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Hypergrammar-based parallel multi-frontal solver for grids with point singularities

Gurgul P., Paszyński M., Paszyńska A.

Computer Science

2015

Vol. 16 (1)

75--102

This paper describes the application of hypergraph grammars to drive a linear computational cost solver for grids with point singularities. 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. 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.

Grammar based multi-frontal solver for isogeometric analysis in 1d

Kuźnik K., Paszyński M, Calo V.

Computer Science

2013

Vol. 14 (4)

589--613

In this paper, we present a multi-frontal direct solver for one-dimensional iso-geometric finite element method. The solver implementation is based on the graph grammar (GG) model. The GG model allows us to express the entire solver algorithm, including generation of frontal matrices, merging, and eliminations as a set of basic undividable tasks called graph grammar productions. Having the solver algorithm expressed as GG productions, we can find the partial order of execution and create a dependency graph, allowing for scheduling of tasks into shared memory parallel machine. We focus on the implementation of the solver with NVIDIA CUDA on the graphic processing unit (GPU). The solver has been tested for linear, quadratic, cubic, and higher-order B-splines, resulting in logarithmic scalability.