COMP 5055 Parallel Computing Parallel Implementation of the Conjugate Gradient Method

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COMP 5055 Parallel ComputingParallel
Implementation of the Conjugate Gradient Method

• Lola Bautista

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Outline

• Introduction
• Sequential Algorithm
• Parallel Implementation
• Results

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Introduction The Conjugate Gradient Method

• Iterative method to solve linear systems of the
  form Axb.
• Matrix A must be symmetric positive definite

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Sequential Algorithm
ConjugateGradient(A,b,x,iter,tol)
residb-Ax dirresid for k1 to iter
do zAdir a ltresid,residgt ß ltdir,zgt µ
a/ß xx µdir residresid- µz a_1
ltresid,residgt if(a_1vtol)then break else ß_1
a_1/a dr ßd end end
Matrix-Vector
Matrix-Vector
Dot Product
Dot Product
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Input Matrix CSR Format
N10
A
col
start
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Rowwise Decomposition of Start vector
A
col
p0
p1
p2
p3
p4
p5
p6
p7
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Rowwise Decomposition of b vector
p0
p1
p2
p3
p4
p5
p6
p7
b
0
1
2
3
4
5
6
7
8
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MPI Communication Functions
MPI_Send MPI_Recv
Rowwise Decomposition
DotProduct(a,b)
for i1 to rows do Ciajbi end
MPI_Reduce
MPI_Bcast
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Results (N 100)
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Results (N 100,000)
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Results (N 1,000,000)
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Results Speedup
Conclusion Optimize implementation to reduce
communications