A robust hierarchical solver for illconditioned systems with applications to ice sheet modeling
Abstract
A hierarchical solver is proposed for solving sparse illconditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferredcompression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferredcompression technique introduces minimal overhead and does not affect parallelism. As a result, the new solver achieves linear computational complexity under mild assumptions and excellent parallel scalability. To demonstrate the performance of the new solver, we focus on applying it to solve sparse linear systems arising from ice sheet modeling. The strong anisotropic phenomena associated with the thin structure of ice sheets creates serious challenges for existing solvers. To address the anisotropy, we additionally developed a customized partitioning scheme for the solver, which captures the strongcoupling direction accurately. In general, the partitioning can be computed algebraically with existing software packages, and thus the new solver is generalizable for solving other sparse linear systems. Our results show that ice sheet problems of about 300 million degrees of freedom have been solved in just a few minutes using a thousand processors.
 Authors:

 Stanford Univ., CA (United States)
 Sandia National Lab. (SNLCA), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States); Sandia National Lab. (SNLCA), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1559514
 Alternate Identifier(s):
 OSTI ID: 1547944
 Report Number(s):
 SAND20199481J
Journal ID: ISSN 00219991; 678447; TRN: US2000355
 Grant/Contract Number:
 AC0494AL85000; NA00023731; NA0003525
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 396; Journal Issue: C; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
Citation Formats
Chen, Chao, Cambier, Leopold, Boman, Erik Gunnar, Rajamanickam, Sivasankaran, Tuminaro, Raymond S., and Darve, Eric. A robust hierarchical solver for illconditioned systems with applications to ice sheet modeling. United States: N. p., 2019.
Web. doi:10.1016/j.jcp.2019.07.024.
Chen, Chao, Cambier, Leopold, Boman, Erik Gunnar, Rajamanickam, Sivasankaran, Tuminaro, Raymond S., & Darve, Eric. A robust hierarchical solver for illconditioned systems with applications to ice sheet modeling. United States. doi:10.1016/j.jcp.2019.07.024.
Chen, Chao, Cambier, Leopold, Boman, Erik Gunnar, Rajamanickam, Sivasankaran, Tuminaro, Raymond S., and Darve, Eric. Fri .
"A robust hierarchical solver for illconditioned systems with applications to ice sheet modeling". United States. doi:10.1016/j.jcp.2019.07.024. https://www.osti.gov/servlets/purl/1559514.
@article{osti_1559514,
title = {A robust hierarchical solver for illconditioned systems with applications to ice sheet modeling},
author = {Chen, Chao and Cambier, Leopold and Boman, Erik Gunnar and Rajamanickam, Sivasankaran and Tuminaro, Raymond S. and Darve, Eric},
abstractNote = {A hierarchical solver is proposed for solving sparse illconditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferredcompression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferredcompression technique introduces minimal overhead and does not affect parallelism. As a result, the new solver achieves linear computational complexity under mild assumptions and excellent parallel scalability. To demonstrate the performance of the new solver, we focus on applying it to solve sparse linear systems arising from ice sheet modeling. The strong anisotropic phenomena associated with the thin structure of ice sheets creates serious challenges for existing solvers. To address the anisotropy, we additionally developed a customized partitioning scheme for the solver, which captures the strongcoupling direction accurately. In general, the partitioning can be computed algebraically with existing software packages, and thus the new solver is generalizable for solving other sparse linear systems. Our results show that ice sheet problems of about 300 million degrees of freedom have been solved in just a few minutes using a thousand processors.},
doi = {10.1016/j.jcp.2019.07.024},
journal = {Journal of Computational Physics},
number = C,
volume = 396,
place = {United States},
year = {2019},
month = {11}
}
Web of Science