1,570 research outputs found
PT-Scotch: A tool for efficient parallel graph ordering
The parallel ordering of large graphs is a difficult problem, because on the
one hand minimum degree algorithms do not parallelize well, and on the other
hand the obtainment of high quality orderings with the nested dissection
algorithm requires efficient graph bipartitioning heuristics, the best
sequential implementations of which are also hard to parallelize. This paper
presents a set of algorithms, implemented in the PT-Scotch software package,
which allows one to order large graphs in parallel, yielding orderings the
quality of which is only slightly worse than the one of state-of-the-art
sequential algorithms. Our implementation uses the classical nested dissection
approach but relies on several novel features to solve the parallel graph
bipartitioning problem. Thanks to these improvements, PT-Scotch produces
consistently better orderings than ParMeTiS on large numbers of processors
High performance interior point methods for three-dimensional finite element limit analysis
The ability to obtain rigorous upper and lower bounds on collapse loads of various structures makes finite element limit analysis an attractive design tool. The increasingly high cost of computing those bounds, however, has limited its application on problems in three dimensions. This work reports on a high-performance homogeneous self-dual primal-dual interior point method developed for three-dimensional finite element limit analysis. This implementation achieves convergence times over 4.5× faster than the leading commercial solver across a set of three-dimensional finite element limit analysis test problems, making investigation of three dimensional limit loads viable. A comparison between a range of iterative linear solvers and direct methods used to determine the search direction is also provided, demonstrating the superiority of direct methods for this application. The components of the interior point solver considered include the elimination of and options for handling remaining free variables, multifrontal and supernodal Cholesky comparison for computing the search direction, differences between approximate minimum degree [1] and nested dissection [13] orderings, dealing with dense columns and fixed variables, and accelerating the linear system solver through parallelization. Each of these areas resulted in an improvement on at least one of the problems in the test set, with many achieving gains across the whole set. The serial implementation achieved runtime performance 1.7× faster than the commercial solver Mosek [5]. Compared with the parallel version of Mosek, the use of parallel BLAS routines in the supernodal solver saw a 1.9× speedup, and with a modified version of the GPU-enabled CHOLMOD [11] and a single NVIDIA Tesla K20c this speedup increased to 4.65×
Engineering Data Reduction for Nested Dissection
Many applications rely on time-intensive matrix operations, such as
factorization, which can be sped up significantly for large sparse matrices by
interpreting the matrix as a sparse graph and computing a node ordering that
minimizes the so-called fill-in. In this paper, we engineer new data reduction
rules for the minimum fill-in problem, which significantly reduce the size of
the graph while producing an equivalent (or near-equivalent) instance. By
applying both new and existing data reduction rules exhaustively before nested
dissection, we obtain improved quality and at the same time large improvements
in running time on a variety of instances. Our overall algorithm outperforms
the state-of-the-art significantly: it not only yields better elimination
orders, but it does so significantly faster than previously possible. For
example, on road networks, where nested dissection algorithms are typically
used as a preprocessing step for shortest path computations, our algorithms are
on average six times faster than Metis while computing orderings with less
fill-in
PACE Solver Description: Tweed-Plus: A Subtree-Improving Heuristic Solver for Treedepth
This paper introduces Tweed-Plus, a heuristic solver for the treedepth problem. The solver uses two well-known algorithms to create an initial elimination tree: nested dissection (making use of the Metis library) and the minimum-degree heuristic. After creating an elimination tree of the entire input graph, the solver continues to apply nested dissection and the minimum-degree heuristic to parts of the graph with the aim of replacing subtrees of the elimination tree with alternatives of lower depth
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