1. 05/23/11
Evaluation and Benchmarking of Highly Scalable Parallel Numerical Libraries
Christos Theodosiou
User and Application Support
Scientific Computing AUTH

2. Presentation Outline Problem Description Serial Implementation
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Presentation Outline
Problem Description
Serial Implementation
Parallel implementation
Numerical Results
Conclusions 2

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Problem Description 3

4. Linear Algebra Linear System Solution (LAPACK/SCALAPACK) Ax=b
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Linear Algebra
Linear System Solution (LAPACK/SCALAPACK) Ax=b
Matrix-Matrix Multiplication (BLAS/PBLAS) Ax-b=0 4

5. Linear Algebra Libraries
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Linear Algebra Libraries
Serial Implementation
BLAS (Basic Linear Algebra Subprograms)
LAPACK (Linear Algebra PACKage)
Parallel Implementation
BLACS (Basic Linear Algebra Communication Subprograms)
PBLAS (Parallel BLAS)
SCALAPACK (Scalable LAPACK) 5

6. 05/23/11
Example Case 6

7. 05/23/11
Example Case N M 7

8. 05/23/11
Example Case N
NRHS M 8

9. Serial Implementation (Ax=b)
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Serial Implementation (Ax=b)
DGESV: Computes the solution to a real system of linear equations A*X=B
DGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )
N : The number of linear equations. NRHS: The number of right hand sides. A: On entry, the N-by-N matrix A On exit, the factors L and U of A. LDA: The leading dimension of the matrix A. IPIV: The pivot indices that define the permutation matrix P B: On entry, the matrix of right hand side matrix B. On exit, the N-by-NRHS solution matrix X. LDB: The leading dimension of the array B. INFO: If equal to zero the solve was successful. 9

10. Serial Implementation (Ax-b=0)
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Serial Implementation (Ax-b=0)
DGEMM: Perform one of the matrix-matrix operations C = a*A*B + b*C
DGEMM( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC )
TRANSA: Specifies if “normal” or “transpose” matrix A will be used. TRANSB: Specifies if “normal” or “transpose” matrix B will be used. M: The number of rows of the matrix A and C. N: The number of columns of the matrix B and C. K: The number of columns of the matrix A and rows of the matrix B. ALPHA: The scalar alpha. A: The M-by-K matrix A. LDA: The leading dimension of the matrix A. B: The K-by-N matrix B. LDB: The leading dimension of the matrix A. BETA: The scalar beta. C: The M-by-N matrix C. LDC: The leading dimension of the matrix A. 10

11. Example Case (Parallel Implementation)
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Example Case (Parallel Implementation) 11

12. Example Case (Parallel Implementation)
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Example Case (Parallel Implementation) N
NRHS M 12

13. Example Case (Parallel Implementation)
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Example Case (Parallel Implementation)
2 x 2 = 4 cpus N
NRHS M 13

14. Example Case (Parallel Implementation)
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Example Case (Parallel Implementation)
3 x 2 = 6 cpus N
NRHS M 14

15. Parallel Implementation (Ax=b)
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Parallel Implementation (Ax=b)
PDGESV: Computes the solution to a real system of linear equations A*X=B
DGESV ( N, NRHS, A, LDA, IPIV, B, LDB, INFO ) PDGESV( N, NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO )
IA: The row index in the global array A. JA: The column index in the global array A. DESCA: The array descriptor for the distributed matrix A. IB: The row index in the global array B. JB: The column index in the global array B. DESCB: The array descriptor for the distributed matrix B. 15

16. Parallel Implementation (Ax-b=0)
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Parallel Implementation (Ax-b=0)
PDGEMM: Perform one of the matrix-matrix operations C = a*A*B + b*C
DGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC ) PDGEMM( TRANSA, TRANSB, M, N, K, ALPHA, A, IA, JA, DESCA, B, IB, JB, DESCB, BETA, C, IC, JC, DESCC )
IA: The row index in the global array A. JA: The column index in the global array A. DESCA: The array descriptor for the distributed matrix A. IB: The row index in the global array B. JB: The column index in the global array B. DESCB: The array descriptor for the distributed matrix B. IC: The row index in the global array C. JC: The column index in the global array C. DESCC: The array descriptor for the distributed matrix C. 16

17. Serial Implementation
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Serial Implementation
Standard BLAS
Goto BLAS
ATLAS BLAS
ACML (AMD Core Math Library)
Intel MKL 17

18. Serial Implementation Results (Ax-b=0)
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Serial Implementation Results (Ax-b=0)
* Intel Xeon E5345 @ 2.33GHz 18

19. Serial Implementation Results (Ax=b)
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Serial Implementation Results (Ax=b)
* Intel Xeon E5345 @ 2.33GHz 19

20. Parallel Implementation Results (Ax-b=0)
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Parallel Implementation Results (Ax-b=0)
* Intel Xeon E5345 @ 2.33GHz 20

21. Parallel Implementation Results (Ax=b)
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Parallel Implementation Results (Ax=b)
* Intel Xeon E5345 @ 2.33GHz 21

22. Conclusions Optimized Linear Algebra Libraries improve performance
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Conclusions
Optimized Linear Algebra Libraries improve performance
Scale becomes better as the problems get bigger
Distributed Memory Libraries can treat larger problems than Shared Memory Libraries 22