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We develop a method for improving the parallel scalability of the recentlydeveloped parallel selected inversion algorithm [Jacquelin, Lin and Yang 2014],named PSelInv, on massively parallel distributed memory machines. In thePSelInv method, we compute selected elements of the inverse of a sparse matrixA that can be decomposed as A = LU, where L is lower triangular and U is uppertriangular. Updating these selected elements of A-1 requires restrictedcollective communications among a subset of processors within each column orrow communication group created by a block cyclic distribution of L and U. Wedescribe how this type of restricted collective communication can beimplemented by using asynchronous point-to-point MPI communication functionscombined with a binary tree based data propagation scheme. Because multiplerestricted collective communications may take place at the same time in theparallel selected inversion algorithm, we need to use a heuristic to preventprocessors participating in multiple collective communications from receivingtoo many messages. This heuristic allows us to reduce communication loadimbalance and improve the overall scalability of the selected inversionalgorithm. For instance, when 6,400 processors are used, we observe over 5xspeedup for test matrices. It also mitigates the performance variabilityintroduced by an inhomogeneous network topology.

Enhancing the scalability and load balancing of the parallel selected inversion algorithm via tree-based asynchronous communication