Sparse Matrix Dense Vector Multiplication
Sparse Matrix Dense Vector Multiplication. by Pedro A. Escallon Parallel Processing Class Florida Institute of Technology April 2002. The Problem. Improve the speed of sparse matrix - dense vector multiplication using MPI in a beowolf parallel computer. What To Improve.
Sparse Matrix Dense Vector Multiplication
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Presentation Transcript
Sparse Matrix Dense Vector Multiplication by Pedro A. Escallon Parallel Processing Class Florida Institute of Technology April 2002
The Problem • Improve the speed of sparse matrix - dense vector multiplication using MPI in a beowolf parallel computer.
What To Improve • Current algorithms use excessive indirect addressing • Current optimizations depend on the structure of the matrix (distribution of the nonzero elements)
Sparse Matrix Representations • Coordinate format • Compressed Sparse Row (CSR) • Compressed Sparse Column (CSC) • Modified Sparse Row (MSR)
Compressed Sparse Row (CSR) rS ndx val
CSR Code void sparseMul(int m, double *val, int *ndx, int *rS, double *x, double *y) { int i,j; for(i=0;i<m;i++) { for(j=rowStart[i];j<rS[i+1];j++) { y[i]+=(*val++)*x[*ndx++]; } } }
Goals • Eliminate indirect addressing • Remove the dependency on the distribution of the nonzero elements • Further compress the matrix storage • Most of all, to speed up the operation
Data Structure typedef struct { int rCol; double val; } dSparS_t; {rCol,val}
hdr.size Process A local_size residual < p local_size – hdr.size / p residual = hdr.size % p
Scatter A local_A local_size
Multiplication Code if( (index=local_A[0].rCol) > 0 ) local_Y[0].val = local_A[0].val * X[index]; else local_Y[0].val = local_A[0].val * X[0]; local_Y[0].rCol = -1; k=1; h=0; while(k<local_size) { while((0<(index=local_A[k].rCol)) && (k<local_size)) local_Y[h].val += local_A[k++].val * X[index]; if(k<local_size) { local_Y[h++].rCol = -index-1; local_Y[h].val = local_A[k++].val * X[0]; } } local_Y[h].rCol = local_Y[-1+h++].rCol+1; while(h < stride) local_Y[h++].rCol = -1;
Multiplication local_Y stride doamin Range local_A * = local_size X
Algorithm local_A X Y.val Y.rCol
Gather stride split element range local_Y gatherBuffer residual
Consolidation of Split Rows nCols gatherBuffer += Y residual
Results (vavasis3) vavasis3.rua - Total non-zero values: 1,683,902 - p = 10
Results (vavasis3) vavasis3.rua - Total non-zero values: 1,683,902 - p = 8 vavasis3.rua - Total non-zero values: 1,683,902 - p = 1
Results (vavasis3) vavasis3.rua - Total non-zero values: 1,683,902 - p = 4 vavasis3.rua - Total non-zero values: 1,683,902 - p = 2
Results (vavasis3) vavasis3.rua - Calculated Results
Results (bayer02) bayer02.rua - Total non-zero values: 63,679 - p = 10
Results (bayer02) bayer02.rua - Total non-zero values: 63,679 - p = 8 bayer02.rua - Total non-zero values: 63,679 - p = 1
Results (bayer02) bayer02.rua - Total non-zero values: 63,679 - p = 4 bayer02.rua - Total non-zero values: 63,679 - p = 2
Results (bayer02) bayer02.rua - Calculated Results
Conclusions • The proposed representation speeds up the matrix calculation • Data mismatch solution before gather should be improved • There seems to be a communication penalty for using moving structured data
Bibliography • “Optimizing the Performance of Sparse Matrix-Vector Multiplication” dissertation by Eun-Jin Im. • “Iterative Methods for Sparse Linear Systems” by Yousef Saad • “Users’ Guide for the Harwell-Boeing Sparse Matrix Collection” by Iain S. Duff
