Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

Chávez, Gustavo; Turkiyyah, George; Zampini, Stefano; Keyes, David

doi:10.1016/j.cam.2017.11.035

Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

Journal Article · Wed Dec 06 00:00:00 EST 2017 · Journal of Computational and Applied Mathematics

DOI:https://doi.org/10.1016/j.cam.2017.11.035· OSTI ID:1580345

^[1]; Turkiyyah, George ^[2]; Zampini, Stefano ^[1]; Keyes, David ^[1]

King Abdullah Univ. of Science and Technology, Thuwal (Saudi Arabia)
American Univ. of Beirut, Beirut (Lebanon)

We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: AC02-05CH11231

OSTI ID:: 1580345

Journal Information:: Journal of Computational and Applied Mathematics, Journal Name: Journal of Computational and Applied Mathematics Journal Issue: C Vol. 344; ISSN 0377-0427

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (36)

Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation Engquist, Björn; Ying, Lexing Communications on Pure and Applied Mathematics, Vol. 64, Issue 5 https://doi.org/10.1002/cpa.20358	journal	February 2011
A fast, memory efficient and robust sparse preconditioner based on a multifrontal approach with applications to finite‐element matrices Aminfar, AmirHossein; Darve, Eric International Journal for Numerical Methods in Engineering, Vol. 107, Issue 6 https://doi.org/10.1002/nme.5196	journal	February 2016
Hierarchical Matrices Hackbusch, Wolfgang Encyclopedia of Applied and Computational Mathematics https://doi.org/10.1007/978-3-540-70529-1_281	book	January 2015
Domain decomposition based $${\mathcal H}$$ -LU preconditioning Grasedyck, Lars; Kriemann, Ronald; Le Borne, Sabine Numerische Mathematik, Vol. 112, Issue 4 https://doi.org/10.1007/s00211-009-0218-6	journal	March 2009
A Sparse Matrix Arithmetic Based on $\Cal H$ -Matrices. Part I: Introduction to ${\Cal H}$ -Matrices Hackbusch, W. Computing, Vol. 62, Issue 2 https://doi.org/10.1007/s006070050015	journal	April 1999
Parallel black box $$\mathcal {H}$$ -LU preconditioning for elliptic boundary value problems Grasedyck, Lars; Kriemann, Ronald; Le Borne, Sabine Computing and Visualization in Science, Vol. 11, Issue 4-6 https://doi.org/10.1007/s00791-008-0098-9	journal	April 2008
$${{\fancyscript{H}}} $$ H -LU factorization on many-core systems Kriemann, Ronald Computing and Visualization in Science, Vol. 16, Issue 3 https://doi.org/10.1007/s00791-014-0226-7	journal	June 2013
A bibliography on semiseparable matrices* Vandebril, R.; Barel, M. Van; Golub, G. Calcolo, Vol. 42, Issue 3-4 https://doi.org/10.1007/s10092-005-0107-z	journal	December 2005
An $$\mathcal O (N \log N)$$ O ( N log N ) Fast Direct Solver for Partial Hierarchically Semi-Separable Matrices: With Application to Radial Basis Function Interpolation Ambikasaran, Sivaram; Darve, Eric Journal of Scientific Computing, Vol. 57, Issue 3 https://doi.org/10.1007/s10915-013-9714-z	journal	April 2013
On the Numerical Rank of the Off-Diagonal Blocks of Schur Complements of Discretized Elliptic PDEs Chandrasekaran, S.; Dewilde, P.; Gu, M. SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 5 https://doi.org/10.1137/090775932	journal	January 2010
Improving Multifrontal Methods by Means of Block Low-Rank Representations Amestoy, Patrick; Ashcraft, Cleve; Boiteau, Olivier SIAM Journal on Scientific Computing, Vol. 37, Issue 3 https://doi.org/10.1137/120903476	journal	January 2015
Fast algorithms for hierarchically semiseparable matrices Xia, Jianlin; Chandrasekaran, Shivkumar; Gu, Ming Numerical Linear Algebra with Applications, Vol. 17, Issue 6 https://doi.org/10.1002/nla.691	journal	November 2010
Predicting Performance and Selecting Modes of Parallelism: A Case Study Using Cyclic Reduction on Three Parallel Machines Saghi, G.; Siegel, H. J.; Gray, J. L. Journal of Parallel and Distributed Computing, Vol. 19, Issue 3 https://doi.org/10.1006/jpdc.1993.1106	journal	November 1993
Parallel -Matrix Arithmetics on Shared Memory Systems Kriemann, R. Computing, Vol. 74, Issue 3 https://doi.org/10.1007/s00607-004-0102-2	journal	December 2004
A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework Bastian, P.; Blatt, M.; Dedner, A. Computing, Vol. 82, Issue 2-3 https://doi.org/10.1007/s00607-008-0003-x	journal	April 2008
PetIGA: A framework for high-performance isogeometric analysis Dalcin, L.; Collier, N.; Vignal, P. Computer Methods in Applied Mechanics and Engineering, Vol. 308 https://doi.org/10.1016/j.cma.2016.05.011	journal	August 2016
Preconditioning by incomplete block cyclic reduction Rodrigue, Garry; Wolitzer, Donald Mathematics of Computation, Vol. 42, Issue 166 https://doi.org/10.1090/S0025-5718-1984-0736452-6	journal	May 1984
