Interactively Exploring the Connection between Nested Dissection Orderings for Parallel Cholesky Factorization and Vertex Separators

2014 ◽  
Author(s):  
H. Martin Bucker ◽  
M. Ali Rostami
Keyword(s):  
Cholesky Factorization ◽  
Nested Dissection ◽  
Vertex Separators

scholarly journals Practical Compact Routing on Sparse Graphs

2021 ◽  
Author(s):  
Kai Hormann ◽  
Craig Gotsman
Keyword(s):  
Routing Algorithm ◽  
Nested Dissection ◽  
Sparse Graphs ◽  
Compact Routing ◽  
Vertex Graph ◽  
Vertex Separators ◽  
Practical Algorithm ◽  
Routing Tables ◽  
Basic Version ◽  
Advanced Version

We describe a simple and practical algorithm for compact routing on graphs which admit compact and balanced vertex separators. Using a recursive nested dissection of then-vertex graph based on these separators, we construct routing tables with as few as O(log n) entries per vertex in a preprocessing step. They support handshaking-based routing on the graph with moderate stretch, where the handshaking can be implemented similarly to a DNS lookup. We describe a basic version of the algorithm that requires modifiable headers and a more advanced version which eliminates this need and provides better stretch. A number of algorithmic parameters control a graceful tradeoff between the size of the routing tables and the stretch. Our routing algorithm is most effective on planar graphs and unit disk graphs of moderate edge/vertex density.

scholarly journals Practical Compact Routing on Sparse Graphs

2021 ◽  
Author(s):  
Kai Hormann ◽  
Craig Gotsman
Keyword(s):  
Routing Algorithm ◽  
Nested Dissection ◽  
Sparse Graphs ◽  
Compact Routing ◽  
Vertex Graph ◽  
Vertex Separators ◽  
Practical Algorithm ◽  
Routing Tables ◽  
Basic Version ◽  
Advanced Version

We describe a simple and practical algorithm for compact routing on graphs which admit compact and balanced vertex separators. Using a recursive nested dissection of then-vertex graph based on these separators, we construct routing tables with as few as O(log n) entries per vertex in a preprocessing step. They support handshaking-based routing on the graph with moderate stretch, where the handshaking can be implemented similarly to a DNS lookup. We describe a basic version of the algorithm that requires modifiable headers and a more advanced version which eliminates this need and provides better stretch. A number of algorithmic parameters control a graceful tradeoff between the size of the routing tables and the stretch. Our routing algorithm is most effective on planar graphs and unit disk graphs of moderate edge/vertex density.

Implementing Sparse Matrix Ordering Using Hypergraph Partitioning

2013 ◽  
Vol 706-708 ◽  
pp. 1890-1893
Author(s):  
Lu Yao ◽  
Yi Yang ◽  
Zheng Hua Wang ◽  
Wei Cao
Keyword(s):  
Sparse Matrix ◽  
Factorization Method ◽  
Cholesky Factorization ◽  
The Novel ◽  
Nested Dissection ◽  
Vertex Separator ◽  
Hypergraph Partitioning ◽  
Ordering Algorithms ◽  
Sparse Matrix Ordering ◽  
Matrix Ordering

Matrix ordering is a key technique when applying Cholesky factorization method to solving sparse symmetric positive definite system Ax = b. Much effort has been devoted to the development of powerful heuristic ordering algorithms. This paper implements a sparse matrix ordering scheme based on hypergraph partitioning. The novel nested dissection ordering scheme achieve the vertex separator by hypergraph partitioning. Experimental results show that the novel scheme produces results that are substantially better than METIS.

scholarly journals Fast kernel k-means clustering using incomplete Cholesky factorization

2021 ◽  
Vol 402 ◽  
pp. 126037
Author(s):  
Li Chen ◽  
Shuisheng Zhou ◽  
Jiajun Ma ◽  
Mingliang Xu
Keyword(s):  
Cholesky Factorization ◽  
Incomplete Cholesky Factorization

scholarly journals High-performance sampling of generic determinantal point processes

2020 ◽  
Vol 378 (2166) ◽  
pp. 20190059 ◽  
Author(s):  
Jack Poulson
Keyword(s):  
Point Process ◽  
Spectral Decomposition ◽  
High Performance ◽  
Point Processes ◽  
Cholesky Factorization ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
Decomposition Approach ◽  
Map Inference ◽  
Sampling Schemes

