ?????????? ?.?.
??????? ???????? ????????????? ???????????? ?????????? ???????? MPI - ??????????? ????????? ??? ???????? ?????????? ????? ?????????? ??????. MPI ????? ????????? ??????????
??? ?????? ????????? ? ? ??????? ?????????
???????????????. ?????? ??? ?????????? ?????
????????? ???????:
? ??????????? ?????? ???????? ???????? (MPI_
COMM_RANK)
? ??????????? ?????? ?????????? ?????????
(MPI_COMM_SIZE)
? ??????????? ? ????????????? ???????? ???????? ? ????? ?????? (MPI_(I)SEND, MPI_(I)Recv)
? ?????? (MPI_Barrier)
?????????? MPI ???????? ???????? ??? ????????????????? ?????????? ?? ???????????? ??????????.
? ?????? ???????????? ????????? ??????? ????????? ????? ??????-???????????? ???, ? ??????????? ? ??????? ??????? ???????? ????? ?????. ??? ?????? ? ???? ?????????? ??????? ??????????? ??????. ???????? ???? ???????????
?????? ??????? ? ???, ??? ??? ?????? ? ????????,
??????????? ????????? ??????? ???????, ?????????? ??????? ??????? ?????? ????????? ???????? ? ?????????????? ??????????, ?? ???????
????? ???????????? ??????? ????????? ?????????, ???? ???? ???????.
?????????? ????????? ???????? ???????? ??????????? ??????: compressed sparse row (CSR)
format, compressed sparse column format (CSC),
Coordinate Format. ??? ??? ???? ?????? ???????? ???????????????? ? ??????????? ?? ???? ????????. ????????, ??? ????????? ?? ???????????
??????? ????? ????? ?????? ?????? CSC, ??? ???
???????? ? ???? ?????? ??????????? ?? ???????
???????. ????????, ???????
? 1 0 0 0?
?
?
0 0 7 0?
B=?
? ?3 0 0 4 ?
?
?
? 0 5 0 2?
? ??????? CSC ??????????? ? ???? ??????? ????????: Values = [1, -3, 5, 7, 4, 2], Rows = [1, 3, 4, 2, 3, 4],
Pointer B = [1, 3, 4, 5], Pointer E = [3, 4, 5, 7].
??? ??? ???? ?????? ???????? ???????????????? ? ??????????? ?? ???? ????????. ????????,
??? ????????? ?? ??????????? ??????? ?????
1 ????? | ??? 3 | 2011 | ??????
?????????????? ??????????
????? ?????? ?????? CSC, ??? ??? ???????? ? ????
?????? ??????????? ?? ??????? ???????.
? ????????? ????? ??????? ???????????? ?????????? ?????????? ?? ??????????? ???????????. ?????????? ???????? ?????????? ??????????????? ?????????? CUDA. MATLAB 2011a
???????????? ?????? ? CUDA ?? ??????????? ? ?????????????? ???? 2.0.
7. ?????????? ? ??????????
????????, ????????? ? ?. 5.3, ??? ???????????????? ? ???? ?????????? ???? MVDOM ?? ????? FORTRAN ? ? ????? Matlab. ? ??????? 1 ???????????? ????????? ?????? ????? ??? ???? ??????
(???? 1 ? N = 101, K=500, M=32; ???? 2 ? N = 101,
K=1000, M=32). ????? ????????????? ?? ??????? Ubuntu 10.04, Intel Core 2 Duo 3GHz, 2 Gb
RAM, Intel Fortran Compiler 11.1 with MKL 10.2.
?????????????? ??? ??????????? (gfortran ? ifort),
????????? ?????? ??????????? ? ??????? ??????????? ??????.
???????, ??? MKL ?????????? ? ???????? ???
????????? ???? ?????????? (? ????? ?????? 2), ???
???????? ? ??????????? ?????????. ???????????
????????? ????????? ????? ????? ??????? ??????????? ??????, ???????????? ??? ??????????
