Finite Difference Method
We consider the elliptic partial differential equation, knows as Poisson’s equation,
u xx  u yy  f
where ( x, y )    [a, b]  [c, d ]
To solve Poisson’s equation by difference method, the region  is partitioned into a
grid consisting of n x m rectangles with sides h and k. The mesh point are given by
xi  a  ih , y j  c  jk ,
i  0,1,, n , j  0,1, , m
By using central difference approximations for the special derivatives, the finite
ui 1, j  2ui , j  ui 1, j
difference equation is
h
2

ui , j 1  2ui , j  ui , j 1
k2
 fi, j
Often it is desirable to set h=k, the equation becomes
 4u i , j  u i , j 1  u i , j 1  u i 1, j  u i 1, j  hf i , j
(i+1,j)
(i,j-1)
(i,j)
(i,j+1)
(i-1,j)
We can also solve this equation by parallel computing. Suppose first that the parallel
system consists of a mesh connected array of p processors Pi , j arranged in a
two-dimensional lattice. Suppose that N 2  mp . Then it is natural to assign m
unknowns to each processor. Each processor will proceed to compute iterates for the
unknowns it holds. On a local memory system, at the end of each iteration the new
iterates at certain grid points will need to be transmitted to adjacent processors.
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We denote by ” internal boundary values “ those values of u ij that are needed by
other processors at the next iteration and must be transmitted.
xxxxxxxxxxxxxxxxxxxxxxx
x
x
x
x
Pij
x
x
x
x
x
x.
xxxxxxxxxxxxxxxxxxxxxxxx
u41
u31
u 43
u33
u42
u32
P21
u21
u11
u44
u34
P22
u 23
u13
u22
u12
P11
u24
u14
P12
For example, for the situation the computation in processor P22 would be
k 1
k 1
, u34
compute u33
; send to P12
k 1
k 1
k 1
, u 43
compute u 43
, send u33
to P21

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From the above, we know that there are different types of connection between the
processors. We choose the type of connection depend on the structure of the matrix.
Example:
A star
A ring
A cube
Example 9
We have poisson equation
u xx  u yy  1
where ( x, y )    [0,1]  [0,1]
The region  is partitioned into a grid consisting of 240x240 rectangles.
Result:
No of processor
1
4
9
16
Time used in each processor calculating x
17.64
12.17
5.57
3.27
Total time used in this function
21.08
13.4
7.42
4.51
(1) In the Scientific Computing Lab, the result shows that the total time used in this
method is decreasing if more processors are used. E.g. When 9 computers are used,
the total time used is 7.42s. It accelerates about triple times comparing with that when
single processor is used. The time needed to calculate x becomes shortened
5.24+0.23=5.57s. It is because each computer just needs to calculate 1/9 part of x. The
time used in ‘Send’ and ‘Receive’ is very small and the time mainly used in
calculation. So we find that this method can apply parallel algorithm.
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The speedup of parallel algorithm is
S p =execution time for a single processor / execution time using p processors
Ep 
Sp
p
p
Speed up, S p
Efficiency, E p
1
4
9
16
1
1.5731
2.841
4.6741
1
0.3933
0.3157
0.2921
We find that the efficiency of the algorithm is decreasing slowly. Since the number of
processor is limited in our network of work-stations, we need to use PC cluster to see
the development of curve when more processors are used.
The upper line shows the ideal case where Sp=1.
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(2) In TDG Cluster, the result is similar. The number of processor is increased and
we can use up to 100 CPUs.
No of processor, p
1
Total time used in this function 39.68
Speed up, S p
Efficiency, E p
4
9
16
25
36
64
100
16.17
7.8
4.25
2.86
2.28
1.92
1.85
1
2.4539 5.0872 9.3365 13.874 17.4035 20.0667 21.4486
1
0.6135 0.5652 0.5835
0.555
0.4834
0.3135
From the table, we know that the efficiency is decreasing gradually. It means that the
development of Sp is increased slowly.
We notice that the curve of speed up becomes a horizontal line at the end. It is
because when we use more than 70 processors, the time used in calculation is also
around 2 seconds and the time used in message passing is no changed, so the total
time is almost the same.
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0.2145
Conclusion
In conclusion, we know that the jacobi method can apply parallel algorithm. We can
connect several processors to calculate the x at the same time, it will decrease the time
used. But the Gauss-Seidel method need the latest information as new as possible, so
the processors can not run the program at the same time and it can not apply parallel
algorithm.
For jacobi method, we need to notice the time used in message passing. It is because
the time need to broadcast or gather the x is very long. The solution is we can apply
the property of the matrix’s structure. Ex. For the sparse matrix, we just need to send
the necessary data to the adjacent processor, it will decrease the time used in
transferring data. In addition, we just use this parallel algorithm if the program need
extensive time to solve in one processor, i.e. the matrix’s size is very large or it need
many iterates to convergent, then the efficiency will be more evident.
The main factors that cause degradation from perfect speedup are:
(i)
Lack of a perfect degree of parallelism in the algorithm and/or lack of perfect
load balance;
(ii)
Communication, contention, and synchronization time
By load balancing we mean the assignment of tasks to the processors of the system so
as to keep each processor doing useful work as much as possible.
We expect to need a parallel machine only for those problems too large to run in a
feasible time on a single processor.
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Bibliography
[1]
K. A. Gallivan, Michael T. Heath, Esmond Ng, James M. Ortega,
Barry W. Peyton, R. J. Plemmons, Charles H. Romine, A. H. Sameh,
Robert G. Voigt, Parallel Algorithms for Matrix Computations, Society for
Industrial and Applied Mathematics, 1990
[2]
U. Schendel, Introduction to Numerical Methods for Parallel Computers, Ellis
Horwood Limited, 1984
[3]
Marc Snir, MPI--the complete reference, Cambridge, Mass, 1998
[4]
C. T. Kelley, Iterative methods for linear and nonlinear equations, Society for
Industrial and Applied Mathematics, 1995
[5]
Anne Greenbaum, Iterative Method for Solving Linear Systems, Society of
Industrial and Applied Mathematics, 1997
[6]
Wolfgang Hackbusch, Iterative Solution of Large Sparse Systems of Equations,
Springer-Verlag, 1994
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Appendix
Total time used in different method
Example 1 Jacobi method (matrix size:6x6)
Gauss-Seidel method (matrix size:6x6)
Example 2: Jacobi method (matrix size:6x6)
Comparing the cases when 1,3 clusters are used.
Example 3: Jacobi method (Sparse matrix:57600x57600)
Comparing the cases when 1,2,4,8 clusters are used.
Example 4: Jacobi method (Sparse matrix:921600x921600)
Comparing the cases when 1,2,4,8 clusters are used.
Example 5: Gauss-Seidel method (matrix size:6x6)
Comparing the cases when 1,3 clusters are used.
Example 6: Gauss-Seidel method (Sparse matrix:57600x57600)
Comparing the cases when 1,5 clusters are used.
Example 7: Conjugate gradient method (Sparse matrix) (diagonal=5)
Comparing the cases when 1,2,4 clusters are used.
Example 8: Conjugate gradient method (Sparse matrix) (diagonal=4)
Comparing the cases when 1,2,4,8 clusters are used.
Example 9: Finite different method (Poisson’s matrix)
Comparing the cases when 1,4,9,16,25,36,64,100 clusters are used.
Profile Report
(1) Network of work-stations
(2) PC Cluster
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