Fast and effective reordering of columns within supernodes using partition refinement
Abstract
In this paper, we consider the problem of computing a triangular factorization of a sparse symmetric matrix using Gaussian elimination. We assume that the sparse matrix has been permuted using a fillreducing ordering algorithm. When the matrix is symmetric positive definite, the sparsity structure of the triangular factor can be determined once the fillreducing ordering has been computed. Thus, an efficient numerical factorization scheme can be designed so that only the nonzero entries are stored and operated on. On modern architectures, the positions of the nonzero entries in the triangular factor play a big role in determining the efficiency. It is desirable to have dense blocks in the factor so that the computation can be cast in terms of level3 BLAS as much as possible. On architectures with GPUs, for example, it is also desirable for these dense blocks to be as large as possible in order to reduce the times to transfer data between the main CPU and the GPUs. We address the problem of locally refining the ordering so that the number of dense blocks is reduced and the sizes of these dense blocks are increased in the triangular factor.
 Authors:

 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Applied Mathematics Department
 Dalton State College, GA (United States)
 Publication Date:
 Research Org.:
 Lawrence Berkeley National LaboratoryNational Energy Research Scientific Computing Center (NERSC)
 Sponsoring Org.:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 OSTI Identifier:
 1545786
 Resource Type:
 Conference
 Journal Name:
 Eighth SIAM Workshop on Combinatorial Scientific Computing, 2018
 Additional Journal Information:
 Conference: Fast and effective reordering of columns within supernodes using partition refinement
 Publisher:
 Society for Industrial and Applied Mathematics (SIAM)
 Country of Publication:
 United States
 Language:
 English
Citation Formats
Jacquelin, Mathias, Ng, Esmond G., and Peyton, Barry W. Fast and effective reordering of columns within supernodes using partition refinement. United States: N. p., 2018.
Web. doi:10.1137/1.9781611975215.8.
Jacquelin, Mathias, Ng, Esmond G., & Peyton, Barry W. Fast and effective reordering of columns within supernodes using partition refinement. United States. doi:10.1137/1.9781611975215.8.
Jacquelin, Mathias, Ng, Esmond G., and Peyton, Barry W. Mon .
"Fast and effective reordering of columns within supernodes using partition refinement". United States. doi:10.1137/1.9781611975215.8.
@article{osti_1545786,
title = {Fast and effective reordering of columns within supernodes using partition refinement},
author = {Jacquelin, Mathias and Ng, Esmond G. and Peyton, Barry W.},
abstractNote = {In this paper, we consider the problem of computing a triangular factorization of a sparse symmetric matrix using Gaussian elimination. We assume that the sparse matrix has been permuted using a fillreducing ordering algorithm. When the matrix is symmetric positive definite, the sparsity structure of the triangular factor can be determined once the fillreducing ordering has been computed. Thus, an efficient numerical factorization scheme can be designed so that only the nonzero entries are stored and operated on. On modern architectures, the positions of the nonzero entries in the triangular factor play a big role in determining the efficiency. It is desirable to have dense blocks in the factor so that the computation can be cast in terms of level3 BLAS as much as possible. On architectures with GPUs, for example, it is also desirable for these dense blocks to be as large as possible in order to reduce the times to transfer data between the main CPU and the GPUs. We address the problem of locally refining the ordering so that the number of dense blocks is reduced and the sizes of these dense blocks are increased in the triangular factor.},
doi = {10.1137/1.9781611975215.8},
journal = {Eighth SIAM Workshop on Combinatorial Scientific Computing, 2018},
number = ,
volume = ,
place = {United States},
year = {2018},
month = {1}
}