???????????? ????? (20). ?? ???. 1 ????????? ??????????? ??????? ????? ?? K. ??????????? ??????? ???????????? ?????????? ??????? ? ??????????, ??????? ????? ????? ?????? ???????, ? ??
???????????.
????????? ???????????? ifort ? gfortran ??????????, ??? ?? ????????????? ???????? ????????? ? ????????? ????????????? ifort.
????????? ??? MDOM ??????????? ? ????? DISORT. ????????? ??????? ????? ????????? ???????? ??????????????, ??? ??? ??????? ?????????? ?????????? ???????? ??????
??????? 1
????????? ??????? ????? ??? ???? ??????
Runtime
I, sec
Runtime
II, sec
gfortran + LAPACK
240
530
gfortran + LAPACK + optimization
230
505
ifort + LAPACK
210
490
ifort + MKL
115
250
ifort + MKL + optimization
105
230
ifort + MKL + optimization +
sparse matrix
33
44
MATLAB 2010a
27
45
Features
??????? ?????????-???????????
??????? ?? ???????? ??????????
?????????????? ??????????
????????? ????????????
200
?????????? ? ??????????
ifort + mkl
ifort+ lapack
gfortran + lapack
f95 + lapack
180
with sparse matrixes
without sparse matrixes
100
80
60
40
20
0
0
200
400
600
K
800
1000
1200
???. 1. ??????????? ??????? ????? ?? K.
???????????. ?? ??????????? ???????? ??????? ???????? ??????? ???????, ??? ??? ???????????? ????? ????????, ????????, ??????????? ??????????? ???????. ??????????? ??????????? ????? ??????????? MDOM ? DISORT
?????????? 10-4. ??? ??????? M ???????????
?????? ?????????????. ??? ????? ?????????
?????? ?????????? ??????????? ??????? ???
??????? M, ? ??? ????????? ??????? ????????????? ???????, ??????????? ??? ?? ???????????? ??????????.
??? DISORT ???????? ?? ?????? [ftp://climate1.
gsfc.nasa.gov/wiscombe/Multiple Scatt]. ????????? ????????, ??? ????? ????? ????? ???? ??????? ?? ??????? ? ??????? ????????? ??????????? ????????? ZEROIT. ? ?????? ?????????? ??????? ?????????? (g<0.9) ????? ????? MDOM ? DISORT
??????????? ??????????. ? DISORT2.0 ??????????? ??????-M ????? ? TMS ?????. ??? ??????????? [20] ?????????, ??? ????? ????????? ???????????? ????? ? ??????????? ??????????? ???????? ??????????? ????? ???????????. ?????????? ????????? ???????? ???????????? ?? ???. 2. ???????? ????? MDOM (10 ??????) ??? ???? ?? ??????? ????, ???
? DISORT (100 ??????).
?0=10.0, ?0=45.0o, ?=1.00
0.18
0.08
0.04
90
0.4
110
120
130 140
?, ╟
a
150
160
170
180
0
90
120
150
180
? ? Sighting angle, degrees
x 10
?5
?10
90
?3
4
0
MVDOM
PSTAR
120
150
180
? ? Sighting angle, degrees
x 10
?6
2
?2
?4
90
0.1
100
0
0
0.3
0.2
0.06
0.2
0.1
U, r.u.
L ? НРМ.ЕД
L, НРМ.ЕД
0.5
0.1
0.3
MDOM
DISORT
0.6
0.12
Upward radiation ?3
x 10
5
0.4
2
0.7
0.14
a
?0=10.0, ?0=45.0o, ?=1.00
0.8
MDOM
DISORT
0.16
?????????? ????????? ???????????? ???????????? ? ??????? 2.
?????? ?????????? ???????? ? ? ?????????
?????? ? ???????????? ???? PSTAR ? MVDOM
??? ?????? ??????? ??????? Water Cloud C1,
?0=85?, ?0=10.0. ?????????? ??????? ????????????
