Xin, ZixingXia, Jianlinde Hoop, Maarten V.Cauley, StephenBalakrishnan, Venkataramanan2017-11-172017-11-172017Xin, Zixing, Xia, Jianlin, de Hoop, Maarten V., et al.. "A Distributed-Memory Randomized Structured Multifrontal Method for Sparse Direct Solutions." <i>SIAM Journal on Scientific Computing,</i> 39, no. 4 (2017) Society for Industrial and Applied Mathematics: C292-C318. https://doi.org/10.1137/16M1079221.https://hdl.handle.net/1911/98834We design a distributed-memory randomized structured multifrontal solver for large sparse matrices. Two layers of hierarchical tree parallelism are used. A sequence of innovative parallel methods are developed for randomized structured frontal matrix operations, structured update matrix computation, skinny extend-add operation, selected entry extraction from structured matrices, etc. Several strategies are proposed to reuse computations and reduce communications. Unlike an earlier parallel structured multifrontal method that still involves large dense intermediate matrices, our parallel solver performs the major operations in terms of skinny matrices and fully structured forms. It thus significantly enhances the efficiency and scalability. Systematic communication cost analysis shows that the numbers of words are reduced by factors of about $O(\sqrt{n}/r)$ in two dimensions and about $O(n^{2/3}/r)$ in three dimensions, where $n$ is the matrix size and $r$ is an off-diagonal numerical rank bound of the intermediate frontal matrices. The efficiency and parallel performance are demonstrated with the solution of some large discretized PDEs in two and three dimensions. Nice scalability and significant savings in the cost and memory can be observed from the weak and strong scaling tests, especially for some 3D problems discretized on unstructured meshes.engArticle is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.Journal article16m1079221https://doi.org/10.1137/16M1079221