The analysis of a nested dissection algorithm
Numerische Mathematik
SIAM Journal on Scientific and Statistical Computing
The role of elimination trees in sparse factorization
SIAM Journal on Matrix Analysis and Applications
Elimination structures for unsymmetric sparse LU factors
SIAM Journal on Matrix Analysis and Applications
Fast Nested Dissection for Finite Element Meshes
SIAM Journal on Matrix Analysis and Applications
Geometric Separators for Finite-Element Meshes
SIAM Journal on Scientific Computing
A New Implementation of Sparse Gaussian Elimination
ACM Transactions on Mathematical Software (TOMS)
The Multifrontal Solution of Indefinite Sparse Symmetric Linear
ACM Transactions on Mathematical Software (TOMS)
Computer Solution of Large Sparse Positive Definite
Computer Solution of Large Sparse Positive Definite
Fast Stable Solver for Sequentially Semi-separable Linear Systems of Equations
HiPC '02 Proceedings of the 9th International Conference on High Performance Computing
Some Fast Algorithms for Sequentially Semiseparable Representations
SIAM Journal on Matrix Analysis and Applications
A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
Parallel black box $$\mathcal {H}$$-LU preconditioning for elliptic boundary value problems
Computing and Visualization in Science
A Fast Direct Solver for a Class of Elliptic Partial Differential Equations
Journal of Scientific Computing
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In this paper we develop a fast direct solver for large discretized linear systems using the supernodal multifrontal method together with low-rank approximations. For linear systems arising from certain partial differential equations such as elliptic equations, during the Gaussian elimination of the matrices with proper ordering, the fill-in has a low-rank property: all off-diagonal blocks have small numerical ranks with proper definition of off-diagonal blocks. Matrices with this low-rank property can be efficiently approximated with semiseparable structures called hierarchically semiseparable (HSS) representations. We reveal the above low-rank property by ordering the variables with nested dissection and eliminating them with the multifrontal method. All matrix operations in the multifrontal method are performed in HSS forms. We present efficient ways to organize the HSS structured operations along the elimination. Some fast HSS matrix operations using tree structures are proposed. This new structured multifrontal method has nearly linear complexity and a linear storage requirement. Thus, we call it a superfast multifrontal method. It is especially suitable for large sparse problems and also has natural adaptability to parallel computations and great potential to provide effective preconditioners. Numerical results demonstrate the efficiency.