?? ???. 3. ????????? ????????, ??? ? ?????? ??????? ???? ?????? ??????????? ? ????????? ?????????. ??????? ????? ??? MVDOM ??????????
50 ??????, ??? PSTAR ~150 ??????.
Q, r.u.
120
I, r.u.
Run?time, sec.
140
????? ?????, ???.
1.00
1.86
2.12
2.34
V, r.u.
160
??????? 2
0
0
10
20
30
40
?, ╟
50
b
60
70
80
???. 2. ????????? MDOM ? DISORT: a) ?????????? ?
b) ????????? ?????????.
90
?2
90
120
150
180
? ? Sighting angle, degrees
120
150
180
? ? Sighting angle, degrees
b
???. 3. ????????? MVDOM ? Pstar: a) ?????????? ?
b) ????????? ?????????
?????? | 2011 | ??? 3 | ????? 1
75
76 ????????? ?.?., ????? ?.?.,
????????? ?.?., ?????????? ?.?.
? ?????? ?????????????? ????????? ???????????? ????? ?? ?????? ??? ?????????? ?????? ?????????? ?????????? ????? 100 ??, ?? ????
????? ???????? ?????? ???? ??????????. ? ????????? ????? ???????? ?????????? ??????????
????????????????? ?????????? ???????? MPI,
OpenMP ? CUDA.
??? ??????? ???????????? ?????????? MPI
???? ???????? ????????? ???????????, ???????
???????, ??? ??? ?????????? ????????? ??????
?????? (?????? ??????? ???????????? ????? ??????????) ??????? ???????? ????? ??????? ????????? ?? ???? ???????? ???? ?????????. ?????? ???
??????? ?????? ?????? ?????? ????????? ???????????? (???. 4). ????? ????, ????????????? ?????????? ??????? ?? ????????? ?????? ????? ??????????. ????? ???????, ????????????? ???????????????? ?????????? MKL, ?????????? ? ?????????, ????????????????? ????? ????, ?????????????? ????? ????????.
??? ????????????? ????????? OpenMP ??????? ??????? ??????? ????????? (? ?????? ?????? ???) ?????????. ????? ?????? ????????????? ? ???????????????? ????????? ?? ????? ???????????, ??? ??? ??? ???????? ???????? ? ?????? ? ???? ?? ?????????, ? ????????? ????? ????
(????????, ???????????? ???????????? A(1:N) =
B(1:N)+C) ?????????? ifort ????? ??????????????
??????????????.
? ?????? [21] ??????????? ???????? ???????????? ????????????? ?????? ?????? ?????? ?????????? ? ???, ????? ???????? ???????? ?????? ????? ?????????? ???? ??????????. ? ?????? ?????????? ????????? ??????? ???? ??? ???? , ??????????? ?? ????????? ???????? ???????, ????????????? ?????????? ??????????? ? ????? ??????
???. 4. ????????????? ????????????? ?????????? MPI.
1 ????? | ??? 3 | 2011 | ??????
?????????????? ??????????
?????????? ?? ??? ????????? BLAS ? LAPACK.
? ???? ?????? ??????????? ????????????? ??????
????? ?? ???? ?????????? CUDA. ??????????? ??
????????????? ??????? ??????????? ? ????????
??? ??????? ? ????? MATLAB ????????? ??? ????????? ????????? ?????????? ? GPU - ?????? ????????? ? CPU?. ????????????? CUDA ? ????? MATLAB ??????? ? ????????? ?? 20 %. ?????
?????????????? ????????? ?? ????????? ? ??????? ????????? ?? ???????? ?????????? ?? ?????? ?????-????? ??????????? ????? ?????????. ???????? ?????????? ????????? ? ?????????? ??????????? ???????? ? ?? ?????????????? ? ????? MATLAB ?? ??????????? ??????????.
?????????? ?? ?????? ????????? ????????????
