A direct solver with O(N) complexity for integral equations on one-dimensional domains

Gillman, Adrianna; Young, Patrick; Martinsson, Per-Gunnar

doi:10.1007/s11464-012-0188-3

Cited by 115 publications

(200 citation statements)

References 27 publications

Supporting

Mentioning

199

Contrasting

Order By: Relevance

“…The H 2 -matrix format [14] is its nested counterpart. HODLR matrices [6] are based on the weak admissibility condition and HSS [41,15] and the closely related HBS [24] formats additionally possess nested basis.…”

mentioning

confidence: 99%

On the Complexity of the Block Low-Rank Multifrontal Factorization

Amestoy¹,

Buttari²,

L’Excellent³

et al. 2017

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. Matrices coming from elliptic Partial Differential Equations have been shown to have a lowrank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products and this property can be efficiently exploited in multifrontal solvers to provide a substantial reduction of their complexity. Among the possible low-rank formats, the Block Low-Rank format (BLR) is easy to use in a general purpose multifrontal solver and has been shown to provide significant gains compared to full-rank on practical applications. However, unlike hierarchical formats, such as H and HSS, its theoretical complexity was unknown. In this paper, we extend the theoretical work done on hierarchical matrices in order to compute the theoretical complexity of the BLR multifrontal factorization. We then present several variants of the BLR multifrontal factorization, depending on the strategies used to perform the updates in the frontal matrices and on the constraints on how numerical pivoting is handled. We show how these variants can further reduce the complexity of the factorization. In the best case (3D, constant ranks), we obtain a complexity of the order of O(n 4/3 ). We provide an experimental study with numerical results to support our complexity bounds.

show abstract

mentioning

confidence: 99%

On the Complexity of the Block Low-Rank Multifrontal Factorization

Amestoy¹,

Buttari²,

L’Excellent³

et al. 2017

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…In this section, we very briefly describe Hierarchical matrices [7] (H-matrices) and Hierarchically Semi-Separable matrices [35] (HSS matrices) because they have been used in the context of a multifrontal solver. The Hierarchically Block-Separable (HBS) format introduced by Gillman et al [16] is highly relevant too but will not be discussed in details as it is essentially equivalent to HSS. All these structures can be used to approximate fronts and thus globally reduce the memory consumption as well as the flop count of the multifrontal process.…”

Section: Matrix Representationsmentioning

confidence: 99%

Improving Multifrontal Methods by Means of Block Low-Rank Representations

Amestoy¹,

Ashcraft²,

Boiteau³

et al. 2015

SIAM J. Sci. Comput.

157

170

View full text Add to dashboard Cite

“…Fast factorization algorithms for HSS matrices have been developed in [7,39]. HBR matrices have a similar structure but emphasize the telescoping nature of the matrix factorization [11] to use in the construction of direct solvers for integral equations [8].…”

Section: Hierarchical Matrices As Algebraic Variants Of Fmmmentioning

confidence: 99%

Communication Complexity of the Fast Multipole Method and its Algebraic Variants

Yokota

Turkiyyah

Keyes

2014

JSFI

View full text Add to dashboard Cite

A combination of hierarchical tree-like data structures and data access patterns from fast multipole methods and hierarchical low-rank approximation of linear operators from H-matrix methods appears to form an algorithmic path forward for efficient implementation of many linear algebraic operations of scientific computing at the exascale. The combination provides asymptotically optimal computational and communication complexity and applicability to large classes of operators that commonly arise in scientific computing applications. A convergence of the mathematical theories of the fast multipole and H-matrix methods has been underway for over a decade. We recap this mathematical unification and describe implementation aspects of a hybrid of these two compelling hierarchical algorithms on hierarchical distributed-shared memory architectures, which are likely to be the first to reach the exascale. We present a new communication complexity estimate for fast multipole methods on such architectures. We also show how the data structures and access patterns of H-matrices for low-rank operators map onto those of fast multipole, leading to an algebraically generalized form of fast multipole that compromises none of its architecturally ideal properties.

show abstract

A direct solver with O(N) complexity for integral equations on one-dimensional domains

Cited by 115 publications

References 27 publications

On the Complexity of the Block Low-Rank Multifrontal Factorization

On the Complexity of the Block Low-Rank Multifrontal Factorization

Improving Multifrontal Methods by Means of Block Low-Rank Representations

Communication Complexity of the Fast Multipole Method and its Algebraic Variants

Contact Info

Product

Resources

About