A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling

Xia, Jianlin; Xi, Yuanzhe; Gu, Ming

doi:10.1137/110831982

Cited by 76 publications

(174 citation statements)

References 41 publications

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“…This still needs selected entries of F i that will be used to form the B generators and leaf level D generators [24,35]. Given the row and column index sets, we can extract the entries of F i from F 0 i , U c1 , and U c2 in (3.3).…”

Section: Extracting Selected Entries From Hss Update Matricesmentioning

confidence: 99%

“…A simplified version in [28] uses dense U i instead. To enhance the efficiency and flexibility, randomized HSS construction [24,35] is used in [31]. The basic idea is to compress the off-diagonal blocks of F i via randomized sampling [21].…”

mentioning

confidence: 99%

“…Due to the feature that the row basis matrix F − i |ŝ i is a submatrix of F − i , this compression is also referred to as a structure-preserving rank-revealing factorization [35]. More details for generating the other HSS generators can be found in [24,35]. The overall HSS construction follows a bottom-up sweep of the HSS tree.…”

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confidence: 99%

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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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Section: Extracting Selected Entries From Hss Update Matricesmentioning

confidence: 99%

mentioning

confidence: 99%

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

Xin¹,

Xia²,

Hoop³

et al. 2017

SIAM J. Sci. Comput.

Self Cite

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“…For matrices which are explicitly available, we can directly use the methods in [24,39]. These methods traverse the HSS tree T in a bottom-up order, and only require the matrix to be multiplied with O(r) random vectors.…”

Section: Adaptive and Matrix-free Hss Constructionsmentioning

confidence: 99%

A Fast Randomized Eigensolver with Structured LDL Factorization Update

Xi¹,

Xia²,

Chan³

2014

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper, we propose a structured bisection method with adaptive randomized sampling for finding selected or all of the eigenvalues of certain real symmetric matrices A. For A with a low-rank property, we construct a hierarchically semiseparable (HSS) approximation and show how to quickly evaluate and update its inertia in the bisection method. Unlike some existing randomized HSS constructions, the methods here do not require the knowledge of the off-diagonal (numerical) ranks in advance. Moreover, for A with a weak rank property or slowly decaying offdiagonal singular values, we show an idea of aggressive low-rank inertia evaluation, which means that a compact HSS approximation can preserve the inertia for certain shifts. This is analytically justified for a special case, and numerically shown for more general ones. A generalized LDL factorization of the HSS approximation is then designed for the fast evaluation of the inertia. A significant advantage over standard LDL factorizations is that the HSS LDL factorization (and thus the inertia) of A − sI can be quickly updated with multiple shifts s in bisection. The factorization with each new shift can reuse about 60% of the work. As an important application, the structured eigensolver can be applied to symmetric Toeplitz matrices, and the cost to find one eigenvalue is nearly linear in the order of the matrix. The numerical examples demonstrate the efficiency and the accuracy of our methods, especially the benefit of low-rank inertia evaluations. The ideas and methods can be potentially adapted to other HSS computations where shifts are involved and to more problems without a significant low-rank property.

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“…Recently in [81] a super-fast and stable Toeplitz solver has been designed relying on the reduction to Cauchy matrices described in the previous section and on the properties of rank structured matrices.…”

Section: Divide and Conquer Techniques: Super-fast Algorithmsmentioning

confidence: 99%

Matrix structures and applications

Бини¹

2014

Cours du CIRM

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A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling

Cited by 76 publications

References 41 publications

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

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

A Fast Randomized Eigensolver with Structured LDL Factorization Update

Matrix structures and applications

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