????? ??????? ? ????????????? ?????????? ??????? ??????, ??????????? ? ???????. ???????
??????? ?? ????????????? CUDA ????????? ??
????? ???????.
8. ??????????
????????? ????????? ???????? ????????? ????? ????????????? ???????? ? ?????????? ?????????. ??? ?????????? ????????? ??????? ??
?????????? ?????????? ???????????? ?????????? ????????? ???????. ???????? ?????????? ?????????? ??? ??????????? ??????????? ? ?????????????? ?????????? MKL. ????????????? ?????? ??????????, ??? ????? ????????? ???????????? ????? ???????? ? ???????? ?? ??????? ???
??????? ??????????? ????????? ? ??????? ?????????? ??????.
??????????
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S, Hess M. An efficient inversion algorithm for
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103:193-208.
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discrete ordinate radiative transfer model for
atmospheric remote-sensing retrieval. J Quant Spectrosc
Radiat Transf, 2001, 68:689-735.
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for Japanese GOSAT project. Proceedings of the XXIV
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approximation. J Quant Spectrosc Radiat Transf, 1988, 40:51?69.
?????????????? ??????????
??????? ?????????-???????????
??????? ?? ???????? ??????????
5. Budak VP, Klyuykov DA, Korkin SV. Complete
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horizontally homogeneous slabs. J Quant Spectrosc Radiat
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6. Wang MC, Guth E. On the theory of multiple
scattering, particularly of charged particles. Phys.Rev.,
1951, 84:1092?111.
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York, Macmillan, 1962, 276 p.
8. Eddington AS. On the radiative equilibrium of the
stars. Month Not R Astroph Soc, 1916, 77:16?35.
9. Milne EA. The reflection effect in eclipse binaries.
Mon Not R Astroph Soc, 1926, 87:43-55.
10.Ku??er I, Ribari? M. Matrix formalism in the theory
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through an arbitrary thick scattering atmosphere. J Quant
Spectrosc Radiat Transf, 1980, 24:391?406.
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NJ, Van Nostrand, 1963, 171 p.
17.Kisselev V. Peaked phase function approximation in the
solution of radiative transfer equation. Proc of SPIE, 2005,
5829:63-73.
18.Budak VP, Kozelskii AV, Savitskii EN. Improvement of
the spherical harmonics method convergence at strongly
anisotropic scattering. Atm Ocean Opt J, 2004, 17:28-33.
19.Press W, Teukolsky S, Vetterling W, Flannery B.
Numerical Recipes in Fortran 77: The Art of Scientific Computing.
Cambridge: University Press, 2007, 818 p.
20. Mobley CD, Sundman LK, Boss E. Phase function
effects on oceanic light fields. Appl. Opt., 2002, 41:10351050.
????????? ?????? ????????,
??? (??), ??????? ????? ?????? ? ???????? ???????,
111250 ??????, ?????????????????, 14,
???. +7 903 222 0899, v.af@mail.ru
????? ???????? ????????,
?.?.?., ?????????,
??? (??), ???, ??????? ????????????,
111250 ??????, ?????????????????, 13, ?.?,
???. +7 985 763 5239, BudakVP@mpei.ru
????????? ??????? ?????????,
??? (??), ??????? ????? ?????? ? ???????? ???????,
111250 ??????, ?????????????????, 14,
???. +7 905 521 6815, efremenkods@gmail.com
21.Peng K, Gao X, Qu X, Ren N, Chen X, He X,
?????????? ????????? ????????????,
Wang X, Liang J, Tian J. Graphics processing unit
parallel accelerated solution of the discrete ordinates
for photon transport in biological tissues. Appl. Opt.,
2011, 50:3808-3823.
