Volume 3, Issue 3
Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D

Numer. Math. Theor. Meth. Appl., 3 (2010), pp. 276-294.

Published online: 2010-03

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• Abstract

In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems. In earlier works, the local Schur complements were computed exactly using a sparse direct solver. The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling  huge problems. In this work we investigate the use of sparse approximation of the dense local Schur complements. These approximations are computed using a partial incomplete $LU$ factorization. Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems; preliminary experiments on linear systems arising from structural mechanics are also reported.

• Keywords

Hybrid direct/iterative solver, domain decomposition, incomplete/partial factorization, Schur approximation, scalable preconditioner, convection-diffusion, large $3D$ problems, parallel scientific computing, High Performance Computing.

15A06, 15A23

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@Article{NMTMA-3-276, author = {}, title = {Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2010}, volume = {3}, number = {3}, pages = {276--294}, abstract = {

In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems. In earlier works, the local Schur complements were computed exactly using a sparse direct solver. The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling  huge problems. In this work we investigate the use of sparse approximation of the dense local Schur complements. These approximations are computed using a partial incomplete $LU$ factorization. Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems; preliminary experiments on linear systems arising from structural mechanics are also reported.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2010.33.2}, url = {http://global-sci.org/intro/article_detail/nmtma/6000.html} }
TY - JOUR T1 - Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 276 EP - 294 PY - 2010 DA - 2010/03 SN - 3 DO - http://doi.org/10.4208/nmtma.2010.33.2 UR - https://global-sci.org/intro/article_detail/nmtma/6000.html KW - Hybrid direct/iterative solver, domain decomposition, incomplete/partial factorization, Schur approximation, scalable preconditioner, convection-diffusion, large $3D$ problems, parallel scientific computing, High Performance Computing. AB -

In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems. In earlier works, the local Schur complements were computed exactly using a sparse direct solver. The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling  huge problems. In this work we investigate the use of sparse approximation of the dense local Schur complements. These approximations are computed using a partial incomplete $LU$ factorization. Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems; preliminary experiments on linear systems arising from structural mechanics are also reported.

L. Giraud, A. Haidar & Y. Saad. (2020). Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D. Numerical Mathematics: Theory, Methods and Applications. 3 (3). 276-294. doi:10.4208/nmtma.2010.33.2
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