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神戸大学 大学院システム情報学研究科 計算科学専攻 陰山 聡
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【目次】
#contents
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* 前回のレポート（並列化）の解説 [#kc0e3327]
// * 前回のレポート（並列化）の解説 [#kc0e3327]
** 問題設定の復習 [#t00dbb2b]
#ref(problem_thermal.jpg)
- 2次元正方形領域 [0,1]×[0,1] での熱伝導を考える
- 境界をすべて0℃に固定
- 領域全体に一定の熱を加える
- ⇒ このとき，十分な時間が経った後での温度分布はどうなるか？
** 非並列版コードの復習 [#d5317fed]
熱伝導とは注目する一点が、「近所」の温度と等しくなろうという傾向である。
⇒周囲の温度の平均値になろうとする。
熱源があれば、一定の割合で温度が上がろうとする。
ヤコビ法アルゴリズム。
#ref(jacobi_method.jpg)
do j=1, m
do i=1, m
u_new(i,j) = ( u(i-1,j) + u(i+1,j) + u(i,j-1) + u(i,j+1) ) / 4 + f(i,j)
end do
end do
このヤコビ法に基づいたコード heat1.f90を前回の演習で学んだ。
** heat1_wi_comments.f90 （詳細コメント付きバージョン） [#ta5f8f8c]
後の改訂に備えて、まずこのheat1.f90を少し丁寧に復習しよう。
!
! heat1_wi_comments.f90
!
!-----------------------------------------------------------------------
! Time development form of the thermal diffusion equation is
! \partial T(x,y,t) / \partial t = \nabla^2 T(x,y,t) + heat_source.
! In the stationary state,
! \nabla^2 T(x,y) + heat_source = 0.
! The finite difference method with grid spacings dx and dy leads to
! (T(i+1,j)-2*T(i,j)+T(i-1,j))/dx^2
! + (T(i,j+1)-2*T(i,j)+T(i,j-1))/dy^2 + heat_source = 0.
! When dx=dy=h,
! T(i,j) = (T(i+1,j)+T(i-1,j)+T(i,j+1)+T(i,j-1))/4+heat_source*h^2/4.
! This suggests a relaxation method called Jacobi method adopted in
! this code.
!-----------------------------------------------------------------------
program heat1_wi_comments
implicit none
integer, parameter :: m=31 ! mesh size
integer :: nmax=20000 ! max loop counter
integer :: i,j,n
integer, parameter :: SP = kind(1.0)
integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
real(DP), dimension(:,:), allocatable :: u ! temperature field
real(DP), dimension(:,:), allocatable :: un ! work array
real(DP) :: heat=1.0_DP ! heat source term; uniform distribution.
real(DP) :: h ! grid size
! |<----- 1.0 ----->|
! j=m+1+-------u=0-------+ ---
! j=m | | ^
! . | | |
! . | uiform | |
! u=0 heat u=0 1.0
! j-direction . | source | |
!(y-direction) j=2 | | |
! ^ j=1 | | v
! | j=0 +-------u=0-------+ ---
! | i=0 1 2 ... i=m+1
! |
! +-------> i-direction (x-direction)
!
! when m = 7,
! |<-------------------- 1.0 -------------------->|
! | |
! | h h h h h h h h |
! |<--->|<--->|<--->|<--->|<--->|<--->|<--->|<--->|
! +-----+-----+-----+-----+-----+-----+-----+-----+
! i=0 1 2 3 4 5 6 7 8
!
allocate( u(0:m+1,0:m+1) ) ! memory allocation of 2-D array.
