Abstract
We consider the solution of sparse linear systems using direct methods via LU factorization. Unless the matrix is positive definite, numerical pivoting is usually needed to ensure stability, which is costly to implement especially in the sparse case. The Random Butterfly Transformations (RBT) technique provides an alternative to pivoting and is easily parallelizable. The RBT transforms the original matrix into another one that can be factorized without pivoting with probability one. This approach has been successful for dense matrices; in this work, we investigate the sparse case. In particular, we address the issue of fill-in in the transformed system.
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Acknowledgement
We would like to thank Stott Parker for insightful discussions about the one-sided transformation. Partial support for this work was provided through Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research (and Basic Energy Sciences/Biological and Environmental Research/High Energy Physics/Fusion Energy Sciences/Nuclear Physics). We used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
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Baboulin, M., Li, X.S., Rouet, FH. (2015). Using Random Butterfly Transformations to Avoid Pivoting in Sparse Direct Methods. In: Daydé, M., Marques, O., Nakajima, K. (eds) High Performance Computing for Computational Science -- VECPAR 2014. VECPAR 2014. Lecture Notes in Computer Science(), vol 8969. Springer, Cham. https://doi.org/10.1007/978-3-319-17353-5_12
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