Abstract
CPUs have reached their clock rate limits due to physical constrains. Parallel computers are increasingly used to achieve higher computing performances. How computational algorithms and codes can take advantages of parallel computers thus become more and more important. In scientific computing, solving large sparse general linear systems is one of critical problem. The nested dissection recording can reorder a sparse matrix to allow parallelism of triangular factorization on block sub-matrices. The key of this approach is how we solve embedded linear system corresponding to the Schur complement. While existed methods solve this embedded linear system iteratively, we carefully study the hierarchical of the Schur complement so that we can use direct methods to solve the embedded linear system recursively. We demonstrate the advantage of our approach by focusing on an implementation on CPU-GPU hybrid system for symmetric positive definite linear systems. Numerical results suggest the proposed hierarchical Schur method is promising in small or large parallel computer cluster with many-core accelerators.
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