Skip to main content
OSTI.GOVU.S. Department of Energy Office of Scientific and Technical Information
U.S. Department of Energy
Office of Scientific and Technical Information
 

An Algebraic Sparsified Nested Dissection Algorithm Using Low-Rank Approximations

Journal Article · · SIAM Journal on Matrix Analysis and Applications
DOI:https://doi.org/10.1137/19m123806x· OSTI ID:1697983
 [1];  [2];  [3];  [3];  [3];  [1]
  1. Stanford Univ., CA (United States)
  2. Univ. of Texas, Austin, TX (United States)
  3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
+ Show Author Affiliations

Here, we propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND (sparsified Nested Dissection). It is based on nested dissection, sparsification, and low-rank compression. After eliminating all interiors at a given level of the elimination tree, the algorithm sparsifies all separators corresponding to the interiors. This operation reduces the size of the separators by eliminating some degrees of freedom but without introducing any fill-in. This is done at the expense of a small and controllable approximation error. The result is an approximate factorization that can be used as an efficient preconditioner. We then perform several numerical experiments to evaluate this algorithm. We demonstrate that a version using orthogonal factorization and block-diagonal scaling takes fewer CG iterations to converge than previous similar algorithms on various kinds of problems. Furthermore, this algorithm is provably guaranteed to never break down and the matrix stays symmetric positive-definite throughout the process. We evaluate the algorithm on some large problems show it exhibits near-linear scaling. The factorization time is roughly $$\mathcal{O}$$(N), and the number of iterations grows slowly with N.

Research Organization:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program
Grant/Contract Number:
AC04-94AL85000; NA0002373-1; NA-0003525
OSTI ID:
1697983
Report Number(s):
SAND-2019-0313J; 671477
Journal Information:
SIAM Journal on Matrix Analysis and Applications, Vol. 41, Issue 2; ISSN 0895-4798
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English

References (45)

Improving Multifrontal Methods by Means of Block Low-Rank Representations journal January 2015
A hierarchical O(N log N) force-calculation algorithm journal December 1986
Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients journal September 2004
Existence of ?-matrix approximants to the inverse FE-matrix of elliptic operators with L?-coefficients journal July 2003
A Partitioning Strategy for Nonuniform Problems on Multiprocessors journal May 1987
On the Numerical Rank of the Off-Diagonal Blocks of Schur Complements of Discretized Elliptic PDEs journal January 2010
A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations journal January 2006
A robust hierarchical solver for ill-conditioned systems with applications to ice sheet modeling journal November 2019
Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate journal October 2008
On the Compression of Low Rank Matrices journal January 2005
A relaxation method for solving elliptic difference equations journal January 1962
The black-box fast multipole method journal December 2009
Nested Dissection of a Regular Finite Element Mesh journal April 1973
An Efficient Multicore Implementation of a Novel HSS-Structured Multifrontal Solver Using Randomized Sampling journal January 2016
Domain decomposition based $${\mathcal H}$$ -LU preconditioning journal March 2009
A fast algorithm for particle simulations journal December 1987
Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization journal July 1996
A Sparse Matrix Arithmetic Based on $\Cal H$ -Matrices. Part I: Introduction to ${\Cal H}$ -Matrices journal April 1999
Data-sparse Approximation by Adaptive ℋ2-Matrices journal September 2002
PaStiX: a high-performance parallel direct solver for sparse symmetric positive definite systems journal February 2002
Methods of conjugate gradients for solving linear systems journal December 1952
Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations journal May 2015
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs journal January 1998
Randomized algorithms for the low-rank approximation of matrices journal December 2007
Generalized Nested Dissection journal April 1979
Strong rank revealing LU factorizations journal July 2003
Solution of Sparse Indefinite Systems of Linear Equations journal September 1975
Fast Hierarchical Solvers For Sparse Matrices Using Extended Sparsification and Low-Rank Approximation journal January 2017
ILUT: A dual threshold incomplete LU factorization journal July 1994
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems journal July 1986
BILUM: Block Versions of Multielimination and Multilevel ILU Preconditioner for General Sparse Linear Systems journal January 1999
A fast direct solver for elliptic problems on general meshes in 2D journal February 2012
A review of algebraic multigrid journal March 2001
Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems journal March 1992
Efficient Structured Multifrontal Factorization for General Large Sparse Matrices journal January 2013
Randomized Sparse Direct Solvers journal January 2013
Superfast Multifrontal Method for Large Structured Linear Systems of Equations journal January 2010
Fast algorithms for hierarchically semiseparable matrices journal November 2010
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices journal January 2010
Effective and Robust Preconditioning of General SPD Matrices via Structured Incomplete Factorization journal January 2017
Sparse hierarchical solvers with guaranteed convergence journal July 2019
Rank revealing QR factorizations journal April 1987
-matrix approximation of integral operators by interpolation journal October 2002
Some Fast Algorithms for Sequentially Semiseparable Representations journal January 2005
Sparse Hierarchical Solvers with Guaranteed Convergence preprint January 2016

Similar Records

Related Subjects

97 MATHEMATICS AND COMPUTING
sparse linear solver
hierarchical matrix
nested dissection
preconditioner
low-rank