! you can access each element of u
! as u(i,j), where 0<=i<=m+1
! and 0<=j<=m+1.
allocate( un(m,m) ) ! another memory allocation. in
! this case un(i,j) with
! 1<=i<=m and 1<=j<=m.
h = 1.0_DP/(m+1) ! grid spacing.
u(:,:) = 0.0_DP ! initial temperature is zero all over the square.
do n = 1 , nmax ! relaxation loop
do j = 1 , m ! in the y-direction
do i = 1 , m
un(i,j)=(u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1))/4.0_DP+heat*h*h
end do
end do ! same as the following do-loops:
u(1:m,1:m) = un(1:m,1:m) ! do j = 1 , m
! do i = 1 , m
! u(i,j) = un(i,j)
! end do
! end do
if ( mod(n,100)==0 ) print *, n, u(m/2+1,m/2+1) ! temperature at
end do ! the center
end program heat1_wi_comments
** 再びheat1.f90（コメントなしバージョン） [#x10d0f37]
次に詳しいコメントをはずしてもう一度もとのheat1.f90を見てみる。
（ただし、先週のコードから本質的でないところを少しだけ修正している。）
program heat1
implicit none
integer, parameter :: m=31, nmax=20000
integer :: i,j,n
integer, parameter :: SP = kind(1.0)
integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
real(DP), dimension(:,:), allocatable :: u, un
real(DP) :: h, heat=1.0_DP
allocate(u(0:m+1,0:m+1))
allocate(un(m,m))
h=1.0_DP/(m+1)
u(:,:) = 0.0_DP
do n=1, nmax
do j=1, m
do i=1, m
un(i,j)=(u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1))/4.0_DP+heat*h*h
end do
end do
u(1:m,1:m)=un(1:m,1:m)
if (mod(n,100)==0) print *, n, u(m/2+1,m/2+1)
end do
end program heat1
** 先週の課題（演習3-3） [#e4b52e2d]
- heat1.f90 を MPI を用いて並列化せよ
// - 4 または 8 プロセスで実行し，heat1.f90 と出力結果が同じであることを確認せよ
// - 余裕があれば，プロセス数を 1，2，4，8，16 と変えて実行し，計算時間の変化を調べよ。また，加速率を求めよさらに余裕があれば，問題サイズ m を 100，200 と大きくして同様の実験を行い，加速率を調べよ
** 解答例（heat2.f90） [#v581872f]
program heat2
use mpi
implicit none
integer, parameter :: m=31, nmax=20000
integer :: i,j,jstart,jend,n
integer, parameter :: SP = kind(1.0)
integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
real(DP), dimension(:,:), allocatable :: u, un
real(DP) :: h, heat=1.0_DP
integer :: nprocs,myrank,ierr,left,right
integer, dimension(MPI_STATUS_SIZE) :: istat
call mpi_init(ierr)
call mpi_comm_size(MPI_COMM_WORLD,nprocs,ierr)
call mpi_comm_rank(MPI_COMM_WORLD,myrank,ierr)
h = 1.0_DP/(m+1)
jstart = m*myrank/nprocs+1
jend = m*(myrank+1)/nprocs
allocate( u(0:m+1,jstart-1:jend+1))
allocate(un( m, jstart:jend ))
u(:,:) = 0.0_DP
left = myrank-1
if ( myrank==0 ) left = nprocs-1
right = myrank+1
if ( myrank==nprocs-1 ) right = 0
do n=1, nmax
call mpi_sendrecv(u(1,jend),m,MPI_DOUBLE_PRECISION,right,100, &
u(1,jstart-1),m,MPI_DOUBLE_PRECISION,left,100, &
MPI_COMM_WORLD,istat,ierr)
call mpi_sendrecv(u(1,jstart),m,MPI_DOUBLE_PRECISION,left,100, &
u(1,jend+1),m,MPI_DOUBLE_PRECISION,right,100, &
MPI_COMM_WORLD,istat,ierr)
if (myrank==0) u(1:m, 0) = 0.0_DP
if (myrank==nprocs-1) u(1:m,m+1) = 0.0_DP
do j = jstart , jend
do i = 1 , m
un(i,j)=(u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1))/4.0_DP+heat*h*h
end do
end do
u(1:m,jstart:jend)=un(1:m,jstart:jend)
if (jstart<=m/2+1 .and. jend>=m/2+1) then
if (mod(n,100)==0) print *, n, u(m/2+1,m/2+1)
end if
end do
call mpi_finalize(ierr)
end program heat2
** コード解説 （コメント付きソースコード heat2_wi_comments.f90） [#m3d0f5c6]
上のheat2.f90を解説するために詳しいコメントをつけたソースコードheat2_wi_comments.f90を以下に示す。
program heat2_wi_comments
use mpi
implicit none
integer, parameter :: m=31 ! mesh size
integer, parameter :: nmax=20000 ! max loop counter
integer :: i,j,jstart,jend,n
integer, parameter :: SP = kind(1.0)
integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
real(DP), dimension(:,:), allocatable :: u ! temperature field
real(DP), dimension(:,:), allocatable :: un ! work array
real(DP) :: heat=1.0_DP ! heat source term. uniform distribution.
