Finding effective support-tree preconditioners

Maggs, Bruce M.; Parekh, Ojas; Ravi, R.; Woo, Shan Leung Maverick

doi:10.1145/1073970.1073996

Cited by 14 publications

(12 citation statements)

References 22 publications

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“…Among other results, they find ways of constructing preconditioners by adding vertices to the graphs. Maggs et al [44] prove that this technique may be used to construct excellent preconditioners, but it is still not clear if they can be constructed efficiently.…”

Section: Subgraph Preconditionersmentioning

confidence: 99%

“…Each of these takes nearly linear expected time, so the overall expected running time of TreeUltraSparsify is O(m log c n) for some constant c. Appendix A. Gremban's reduction. Gremban [30] (see also [44] There are three ways that sub could decide to create sets W j . The first is if some subset of the children of v return sets F i whose values under η sum to more than φ.…”

Section: E S = Rootedultrasparsify(e T R T P)mentioning

confidence: 99%

See 1 more Smart Citation

Nearly Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems

Spielman¹,

Teng²

2014

SIAM J. Matrix Anal. & Appl.

269

167

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Abstract. We present a randomized algorithm that on input a symmetric, weakly diagonally dominant n-by-n matrix A with m nonzero entries and an n-vector b produces anx such thatc n log(1/ )) for some constant c. By applying this algorithm inside the inverse power method, we compute approximate Fiedler vectors in a similar amount of time. The algorithm applies subgraph preconditioners in a recursive fashion. These preconditioners improve upon the subgraph preconditioners first introduced by Vaidya in 1990. For any symmetric, weakly diagonally dominant matrix A with nonpositive off-diagonal entries and k ≥ 1, we construct in time O(m log c n) a preconditioner B of A with at most 2(n − 1) + O((m/k) log 39 n) nonzero off-diagonal entries such that the finite generalized condition number κ f (A, B) is at most k, for some other constant c. In the special case when the nonzero structure of the matrix is planar the corresponding linear system solver runs in expected time O(n log 2 n + n log n log log n log(1/ )). We hope that our introduction of algorithms of low asymptotic complexity will lead to the development of algorithms that are also fast in practice. 1. Introduction. We design an algorithm with nearly optimal asymptotic complexity for solving linear systems in symmetric, weakly diagonally dominant (SDD 0 ) matrices. The algorithm applies a classical iterative solver, such as the preconditioned conjugate gradient or the preconditioned Chebyshev method, with a novel preconditioner that we construct and analyze using techniques from graph theory. Linear systems in these preconditioners may be reduced to systems of smaller size in linear time by use of a direct method. The smaller linear systems are solved recursively. The resulting algorithm solves linear systems in SDD 0 matrices in time that is asymptotically almost linear in their number of nonzero entries. Our analysis does not make any assumptions about the nonzero structure of the matrix and thus may be applied to the solution of the systems in SDD 0 matrices that arise in any application, such as the

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Section: Subgraph Preconditionersmentioning

confidence: 99%

Section: E S = Rootedultrasparsify(e T R T P)mentioning

confidence: 99%

Nearly Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems

Spielman¹,

Teng²

2014

SIAM J. Matrix Anal. & Appl.

269

167

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“…It also uses insights and results in the work of Joshi, Reif, Gremban, Miller, Boman, Hendrickson, Maggs, Parekh, Ravi, Woo, Bern, Gilbert, Chen, Nguyen, Toledo (Boman and Hendrickson 2003;Bern et al 2006;Joshi 1997;Rief 1998;Gremban 1996;Maggs et al 2005).…”

Section: Definition 3 (Laplacian Primitive) This Primitive Concerns mentioning

confidence: 99%

Numerical Thinking in Algorithm Design and Analysis

Teng

2011

Computer Science

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“…There are other classes of combinatorial preconditioners, for example [21,20,29]. It is not clear whether they can be used effectively in our framework.…”

Section: Combinatorial Sparsification Of the Assembled Sdd Approximatmentioning

confidence: 99%

“…They show that under certain conditions on the continuous problem and on the finiteelement mesh, the approximations are all good. They proposed to use L to construct a combinatorial preconditioner rather then apply multigrid to L. This proposal was based on the observation that over the last decade, several provably good combinatorial graph algorithms for constructing preconditioners for SDD matrices have been developed [46,21,20,7,43,44,18,45,29]. Some of them, as well as some combinatorial heuristics, have been shown to be effective in practice [15,25,36,38,19,35].…”

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confidence: 99%

Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems

Avron¹,

Chen²,

Shklarski³

et al. 2009

SIAM J. Matrix Anal. & Appl.

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Abstract. We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). The solver splits the collection {Ke} of element matrices into a subset of matrices that are approximable by diagonally dominant matrices and a subset of matrices that are not approximable. The approximable Ke's are approximated by diagonally dominant matrices Le's that are assembled to form a global diagonally dominant matrix L. A combinatorial graph algorithm then approximates L by another diagonally dominant matrix M that is easier to factor. Finally, M is added to the inapproximable elements to form the preconditioner, which is then factored. When all the element matrices are approximable, which is often the case, the preconditioner is provably efficient. Approximating element matrices by diagonally dominant ones is not a new idea, but we present a new approximation method which is both efficient and provably good. The splitting idea is simple and natural in the context of combinatorial preconditioners, but hard to exploit in other preconditioning paradigms. Experimental results show that on problems in which some of the Ke's are ill conditioned, our new preconditioner is more effective than an algebraic multigrid solver, than an incomplete-factorization preconditioner, and than a direct solver.

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Finding effective support-tree preconditioners

Cited by 14 publications

References 22 publications

Nearly Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems

Nearly Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems

Numerical Thinking in Algorithm Design and Analysis

Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems

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