Determinantal point processes (DPPs) were introduced by Macchi (Macchi 1975 Adv. Appl. Probab. 7 , 83–122) as a model for repulsive (fermionic) particle distributions. But their recent popularization is largely due to their usefulness for encouraging diversity in the final stage of a recommender system (Kulesza & Taskar 2012 Found. Trends Mach. Learn. 5 , 123–286). The standard sampling scheme for finite DPPs is a spectral decomposition followed by an equivalent of a randomly diagonally pivoted Cholesky factorization of an orthogonal projection, which is only applicable to Hermitian kernels and has an expensive set-up cost. Researchers Launay et al. 2018 ( http://arxiv.org/abs/1802.08429 ); Chen & Zhang 2018 NeurIPS ( https://papers.nips.cc/paper/7805-fast-greedy-map-inference-for-determinantal-point-process-to-improve-recommendation-diversity.pdf ) have begun to connect DPP sampling to LDL H factorizations as a means of avoiding the initial spectral decomposition, but existing approaches have only outperformed the spectral decomposition approach in special circumstances, where the number of kept modes is a small percentage of the ground set size. This article proves that trivial modifications of LU and LDL H factorizations yield efficient direct sampling schemes for non-Hermitian and Hermitian DPP kernels, respectively. Furthermore, it is experimentally shown that even dynamically scheduled, shared-memory parallelizations of high-performance dense and sparse-direct factorizations can be trivially modified to yield DPP sampling schemes with essentially identical performance. The software developed as part of this research, Catamari ( hodgestar.com/catamari ) is released under the Mozilla Public License v.2.0. It contains header-only, C++14 plus OpenMP 4.0 implementations of dense and sparse-direct, Hermitian and non-Hermitian DPP samplers. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

LOW AND HIGH ORDER FINITE ELEMENT METHOD: EXPERIENCE IN SEISMIC MODELING

1994 ◽  
Vol 02 (04) ◽  
pp. 371-422 ◽  
Author(s):  
E. PADOVANI ◽  
E. PRIOLO ◽  
G. SERIANI
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Computational Efficiency ◽  
High Order ◽  
Cholesky Factorization ◽  
Complex Geometries ◽  
Spectral Elements ◽  
Seismic Modeling ◽  
Static Condensation ◽  
Element Method

The finite element method (FEM) is a numerical technique well suited to solving problems of elastic wave propagation in complex geometries and heterogeneous media. The main advantages are that very irregular grids can be used, free surface boundary conditions can be easily taken into account, a good reconstruction is possible of irregular surface topography, and complex geometries, such as curved, dipping and rough interfaces, intrusions, cusps, and holes can be defined. The main drawbacks of the classical approach are the need for a large amount of memory, low computational efficiency, and the possible appearance of spurious effects. In this paper we describe some experience in improving the computational efficiency of a finite element code based on a global approach, and used for seismic modeling in geophysical oil exploration. Results from the use of different methods and models run on a mini-superworkstation APOLLO DN10000 are reported and compared. With Chebyshev spectral elements, great accuracy can be reached with almost no numerical artifacts. Static condensation of the spectral element's internal nodes dramatically reduces memory requirements and CPU time. Time integration performed with the classical implicit Newmark scheme is very accurate but not very efficient. Due to the high sparsity of the matrices, the use of compressed storage is shown to greatly reduce not only memory requirements but also computing time. The operation which most affects the performance is the matrix-by-vector product; an effective programming of this subroutine for the storage technique used is decisive. The conjugate gradient method preconditioned by incomplete Cholesky factorization provides, in general, a good compromise between efficiency and memory requirements. Spectral elements greatly increase its efficiency, since the number of iterations is reduced. The most efficient and accurate method is a hybrid iterative-direct solution of the linear system arising from the static condensation of high order elements. The size of 2D models that can be handled in a reasonable time on this kind of computer is nowadays hardly sufficient, and significant 3D modeling is completely unfeasible. However the introduction of new FEM algorithms coupled with the use of new computer architectures is encouraging for the future.

Incomplete Nested Dissection for Solving n by n Grid Problems

1978 ◽  
Vol 15 (4) ◽  
pp. 662-673 ◽  
Author(s):  
Alan George ◽  
William G. Poole, Jr. ◽  
Robert G. Voigt
Keyword(s):  
Nested Dissection

FSCHOL: An OpenCL-based HPC Framework for Accelerating Sparse Cholesky Factorization on FPGAs

2021 ◽  
Author(s):  
Erfan Bank Tavakoli ◽  
Michael Riera ◽  
Masudul Hassan Quraishi ◽  
Fengbo Ren
Keyword(s):  
Cholesky Factorization ◽  
Sparse Cholesky Factorization
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