real(DP) :: h ! grid spacing; dx=dy=h
integer :: nprocs ! number of MPI processes
integer :: myrank ! my rank number
integer :: left, right ! nearest neighbor processes
integer, dimension(MPI_STATUS_SIZE) :: istat ! used for MPI
integer :: ierr ! used for mpi routines
call mpi_init(ierr)
call mpi_comm_size(MPI_COMM_WORLD,nprocs,ierr)
call mpi_comm_rank(MPI_COMM_WORLD,myrank,ierr)
h = 1.0_DP/(m+1) ! grid spacings. see the comment fig below.
jstart = m*myrank/nprocs+1 ! the process is in charge of this point
jend = m*(myrank+1)/nprocs ! point to this point. see the fig below.
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
! when m = 7, nprocs = 3
!
! |<-------------------- 1.0 -------------------->|
! | |
! | h | h h h h h h h |
! |<--->|<--->|<--->|<--->|<--->|<--->|<--->|<--->|
! | | | | | | | | |
! +-----+-----+-----+-----+-----+-----+-----+-----+
! j=0 1 2 3 4 5 6 7 8
! +-----+-----+-----+-----+-----+-----+-----+-----+
! | jstart | | | | | |
! | | jend | | jstart | |
! o-----rank=0------o | | jend |
! | | | | | |
! | jstart jend | | |
! | | | | | |
! o------- rank=1---------o | |
! | | | |
! o------rank=2-----o
!
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
! Another example, when m=17, nprocs=6
!
! j=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
! +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
! | | | | | | |
! o==rank0=o | o===rank3===o | |
! | | | | | |
! o===rank1===o | o===rank4===o |
! | | | |
! o===rank2===o o===rank5===o
! each process | |
! has 2D array u[i,jstart-1:jend+1]
!
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
allocate( u(0:m+1,jstart-1:jend+1)) ! 2-D array for temperature.
allocate(un(m,jstart:jend)) ! 2-D work array.
u(:,:) = 0.0_DP
left = myrank-1 ! left neighbor.
if (myrank==0) left = nprocs-1 ! a dummy periodic distribution
right = myrank+1 ! right neighbor. ! for code simplicity. this has
if (myrank==nprocs-1) right = 0 ! actually no effect.
do n = 1 , nmax ! main loop for the relaxation, to equilibrium.
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
! jend jstart
! <(left)> | | <(right)>
! - --o--o--o--o--o o--o--o--o--o-- -
! send | ^ recv ^ |
! | | | |
! [2] [3] [1] [4]
! | | | |
! recv v | send | v
! o--o--o--o-- - - --o--o--o--o--o
! | |
! jstart jend
! <(myself)>
!
