Abstract
This paper describes the application of hypergraph grammars to drive a linear computational cost solver for grids with point singularities. Such graph gram- mar productions are the first mathematical formalisms used to describe solver algorithms, and each indicates the smallest atomic task that can be executed in parallel, which is very useful in the case of parallel execution. In particular, the partial order of execution of graph grammar productions can be found, and the sets of independent graph grammar productions can be localized. They can be scheduled set by set into a shared memory parallel machine. The graph- grammar-based solver has been implemented with NVIDIA CUDA for GPU. Graph grammar productions are accompanied by numerical results for a 2D case. We show that our graph-grammar-based solver with a GPU accelerator is, by order of magnitude, faster than the state-of-the-art MUMPS solver.
Highlights
The multi-frontal solver is a state-of-the-art direct solver algorithm for solving systems of linear equations [9, 10]
Such graph grammar productions are the first mathematical formalisms used to describe solver algorithms, and each indicates the smallest atomic task that can be executed in parallel, which is very useful in the case of parallel execution
We focus on the class of matrices generated by the adaptive finite element method [7, 8]
Summary
The multi-frontal solver is a state-of-the-art direct solver algorithm for solving systems of linear equations [9, 10]. The direct solver algorithm can be generalized to the usage of matrix blocks associated with nodes of the computational mesh (called supernodes) rather than particular scalar values [6]. This allows for a reduction of computational cost related to the construction of the elimination tree. It is possible to improve the performance of the multi-frontal solver algorithm by leveraging the knowledge based on the computational mesh instead of a matrix Another question is what is the lowestpossible computational cost that can be obtained for two-dimensional computational meshes containing a singularity (adaptations proceed toward a point). The isogeometric FEM delivers the higher order global continuity of the solution, but it suffers from a computationally more-expensive direct solver algorithm [4]
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