We show in this paper how linear algebra calculations can be enhanced by statistical techniques in the case of a square linear system Ax = b. We study a random transformation of A that enables us to avoid pivoting and then to reduce the amount of communication. Numerical experiments show that this randomization can be performed at a very aordable computational price while providing us with a satisfying accuracy when compared to partial pivoting. This random transformation called Partial Random Buttery Transformation (PRBT) is optimized in terms of data storage and ops count. We propose a solver where PRBT and the LU factorization with no pivoting take advantage of the latest generation of hybrid multicore/GPU machines and we compare its Gop/s performance with a solver implemented in a current parallel library.

International audienceWe reexamine the work of Stumm and Walther on multistage algorithms for adjoint computation. We provide an optimal algorithm for this problem when there are two levels of checkpoints , in memory and on disk. Previously, optimal algorithms for adjoint computations were known only for a single level of checkpoints with no writing and reading costs; a well-known example is the binomial checkpointing algorithm of Griewank and Walther. Stumm and Walther extended that binomial checkpointing algorithm to the case of two levels of checkpoints, but they did not provide any optimality results. We bridge the gap by designing the first optimal algorithm in this context. We experimentally compare our optimal algorithm with that of Stumm and Walther to assess the difference in performance

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Abstract-Finding a good partition of a computational directed acyclic graph associated with an algorithm can help find an execution pattern improving data locality, conduct an analysis of data movement, and expose parallel steps. The partition is required to be acyclic, i.e., the inter-part edges between the vertices from different parts should preserve an acyclic dependency structure among the parts. In this work, we adopt the multilevel approach with coarsening, initial partitioning, and refinement phases for acyclic partitioning of directed acyclic graphs and develop a direct k-way partitioning scheme. To the best of our knowledge, no such scheme exists in the literature. To ensure the acyclicity of the partition at all times, we propose novel and efficient coarsening and refinement heuristics. The quality of the computed acyclic partitions is assessed by computing the edge cut, the total volume of communication between the parts, and the critical path latencies. We use the solution returned by well-known undirected graph partitioners as a baseline to evaluate our acyclic partitioner, knowing that the space of solution is more restricted in our problem. The experiments are run on large graphs arising from linear algebra applications.