Superfast Multifrontal Method for Large Structured Linear Systems of Equations

Xia, Jianlin; Chandrasekaran, Shivkumar; Gu, Ming; Li, Xiaoye S.

doi:10.1137/09074543x

Cited by 195 publications

(238 citation statements)

References 33 publications

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“…Taking into account that estimate (13) reduces to #B σ = O(N 1/2 ) in the case of h-refinements, and to #B σ = O(N ) in the case of p-enrichments, we can revisit the cost estimate (14) as follows:…”

Section: Numerical Resultsmentioning

confidence: 99%

“…This work rests on the observation that, for a large class of problems, the dense Schur complement matrices that arise in the nested dissection method are rankstructured, thus allowing for the use of accelerated matrix algebra, see, e.g., [4,13]. In the case of well-behaved elliptic problems, this property is a consequence of the rapid decay of the underlying Green function.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations

Gatto

Hesthaven

2017

J Sci Comput

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We introduce a preconditioner based on low-rank compression of Schur complements. The construction is inspired by standard nested-dissection and relies on the assumption that the Schur complements can be approximated to high precision by Hierarchical-Block-Separable matrices. We build the preconditioner as an approximate LDM t factorization of a given matrix A, and no knowledge of A in assembled form is required by the construction. The LDM t is amenable to fast inversion and the inverse can be applied fast as well. We investigate the behavior of the preconditioner in the context of DG finite element approximations of elliptic and hyperbolic problems.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations

Gatto

Hesthaven

2017

J Sci Comput

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“…• The third option is to use direct sparse solvers such as CHOLMOD [5] and SuperMF [21]. Theoretically, CHOLMOD has a decomposition complexity of order O(N 3/2 ), with memory use of order O(N log(N )), where N is the number of grid points.…”

Section: Iterative and Direct Solversmentioning

confidence: 99%

Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems

Shen¹,

Yu²

2010

SIAM J. Sci. Comput.

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Abstract. This is the second part in a series of papers on using spectral sparse grid methods for solving higher-dimensional PDEs. We extend the basic idea in the first part [J. Shen and H. Yu, SIAM J. Sci. Comp., 32 (2010), pp. 3228-3250] for solving PDEs in bounded higher-dimensional domains to unbounded higher-dimensional domains and apply the new method to solve the electronic Schrödinger equation. By using modified mapped Chebyshev functions as basis functions, we construct mapped Chebyshev sparse grid methods which enjoy the following properties: (i) the mapped Chebyshev approach enables us to build sparse grids with Smolyak's algorithms based on nested, spectrally accurate quadratures and allows us to build fast transforms between the values at the sparse grid points and the corresponding expansion coefficients; (ii) the mapped Chebyshev basis functions lead to identity mass matrices and very sparse stiffness matrices for problems with constant coefficients and allow us to construct a matrix-vector product algorithm with quasi-optimal computational cost even for problems with variable coefficients; and (iii) the resultant linear systems for elliptic equations with constant or variable coefficients can be solved efficiently by using a suitable iterative scheme. Ample numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithms.

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“…The reason is as follows. For some discretized elliptic PDEs, the frontal matrices are related to (the inverses of) the discretized Green's functions, and the low-rank property of the frontal matrices is related to the geometric separability of the mesh points where the Green's function is evaluated [7,33]. Well-separated points correspond to blocks with small numerical ranks.…”

mentioning

confidence: 99%

“…The class of structured multifrontal methods [30,31,33] is based on the idea that, for certain matrices A, the local dense frontal and update matrices are rank structured. For some cases, F i and U i may be approximated by rank structured forms such as HSS forms.…”

mentioning

confidence: 99%

A Distributed-Memory Randomized Structured Multifrontal Method for Sparse Direct Solutions

Xin¹,

Xia²,

Hoop³

et al. 2017

SIAM J. Sci. Comput.

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Abstract. We design a distributed-memory randomized structured multifrontal solver for large sparse matrices. Two layers of hierarchical tree parallelism are used. A sequence of innovative parallel methods are developed for randomized structured frontal matrix operations, structured update matrix computation, skinny extend-add operation, selected entry extraction from structured matrices, etc. Several strategies are proposed to reuse computations and reduce communications. Unlike an earlier parallel structured multifrontal method that still involves large dense intermediate matrices, our parallel solver performs the major operations in terms of skinny matrices and fully structured forms. It thus significantly enhances the efficiency and scalability. Systematic communication cost analysis shows that the numbers of words are reduced by factors of about O( √ n/r) in two dimensions and about O(n 2/3 /r) in three dimensions, where n is the matrix size and r is an off-diagonal numerical rank bound of the intermediate frontal matrices. The efficiency and parallel performance are demonstrated with the solution of some large discretized PDEs in two and three dimensions. Nice scalability and significant savings in the cost and memory can be observed from the weak and strong scaling tests, especially for some 3D problems discretized on unstructured meshes.

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Superfast Multifrontal Method for Large Structured Linear Systems of Equations

Cited by 195 publications

References 33 publications

Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations

Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations

Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems

A Distributed-Memory Randomized Structured Multifrontal Method for Sparse Direct Solutions

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