A robust hierarchical solver for ill-conditioned systems with applications to ice sheet modeling
Abstract
A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferred-compression 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 strong-coupling 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. (SNL-CA), Livermore, CA (United States)
- Publication Date:
- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1559514
- Alternate Identifier(s):
- OSTI ID: 1547944
- Report Number(s):
- SAND2019-9481J
Journal ID: ISSN 0021-9991; 678447; TRN: US2000355
- Grant/Contract Number:
- AC04-94AL85000
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 396; Journal Issue: C; Journal ID: ISSN 0021-9991
- 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 ill-conditioned 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 ill-conditioned systems with applications to ice sheet modeling. United States. https://doi.org/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 ill-conditioned systems with applications to ice sheet modeling". United States. https://doi.org/10.1016/j.jcp.2019.07.024. https://www.osti.gov/servlets/purl/1559514.
@article{osti_1559514,
title = {A robust hierarchical solver for ill-conditioned 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 ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferred-compression 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 strong-coupling 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 = {Fri Nov 01 00:00:00 EDT 2019},
month = {Fri Nov 01 00:00:00 EDT 2019}
}
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Works referenced in this record:
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
journal, January 1998
- Karypis, George; Kumar, Vipin
- SIAM Journal on Scientific Computing, Vol. 20, Issue 1
Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients
journal, September 2004
- Bebendorf, Mario
- Mathematics of Computation, Vol. 74, Issue 251
A distributed-memory hierarchical solver for general sparse linear systems
journal, May 2018
- Chen, Chao; Pouransari, Hadi; Rajamanickam, Sivasankaran
- Parallel Computing, Vol. 74
Effective and Robust Preconditioning of General SPD Matrices via Structured Incomplete Factorization
journal, January 2017
- Xia, Jianlin
- SIAM Journal on Matrix Analysis and Applications, Vol. 38, Issue 4
Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems
journal, September 1996
- Vaněk, P.; Mandel, J.; Brezina, M.
- Computing, Vol. 56, Issue 3
Data-sparse Approximation by Adaptive ℋ2-Matrices
journal, September 2002
- Hackbusch, W.; Börm, S.
- Computing, Vol. 69, Issue 1, p. 1-35
Improving Multifrontal Methods by Means of Block Low-Rank Representations
journal, January 2015
- Amestoy, Patrick; Ashcraft, Cleve; Boiteau, Olivier
- SIAM Journal on Scientific Computing, Vol. 37, Issue 3
An ice-shelf model test based on the Ross Ice Shelf, Antarctica
journal, January 1996
- MacAyeal, D. R.; Rommelaere, V.; Huybrechts, P.
- Annals of Glaciology, Vol. 23
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices
journal, January 2010
- Xia, Jianlin; Gu, Ming
- SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 5
Preserving Positive Definiteness in Hierarchically Semiseparable Matrix Approximations
journal, January 2018
- Xing, Xin; Chow, Edmond
- SIAM Journal on Matrix Analysis and Applications, Vol. 39, Issue 2
On the Numerical Rank of the Off-Diagonal Blocks of Schur Complements of Discretized Elliptic PDEs
journal, January 2010
- Chandrasekaran, S.; Dewilde, P.; Gu, M.
- SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 5
Improving Multifrontal Methods by Means of Block Low-Rank Representations
journal, January 2015
- Amestoy, Patrick; Ashcraft, Cleve; Boiteau, Olivier
- SIAM Journal on Scientific Computing, Vol. 37, Issue 3
On the Scalability of the Albany/FELIX first-order Stokes Approximation ice Sheet Solver for Large-Scale Simulations of the Greenland and Antarctic ice Sheets
journal, January 2015
- Tezaur, Irina K.; Tuminaro, Raymond S.; Perego, Mauro
- Procedia Computer Science, Vol. 51
A Fast Block Low-Rank Dense Solver with Applications to Finite-Element Matrices
text, January 2014
- Aminfar, Amirhossein; Ambikasaran, Sivaram; Darve, Eric
- arXiv
A new version of the Fast Multipole Method for the Laplace equation in three dimensions
journal, January 1997
- Greengard, Leslie; Rokhlin, Vladimir
- Acta Numerica, Vol. 6
A survey of direct methods for sparse linear systems
journal, May 2016
- Davis, Timothy A.; Rajamanickam, Sivasankaran; Sid-Lakhdar, Wissam M.
