The Design of Sparse Direct Solvers using Object-Oriented Techniques

A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations

“…Oblio is a sparse symmetric direct solver library developed by Dobrian and Pothen as an experimental tool (Dobrian, Kumfert and Pothen, 2000). Their goal was to create a "laboratory for quickly prototyping new algorithmic innovations, and to provide efficient software on serial and parallel platforms".…”

Elimination Structures in Scientific Computing

“…Given an edge (i, j) directed from i to j, we will call i the predecessor of j, and j the successor of i. The elimination ordering must eliminate vertices in a topological [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] ordering of the DAG such that all predecessors of a vertex must be eliminated before it can be eliminated. The requirement that each matrix in the product form of the inverse must have the same nonzero structure as the corresponding columns in the factor is expressed by the fact that the subgraph corresponding to the matrix should be transitively closed.…”

“…Most of the codes that implement minimum-degree ordering heuristics, such as GEN-MMD [50], AMD [2], and Spindle [14,47], use quotient graphs to represent elimination graphs.…”

“…This is not a contradiction since the minimum-degree rule is just a heuristic which is rarely optimal. For further details, we refer the reader to George and Liu's survey [30], to Amestoy, Davis, and Duff's paper on approximate minimum-degree rules [2], and to Kumfert and Pothen's work on minimum-degree variants [14,47]. Heggernes, Eisenstat, Kumfert and Pothen prove upper bounds on the running time of space-efficient minimum-degree variants [44].…”

“…Hence, the storage requirement of this representation may exceed the storage required for the initial representation. Still, the storage requirement is always smaller [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] than the storage required to represent each edge of the reduced matrix explicitly. This representation poses the same degree update problem that symmetric clique covers pose, and the same techniques can be used to address it.…”