Cyclic Reduction Tridiagonal Solvers on GPUs Applied to Mixed-Precision Multigrid Goddeke, Dominik; Strzodka, Robert IEEE Transactions on Parallel and Distributed Systems, Vol. 22, Issue 1 https://doi.org/10.1109/TPDS.2010.61	journal	January 2011
On Direct Methods for Solving Poisson’s Equations Buzbee, B. L.; Golub, G. H.; Nielson, C. W. SIAM Journal on Numerical Analysis, Vol. 7, Issue 4 https://doi.org/10.1137/0707049	journal	December 1970
Superfast Multifrontal Method for Large Structured Linear Systems of Equations Xia, Jianlin; Chandrasekaran, Shivkumar; Gu, Ming SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 3 https://doi.org/10.1137/09074543X	journal	January 2010
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices Xia, Jianlin; Gu, Ming SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 5 https://doi.org/10.1137/090750500	journal	January 2010
A Parallel and Vector Variant of the Cyclic Reduction Algorithm Sweet, Roland A. SIAM Journal on Scientific and Statistical Computing, Vol. 9, Issue 4 https://doi.org/10.1137/0909050	journal	July 1988
Randomized Sparse Direct Solvers Xia, Jianlin SIAM Journal on Matrix Analysis and Applications, Vol. 34, Issue 1 https://doi.org/10.1137/12087116X	journal	January 2013
A Parallel Sweeping Preconditioner for Heterogeneous 3D Helmholtz Equations Poulson, Jack; Engquist, Björn; Li, Siwei SIAM Journal on Scientific Computing, Vol. 35, Issue 3 https://doi.org/10.1137/120871985	journal	January 2013
An Efficient Multicore Implementation of a Novel HSS-Structured Multifrontal Solver Using Randomized Sampling Ghysels, Pieter; Li, Xiaoye S.; Rouet, François-Henry SIAM Journal on Scientific Computing, Vol. 38, Issue 5 https://doi.org/10.1137/15M1010117	journal	January 2016
A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations Chandrasekaran, S.; Gu, M.; Pals, T. SIAM Journal on Matrix Analysis and Applications, Vol. 28, Issue 3 https://doi.org/10.1137/S0895479803436652	journal	January 2006
A Cyclic Reduction Approach to the Numerical Solution of Boundary Value ODEs Amodio, Pierluigi; Paprzycki, Marcin SIAM Journal on Scientific Computing, Vol. 18, Issue 1 https://doi.org/10.1137/S1064827595287225	journal	January 1997
On the Approximate Cyclic Reduction Preconditioner Reusken, Arnold SIAM Journal on Scientific Computing, Vol. 21, Issue 2 https://doi.org/10.1137/S1064827597331655	journal	January 1999
A Parallel Algorithm for Solving Tridiagonal Linear Systems on Distributed-Memory Multiprocessors Lin, Wen-Yang; Chen, Chuen-Liang International Journal of High Speed Computing, Vol. 06, Issue 03 https://doi.org/10.1142/S0129053394000196	journal	September 1994
A Parallel Geometric Multifrontal Solver Using Hierarchically Semiseparable Structure Wang, Shen; Li, Xiaoye S.; Rouet, François-Henry ACM Transactions on Mathematical Software, Vol. 42, Issue 3 https://doi.org/10.1145/2830569	journal	May 2016
A Fast Direct Solution of Poisson's Equation Using Fourier Analysis Hockney, R. W. Journal of the ACM, Vol. 12, Issue 1 https://doi.org/10.1145/321250.321259	journal	January 1965
The Multifrontal Solution of Indefinite Sparse Symmetric Linear Duff, I. S.; Reid, J. K. ACM Transactions on Mathematical Software, Vol. 9, Issue 3 https://doi.org/10.1145/356044.356047	journal	September 1983
SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems Li, Xiaoye S.; Demmel, James W. ACM Transactions on Mathematical Software, Vol. 29, Issue 2 https://doi.org/10.1145/779359.779361	journal	June 2003
Parallel Distributed FFT-Based Solvers for 3-D Poisson Problems in Meso-Scale Atmospheric Simulations Giraud, Luc; Guivarch, Ronan; Stein, Joël The International Journal of High Performance Computing Applications, Vol. 15, Issue 1 https://doi.org/10.1177/109434200101500104	journal	February 2001
Hierarchical Cholesky decomposition of sparse matrices arising from curl–curl-equation Ibragimov, I.; Rjasanow, S.; Straube, K. Journal of Numerical Mathematics, Vol. 15, Issue 1 https://doi.org/10.1515/jnma.2007.031	journal	January 2007
Methods of conjugate gradients for solving linear systems Hestenes, M. R.; Stiefel, E. Journal of Research of the National Bureau of Standards, Vol. 49, Issue 6 https://doi.org/10.6028/jres.049.044	journal	December 1952

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Distributed-memory lattice H-matrix factorization Yamazaki, Ichitaro; Ida, Akihiro; Yokota, Rio The International Journal of High Performance Computing Applications, Vol. 33, Issue 5 https://doi.org/10.1177/1094342019861139	journal	January 2019

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