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
call mpi_sendrecv(u(1,jend), m,MPI_DOUBLE_PRECISION,right,100, & ! [1]
u(1,jstart-1),m,MPI_DOUBLE_PRECISION,left,100, & ! [2]
MPI_COMM_WORLD,istat,ierr)
call mpi_sendrecv(u(1,jstart),m,MPI_DOUBLE_PRECISION,left, 100, & ! [3]
u(1,jend+1),m,MPI_DOUBLE_PRECISION,right,100, & ! [4]
MPI_COMM_WORLD,istat,ierr)
if (myrank==0) u(1:m,0 ) = 0.0_DP ! to keep the boundary
if (myrank==nprocs-1) u(1:m,m+1) = 0.0_DP ! condition (temp=0).
do j = jstart , jend ! Jacobi method.
do i = 1 , m ! no need to calculate the boundary i=0 and m+1.
un(i,j)=(u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1))/4.0_DP+heat*h*h
end do
end do
u(1:m,jstart:jend)=un(1:m,jstart:jend) ! this is actually doubled do-loops.
if (jstart<=m/2+1 .and. jend>=m/2+1) then ! print out the temperature
if (mod(n,100)==0) print *, n, u(m/2+1,m/2+1) ! at the central grid point.
end if
end do
deallocate(un,u) ! or you can call deallocate for two times for un and u.
call mpi_finalize(ierr)
end program heat2_wi_comments
* MPI通信の確認 [#f9e4f872]
前回の演習で説明されたMPI_SENDRECVの挙動を改めて確認しよう。
そのためのソースコード（heat2_sendrecv_check.f90）を以下に用意した。
これは上記のheat2.f90からヤコビ法による熱伝導問題の解法部分を除き、
MPIの通信に関係する部分だけを抜き出したものである。
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
! when m = 7, nprocs = 3
!
! |<-------------------- 1.0 -------------------->|
! | |
! | h | h h h h h h h |
! |<--->|<--->|<--->|<--->|<--->|<--->|<--->|<--->|
! | | | | | | | | |
! +-----+-----+-----+-----+-----+-----+-----+-----+
! j=0 1 2 3 4 5 6 7 8
! +-----+-----+-----+-----+-----+-----+-----+-----+
! | jstart | | | | | |
! | | jend | | jstart | |
! o==== rank=0 =====o | | jend |
! | | | | | |
! | jstart jend | | |
! | | | | | |
! o======= rank=1 ========o | |
! | | | |
! o===== rank=2 ====o
!
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
! Another example, when m=17, nprocs=6
!
! j=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
! +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
! | | | | | | |
! o==rank0=o | o===rank3===o | |
! | | | | | |
! o===rank1===o | o===rank4===o |
! | | | |
! o===rank2===o o===rank5===o
! each process | |
! has 2D array u[i,jstart-1:jend+1]
!
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
program heat2_sendrecv_check
use mpi
implicit none
integer, parameter :: m=17
integer :: i,j,jstart,jend,n
integer, parameter :: SP = kind(1.0)
integer, parameter :: DP = selected_real_kind(2*precision(1.0_SP))
real(DP), dimension(:,:), allocatable :: u, un
integer :: nprocs,myrank,ierr,left,right
integer, dimension(MPI_STATUS_SIZE) :: istat
call mpi_init(ierr)
call mpi_comm_size(MPI_COMM_WORLD,nprocs,ierr)
call mpi_comm_rank(MPI_COMM_WORLD,myrank,ierr)
jstart = m*myrank/nprocs+1
jend = m*(myrank+1)/nprocs
allocate( u(0:m+1,jstart-1:jend+1))
allocate(un( m, jstart:jend ))
u(:,jstart-1:jend+1) = myrank
call print('before',myrank, jstart, jend, u(m/2+1,:))
left = myrank-1
if ( myrank==0 ) left = nprocs-1
right = myrank+1
if ( myrank==nprocs-1 ) right = 0
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
! jend jstart
! <(left)> | | <(right)>
! - --o--o--o--o--o o--o--o--o--o-- -
! send | ^ recv ^ |
! | | | |
! [2] [3] [1] [4]
! | | | |
! | | | |
! recv v | send | v
! o--o--o--o-- - - --o--o--o--o--o
! | |
! jstart jend
! <(myself)>
!
