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SIAM Journal on Scientific and Statistical Computing
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ICCAD '94 Proceedings of the 1994 IEEE/ACM international conference on Computer-aided design
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ACM Transactions on Mathematical Software (TOMS)
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An Approximate Minimum Degree Ordering Algorithm
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
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SIAM Journal on Scientific Computing
Improving the Run Time and Quality of Nested Dissection Ordering
SIAM Journal on Scientific Computing
Hypergraph-Partitioning-Based Decomposition for Parallel Sparse-Matrix Vector Multiplication
IEEE Transactions on Parallel and Distributed Systems
A Supernodal Approach to Sparse Partial Pivoting
SIAM Journal on Matrix Analysis and Applications
Partitioning Rectangular and Structurally Unsymmetric Sparse Matrices for Parallel Processing
SIAM Journal on Scientific Computing
Covering edges by cliques with regard to keyword conflicts and intersection graphs
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Computer Solution of Large Sparse Positive Definite
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IPDPS '01 Proceedings of the 15th International Parallel & Distributed Processing Symposium
A linear-time heuristic for improving network partitions
DAC '82 Proceedings of the 19th Design Automation Conference
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SIAM Journal on Scientific Computing
A column approximate minimum degree ordering algorithm
ACM Transactions on Mathematical Software (TOMS)
Algorithm 837: AMD, an approximate minimum degree ordering algorithm
ACM Transactions on Mathematical Software (TOMS)
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Parallel Computing
On Two-Dimensional Sparse Matrix Partitioning: Models, Methods, and a Recipe
SIAM Journal on Scientific Computing
The university of Florida sparse matrix collection
ACM Transactions on Mathematical Software (TOMS)
Hypergraph-Based Unsymmetric Nested Dissection Ordering for Sparse LU Factorization
SIAM Journal on Scientific Computing
Scientific Programming - A New Overview of the Trilinos Project --Part 1
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A typical first step of a direct solver for the linear system $Mx=b$ is reordering of the symmetric matrix $M$ to improve execution time and space requirements of the solution process. In this work, we propose a novel nested-dissection-based ordering approach that utilizes hypergraph partitioning. Our approach is based on the formulation of graph partitioning by vertex separator (GPVS) problem as a hypergraph partitioning problem. This new formulation is immune to deficiency of GPVS in a multilevel framework and hence enables better orderings. In matrix terms, our method relies on the existence of a structural factorization of the input $M$ matrix in the form of $M=AA^T$ (or $M=AD^2A^T$). We show that the partitioning of the row-net hypergraph representation of the rectangular matrix $A$ induces a GPVS of the standard graph representation of matrix $M$. In the absence of such factorization, we also propose simple, yet effective structural factorization techniques that are based on finding an edge clique cover of the standard graph representation of matrix $M$, and hence applicable to any arbitrary symmetric matrix $M$. Our experimental evaluation has shown that the proposed method achieves better ordering in comparison to state-of-the-art graph-based ordering tools even for symmetric matrices where structural $M=AA^T$ factorization is not provided as an input. For matrices coming from linear programming problems, our method enables even faster and better orderings.