??????????GCR????

1
??????????GCR????

• ???????????? ?? ?
• ???????????? ?? ??
• ???????????????????
• ?? ??
• ???????????? ?? ??

2
??????????
?????????????????????
3
GCR????

• CG???????????????????Eisenstat, 83
• Arnoldi??????????????????? (GMRES???????????)

GCR????

• ?????????????
• ??????????????
• GMRESR?Vorst,91????????????

4
GCR?? ???

• ???????(O(k 2N)????3?)
• ??????????(GMRES??2?)

????! ???????!
5
?????

• ???????efficient GCR(eGCR)???????Yang,95

??????????????????? ??????????????!!
6
????2????????

• Memory efficient GCR(meGCR)?
• ????eGCR??????
• ???????eGCR???????????????
• Unrolled GCR(uGCR)?
• ?????????????eGCR??????

7
Efficient GCR?
8
Efficient GCR?
??????p?????????? ???p?Ap????????????
9
Efficient GCR?
10
Memory efficient GCR?
??????????Ap?????
????eGCR?????
????u??????

11
Unrolled GCR?
?????????????? ???Air0????????????
Air0?dominant???????? ??????????? ???????????
????(BLAS3)??????? ???
????????????????
??????? meGCR???
12
??????
???n? ???-?????
1???????????
???? dmv dp daxpy smv prec bin kmv Dmm
??? 2kn 2n 2n
GCR 3(k-1) 2k 2k-1 k k 0 0 0
eGCR 2k-1 2k k k k 1 1 0
meGCR 2k-1 2k k k k1 1 2 0
uGCR 0 0 0 k k1 1 4k 1
? ????????? ????????????
k ???????(????) n ????? (?????)
13
?????????1(????????????)
Vector of length n Buffer of size k 2
GCR 2k3 0
eGCR 2k3 2
meGCR k3 2
uGCR k2 5
k ???????(????) n ????? (?????)
14
?????????2(???????????????????????)
method ?????????
GCR
eGCR
meGCR
uGCR
k ???????(????) n ????? (?????) z
??????????????
15
????

• ??? HITACHI SR2201
• (????????????)
• CPU 300MFlops 1024PE
• Main memory 256MB/PE
• Communication 300MB/s
• ???????MPI (Message Passing Interface)

16
Problems

• Problem 1
• Toeplitz??
• Problem 2
• ???????????????(2??)
• Problem3
• ???????????????(3??)

17
meGCR??????(??)
????(?) ???????32
?? Problem 1 Problem 2 Problem 3
??? 400,000 160,000 64,000
????? GCR 22.8 4860 37.8
????? eGCR 18.3 3440 27.7
????? meGCR 17.9 3450 28.7
????? (B-ILU(0)) GCR 21.2 938 21.9
????? (B-ILU(0)) eGCR 19.8 812 19.9
????? (B-ILU(0)) meGCR 20.1 825 20.1
18
meGCR??????(????????)
Problem 1 ( n4,000,000 )
Problem 2 ( n160,000 )
Problem 3 ( n512,000 )
???????????32
19
meGCR??????(???B-ILU(0)???)
Problem 1 ( n4,000,000 )
Problem 2 ( n160,000 )
Problem 3 ( n512,000 )
???????????32
20
uGCR??????
??????? 8
????? ????? B-ILU(0)??? B-ILU(0)???
Iteration Time Iteration Time
Problem 1 (n400,000) GCR 46 18.5 17 20.3
Problem 1 (n400,000) eGCR 46 15.4 17 18.2
Problem 1 (n400,000) meGCR 46 15.4 17 19.5
Problem 1 (n400,000) uGCR 55 13.5 25 26.7
Problem 3 (n64,000) GCR 1096 64.8 150 30.6
Problem 3 (n64,000) eGCR 1096 53.6 150 29.6
Problem 3 (n64,000) meGCR 1096 55.7 150 31.7
Problem 3 (n64,000) uGCR 1053 43.0 150 30.1
21
??????

• ???????????
• ???????????
• ???????????
• ??????

• GCR??2?????????????
• Memory efficient GCR?
• ???????????????
• ??????????????????????????????
• Unrolled GCR?
• ??????????????????????

?????????????!
???????????????????