!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
call mpi_sendrecv(u(1,jend), m,MPI_DOUBLE_PRECISION,right,100, & ! [1]
u(1,jstart-1),m,MPI_DOUBLE_PRECISION, left,100, & ! [2]
MPI_COMM_WORLD,istat,ierr)
call mpi_sendrecv(u(1,jstart),m,MPI_DOUBLE_PRECISION, left,100, & ! [3]
u(1,jend+1),m,MPI_DOUBLE_PRECISION,right,100, & ! [4]
MPI_COMM_WORLD,istat,ierr)
! if (myrank==0) u(1:m, 0) = 0.0_DP
! if (myrank==nprocs-1) u(1:m,m+1) = 0.0_DP
call print(' after', myrank, jstart, jend, u(m/2+1,:))
deallocate(u,un)
call mpi_finalize(ierr)
contains
subroutine print(message,myrank,jstart,jend,u)
character(len=*), intent(in) :: message
integer, intent(in) :: myrank
integer, intent(in) :: jstart
integer, intent(in) :: jend
real(DP), dimension(jstart-1:jend+1), intent(in) :: u
integer, dimension(0:m+1) :: iu ! work array of integer
integer :: rank, ierr
character(len=100) :: line ! printed line on stdout
character(len=20) :: format ! print format
iu(:) = -1 ! initialization of the work integer array.
iu(jstart-1:jend+1) = u(jstart-1:jend+1) ! copy the data (double to int).
write(format,'(i3"(i3)")') m+2 ! format string generator.
write(line,format) (iu(i),i=0,m+1) ! integers to a text line.
do rank = 0 , nprocs-1 ! to avoid random printing
call mpi_barrier(MPI_COMM_WORLD,ierr) ! by each process in random order,
if ( rank==myrank ) then ! we here put the loop and barrier.
print *, message//':'//trim(line)
end if
end do
end subroutine print
end program heat2_sendrecv_check
** ジョブスクリプト [#s97db981]
このコード（heat2_sendrecv_check.f90）をscalarに投入するためのジョブスクリプトを
以下のように設定すると、heat2_sendrecv_check.f90のコメントに書いた例と同じ、
プロセス数が6のジョブを投入できる。
#PBS -l cputim_job=00:05:00
#PBS -l memsz_job=2gb
#PBS -l cpunum_job=1
#PBS -T vltmpi
#PBS -b 6
#PBS -q PCL-A
cd /home/users/yourID/YourDirectory
mpirun_rsh -np 6 ${NQSII_MPIOPTS} ./a.out
* &color(#0000ff){【演習】}; [#b33862b1]
heat2_sendrecv_check.f90をmpif90コマンドでコンパイルし、
上記ジョブスクリプトを heat2_sendrecv_check.jsという名前のファイルに保存した上で、
qsub heat2_sendrecv_check.js
としてジョブを投入せよ。
(2) MPIプロセス数やグリッドサイズmを変えて試せ。
** 結果例 [#y9842a4a]
%NQSII(INFO): ------- Output from job:0000 -------
[0] before: 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
[0] after: 5 0 0 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
%NQSII(INFO): ------- Output from job:0001 -------
[1] before: -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
[1] after: -1 -1 0 1 1 1 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
%NQSII(INFO): ------- Output from job:0002 -------
[2] before: -1 -1 -1 -1 -1 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1
[2] after: -1 -1 -1 -1 -1 1 2 2 2 3 -1 -1 -1 -1 -1 -1 -1 -1 -1
%NQSII(INFO): ------- Output from job:0003 -------
[3] before: -1 -1 -1 -1 -1 -1 -1 -1 3 3 3 3 3 -1 -1 -1 -1 -1 -1
[3] after: -1 -1 -1 -1 -1 -1 -1 -1 2 3 3 3 4 -1 -1 -1 -1 -1 -1
%NQSII(INFO): ------- Output from job:0004 -------
[4] before: -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 4 4 4 4 4 -1 -1 -1
[4] after: -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 4 4 4 5 -1 -1 -1
%NQSII(INFO): ------- Output from job:0005 -------
[5] before: -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 5 5 5 5 5
[5] after: -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 4 5 5 5 0
------------------------------
as of &_now; (&counter;)