??? (??), ??????? ????? ?????? ? ???????? ???????,
111250 ??????, ?????????????????, 14,
???. +7 905 541 0495, lem_a@mail.ru
?????? | 2011 | ??? 3 | ????? 1
77
78
information technologies
HARD-SOFTWARE INFLUENCE ON CALCULATING
SPEED OF RADIATIVE TRANSFER EQUATION SOLUTION
ALGORITHM IN MEDIUM SLAB
Afanasiev V. P., Budak V. P., Efremenko D. S., Lubenchenko A. V.
Moscow Power Engineering Institute (Technical University),
Krasnokazarmennaya, 13, 111250 Moscow, Russian Federation
BudakVP@mpei.ru
It is shown that in slab geometry the discretized radiative transfer equation (RTE) has the unique analytical
solution. Since the solution represents as the linear matrix equation, where there are optimized libraries, the
only possible algorithm solution exists. The RTE discretization is probable only after the solution anisotropic
part selection, including all the singularities. The various implementations of the RTE solution algorithms
differ by methods of the anisotropic part selection, among which one the most effective is the small angle
modification of spherical harmonics method. The effect of hard-software on the implementation efficiency of
RTE solution algorithms in the slab is analyzed.
Keywords: radiative transfer equation, slab geometry, anisotropic scattering
PACS 42.68.Ay
Bibliography ? 21 references
Received 17.05.2011, revised 20.05.2011
rensit, 2011, 3(1):69-78
references
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1. Doicu A, Schreier F, Hilgers S, Bargen A, Slijkhuis 15.Karp AH, Greenstadt J, Fillmore JA. J Quant Spectrosc
S, Hess M. J Quant Spectrosc Radiat Transf, 2007,
Radiat Transf, 1980, 24:391?406.
103(2):193-208.
16.Sobolev VV. A Treatise on Radiative Transfer. Princeton, NJ,
2. Spurr RJD, Kurosu TP, Chance KV. J Quant Spectrosc
Van Nostrand, 1963, 171 p.
Radiat Transf, 2001, 68:689-735.
17.Kisselev V. Proc of SPIE, 2005, (5829):63-73.
3. Yokota T, Oguma HM, Morino I, Inoue G. Proc. 24th 18.Budak VP, Kozelskii AV, Savitskii EN. Atm Ocean Opt J,
Intern. Symp. on Space Tech. Science. Miyazaki, Japan, 2004,
p. 887.
4. Nakajima T, Tanaka M. J Quant Spectrosc Radiat Transf,
1988, 40:51?69.
5. Budak VP, Klyuykov DA, Korkin SV. J Quant
Spectrosc Radiat Transf, 2011, 112:1141?1148.
6. Wang MC, Guth E. Phys.Rev., 1951, 84:1092?1110.
7. Krylov VI. Approximate calculation of integrals. New
York, Macmillan, 1962, 276 p.
8. Eddington AS. Month Not R Astroph Soc, 1916, 77:16?
35.
9. Milne EA. Mon Not R Astroph Soc., 1926, 87:43-55.
10.Ku??er I, Ribari? M. Opt Acta, 1959, 6:42?51.
11.Budak VP, Korkin SV. J Quant Spectrosc Radiat Transf, 2008,
109:220?34.
12.Budak VP, Klyuykov DA, Korkin SV. In: Light Scattering
Reviews 5: Single Light Scattering and Radiative Transfer. Ed. A.A.
Kokhanovsky. Berlin, Springer Praxis Books, 2010, pp. 147204.
13.Siewert CE. J Quant Spectrosc Radiat Transf, 2000, 64:227?
254.
1 ????? | ??? 3 | 2011 | ??????
2004, 17:28-33.
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Numerical Recipes in Fortran 77: The Art of Scientific Computing.
Cambridge, University Press, 2007, 818 p.
20. Mobley CD, Sundman LK, Boss E. Appl Opt, 2002,
41:1035-1050.
21.Peng K, Gao X, Qu X, Ren N, Chen X, He X,
Wang X, Liang J, Tian J. Appl Opt, 2011, 50:38083823.
1/--страниц
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