The Role of Elimination Trees in Sparse Factorization

Liu, Joseph W. H.

doi:10.1137/0611010

Cited by 385 publications

(204 citation statements)

References 42 publications

Supporting

Mentioning

199

Contrasting

Unclassified

Order By: Relevance

“…Elimination trees have a wide array of uses and many interesting properties explained thoroughly by Liu [38]. Knowledge necessary for the reading of this paper is summarised here for convenience using the notation presented by Davis and Hager [13] whenever possible.…”

Section: Elimination Treementioning

confidence: 99%

“…When presented such that operations are on C like the algorithm used by Liu [38], the following is obtained:…”

Section: Symbolic Factorisationmentioning

confidence: 99%

“…where |L i | is the number of non-zero entries in column i of L [38]. This can be done as a two step process, with the symbolic Cholesky factorisation predicting which entries of L will be non-zero before calculating their actual value.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Efficient Cholesky Factor Recovery for Column Reordering in Simultaneous Localisation and Mapping

Touchette

Gueaieb

Lanteigne

2016

J Intell Robot Syst

View full text Add to dashboard Cite

Simultaneous Localisation And Mapping problems are inherently dynamic and the structure of the graph representing them changes significantly over time. To obtain the least square solution of such systems efficiently, it is desired to maintain a good column ordering such that fill-ins are reduced. This comes at a cost since general ordering changes require the complete re-computation of the Cholesky factor. While some methods have obtained good results with reordering at loop closing, the changes are not guaranteed to be limited to the scope of the loop, leading to suboptimal performance. In this article, it is shown that the Cholesky factorisation of an updated matrix can be efficiently recovered from the previous factorisation if the permutations are localised. This is experimentally demonstrated on 2D SLAM datasets. A method is then provided to identify when such recovery is advantageous over the complete re-computation of the Cholesky factor. Furthermore, a hybrid algorithm combining factorisation recovery and re-computation of the Cholesky factor is proposed for dynamically evolving problems and tested on SLAM datasets. Steps where reordering occurs can be executed up to 67% faster with the proposed method.

show abstract

Section: Elimination Treementioning

confidence: 99%

“…When presented such that operations are on C like the algorithm used by Liu [38], the following is obtained:…”

Section: Symbolic Factorisationmentioning

confidence: 99%

See 1 more Smart Citation

Efficient Cholesky Factor Recovery for Column Reordering in Simultaneous Localisation and Mapping

Touchette

Gueaieb

Lanteigne

2016

J Intell Robot Syst

View full text Add to dashboard Cite

show abstract

“…Since a poor symbolic factorization leads to an expensive numeric factorization, symbolic factorization has received considerable attention in the literature (Liu, 1989;Duff et al, 1990;.…”

Section: Sparse Solversmentioning

confidence: 99%

Addressing the computational cost of large EIT solutions

2012

View full text Add to dashboard Cite

Abstract. Electrical Impedance Tomography (EIT) is a soft field tomography modality based on the application of electric current to a body and measurement of voltages through electrodes at the boundary. The interior conductivity is reconstructed on a discrete representation of the domain using a FEM mesh and a parametrization of that domain. The reconstruction requires a sequence of numerically intensive calculations. There is strong interest in reducing the cost of these calculations.An improvement in the compute time for current problems would encourage further exploration of computationally challenging problems such as the incorporation of time series data, wide-spread adoption of three-dimensional simulations, and correlation of other modalities such as CT and ultrasound. Multicore processors offer an opportunity to reduce EIT computation times but may require some restructuring of the underlying algorithms to maximize the use of available resources.This work profiles two EIT software packages (EIDORS and NDRM) to experimentally determine where the computational costs arise in EIT as problems scale. Sparse matrix solvers, a key component for the FEM forward problem and sensitivity estimates in the inverse problem, are shown to take a considerable portion of the total compute time in these packages. A sparse matrix solver performance measurement tool, Meagre-Crowd, is developed to interface with a variety of solvers and compare their performance over a range of two-and threedimensional problems of increasing node density. Results show that distributed sparse matrix solvers that operate on multiple cores are advantageous up to a limit that increases as the node density increases. We recommend a selection procedure to find a solver and hardware arrangement matched to the problem and provide guidance and tools to perform that selection.

show abstract

“…We denote the treedepth of a graph G by td(G). The vertex ranking number finds applications in sparse matrix factorization [13,18,20] and VLSI layout problems [19]. This notion also has important connections to the structure of sparse graphs.…”

Section: Introductionmentioning

confidence: 99%

A Faster Parameterized Algorithm for Treedepth

Reidl

Rossmanith

Villaamil

et al. 2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which-given as input an n-vertex graph, a tree decomposition of the graph of width w, and an integer t-decides Treedepth, i.e. whether the treedepth of the graph is at most t, in time 2 O(wt) · n. If necessary, a witness structure for the treedepth can be constructed in the same running time. In conjunction with previous results we provide a simple algorithm and a fast algorithm which decide treedepth in time 2 2 O(t) · n and 2 O(t 2 ) · n, respectively, which do not require a tree decomposition as part of their input. The former answers an open question posed by Ossona de Mendez and Nešetřil as to whether deciding Treedepth admits an algorithm with a linear running time (for every fixed t) that does not rely on Courcelle's Theorem or other heavy machinery. For chordal graphs we can prove a running time of 2 O(t log t) · n for the same algorithm. * Research funded by DFG-Project RO 927/13-1 "Pragmatic Parameterized Algorithms".

show abstract

The Role of Elimination Trees in Sparse Factorization

Cited by 385 publications

References 42 publications

Efficient Cholesky Factor Recovery for Column Reordering in Simultaneous Localisation and Mapping

Efficient Cholesky Factor Recovery for Column Reordering in Simultaneous Localisation and Mapping

Addressing the computational cost of large EIT solutions

A Faster Parameterized Algorithm for Treedepth

Contact Info

Product

Resources

About