Quasi-Optimal Elimination Trees for 2D Grids with Singularities

Abstract

We construct quasi-optimal elimination trees for 2D finite element meshes with singularities. These trees minimize the complexity of the solution of the discrete system. The computational cost estimates of the elimination process model the execution of the multifrontal algorithms in serial and in parallel shared-memory executions. Since the meshes considered are a subspace of all possible mesh partitions, we call these minimizers quasi-optimal. We minimize the cost functionals using dynamic programming. Finding these minimizers is more computationally expensive than solving the original algebraic system. Nevertheless, from the insights provided by the analysis of the dynamic programming minima, we propose a heuristic construction of the elimination trees that has costONelog⁡Ne, whereNeis the number of elements in the mesh. We show that this heuristic ordering has similar computational cost to the quasi-optimal elimination trees found with dynamic programming and outperforms state-of-the-art alternatives in our numerical experiments.

Leveraging Workflows and Clouds for a Multi-frontal Solver for Finite Element Meshes

“…The test data sets were obtained from [12]. The results indicate that the whole process gives correct results.…”

“…The tree is usually generated by planar graph analysis of the linear system of equations, but in the case of the Finite Element Method the elimination tree can also be derived from the mesh. The heuristic for generating suboptimal elimination tree was introduced in [12]. The leaves of an elimination tree represent individual elements of the mesh while the remaining nodes represent subsequent steps of the elimination of a multifrontal solver.…”

“…First, the DAG is symmetrical with respect to the root node of the elimination tree, but the elimination phase that involves matrix factorization has a much higher computational cost than the backward substitution phase. Second, the costs of computation and communication can be precisely calculated analytically by analyzing the mesh and the elimination tree [12]. Finally, well-separated groups (sub-trees) of tasks, which correspond with the structure of the original mesh, can be identified in the workflow structure.…”

“…This component is responsible for creating elimination tree for a given mesh. In our prototype we used an existing implementation [12] which currently is limited to 2D meshes.…”

“…The decomposition of multi-frontal solver into graph of tasks follows the structure of the elimination tree. For adaptive mesh based computations, namely for grids with point and edge singularities, it is possible to construct elimination trees with lightweight computational tasks close to the root of the elimination tree [12,6,1]. This concerns computational meshes in both two and three dimensions.…”