- Acta Numerica, Vol. 25
A Sparse Matrix Arithmetic Based on $\Cal H$ -Matrices. Part I: Introduction to ${\Cal H}$ -Matrices
journal, April 1999
- Hackbusch, W.
- Computing, Vol. 62, Issue 2
A Sparse Matrix Arithmetic Based on $\Cal H$ -Matrices. Part I: Introduction to ${\Cal H}$ -Matrices
journal, April 1999
- Hackbusch, W.
- Computing, Vol. 62, Issue 2
Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet
journal, September 2015
- Isaac, Tobin; Petra, Noemi; Stadler, Georg
- Journal of Computational Physics, Vol. 296
Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations
journal, May 2015
- Ho, Kenneth L.; Ying, Lexing
- Communications on Pure and Applied Mathematics, Vol. 69, Issue 8
A Matrix Dependent/Algebraic Multigrid Approach for Extruded Meshes with Applications to Ice Sheet Modeling
journal, January 2016
- Tuminaro, R.; Perego, M.; Tezaur, I.
- SIAM Journal on Scientific Computing, Vol. 38, Issue 5
Fast algorithms for hierarchically semiseparable matrices
journal, November 2010
- Xia, Jianlin; Chandrasekaran, Shivkumar; Gu, Ming
- Numerical Linear Algebra with Applications, Vol. 17, Issue 6
Albany/FELIX : a parallel, scalable and robust, finite element, first-order Stokes approximation ice sheet solver built for advanced analysis
journal, January 2015
- Tezaur, I. K.; Perego, M.; Salinger, A. G.
- Geoscientific Model Development, Vol. 8, Issue 4
A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations
journal, January 2006
- Chandrasekaran, S.; Gu, M.; Pals, T.
- SIAM Journal on Matrix Analysis and Applications, Vol. 28, Issue 3
On the Numerical Rank of the Off-Diagonal Blocks of Schur Complements of Discretized Elliptic PDEs
journal, January 2010
- Chandrasekaran, S.; Dewilde, P.; Gu, M.
- SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 5
A sparse H-matrix arithmetic: general complexity estimates
journal, December 2000
- Hackbusch, W.; Khoromskij, B. N.
- Journal of Computational and Applied Mathematics, Vol. 125, Issue 1-2
Existence of ?-matrix approximants to the inverse FE-matrix of elliptic operators with L?-coefficients
journal, July 2003
- Bebendorf, Mario; Hackbusch, Wolfgang
- Numerische Mathematik, Vol. 95, Issue 1
Fast Hierarchical Solvers For Sparse Matrices Using Extended Sparsification and Low-Rank Approximation
journal, January 2017
- Pouransari, Hadi; Coulier, Pieter; Darve, Eric
- SIAM Journal on Scientific Computing, Vol. 39, Issue 3
A fast algorithm for particle simulations
journal, December 1987
- Greengard, L.; Rokhlin, V.
- Journal of Computational Physics, Vol. 73, Issue 2
A fast block low-rank dense solver with applications to finite-element matrices
journal, January 2016
- Aminfar, AmirHossein; Ambikasaran, Sivaram; Darve, Eric
- Journal of Computational Physics, Vol. 304
Optimal initial conditions for coupling ice sheet models to Earth system models: PEREGO ET AL.
journal, September 2014
- Perego, Mauro; Price, Stephen; Stadler, Georg
- Journal of Geophysical Research: Earth Surface, Vol. 119, Issue 9
PT-Scotch: A tool for efficient parallel graph ordering
journal, July 2008
- Chevalier, C.; Pellegrini, F.
- Parallel Computing, Vol. 34, Issue 6-8
A Sparse ℋ-Matrix Arithmetic.
journal, February 2000
- Hackbusch, W.; Khoromskij, B. N.
- Computing, Vol. 64, Issue 1