The graph partitioning problem is widely used and studied in many practical and theoretical applications. Today, multilevel strategies represent one of the most effective and efficient generic frameworks for solving t...
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When applying multilevel scheme to solve the graph partitioning problem, shortcomings and limitations exist in the state-of-the-art coarsening schemes depend mainly on finding maximal matchings to obtain the coarse gr...
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ISBN:
(纸本)9783037853849
When applying multilevel scheme to solve the graph partitioning problem, shortcomings and limitations exist in the state-of-the-art coarsening schemes depend mainly on finding maximal matchings to obtain the coarse graphs, which can cause the multilevel algorithms to produce poor-quality solutions. This paper proposes an improved coarsening scheme by improving vertex combining strategy and edge ordering criteria. The new coarsening scheme is more effective in quality, which is proved by both theoretical analysis and experimental results.
Linear ordering problems are combinatorial optimization problems that deal with the minimization of different functionals by finding a suitable permutation of the graph vertices. These problems are widely used and stu...
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Linear ordering problems are combinatorial optimization problems that deal with the minimization of different functionals by finding a suitable permutation of the graph vertices. These problems are widely used and studied in many practical and theoretical applications. In this paper, we present a variety of linear--time algorithms for these problems inspired by the Algebraic Multigrid approach, which is based on weighted-edge contraction. The experimental result for four such problems turned out to be better than every known result in almost all cases, while the short (linear) running time of the algorithms enables testing very large graphs.
We describe the engineering of the distributed-memory multilevel graph partitioner dKaMinPar. It scales to (at least) 8192 cores while achieving partitioning quality comparable to widely used sequential and shared-mem...
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ISBN:
(纸本)9783031396977;9783031396984
We describe the engineering of the distributed-memory multilevel graph partitioner dKaMinPar. It scales to (at least) 8192 cores while achieving partitioning quality comparable to widely used sequential and shared-memory graph partitioners. In comparison, previous distributed graph partitioners scale only in more restricted scenarios and often induce a considerable quality penalty compared to non-distributed partitioners. When partitioning into a large number of blocks, they even produce infeasible solution that violate the balancing constraint. dKaMinPar achieves its robustness by a scalable distributed implementation of the deep-multilevel scheme for graph partitioning. Crucially, this includes new algorithms for balancing during refinement and coarsening.
In this thesis we present results for the topological susceptibility χ YM, and investigate the property of factorization in the 't Hooft large N limit of SU(N) pure Yang-Mills gauge theory. The study of χ YM is ...
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In this thesis we present results for the topological susceptibility χ YM, and investigate the property of factorization in the 't Hooft large N limit of SU(N) pure Yang-Mills gauge theory. The study of χ YM is motivated by the Witten-Veneziano relation, which explains the large mass of the η 0 meson when compared to the rest of light pseudoscalar mesons. A key component in the lattice gauge theory computation of χ YM is the estimation of the topological charge density correlator, which is affected by a severe signal to noise problem. To alleviate this problem, we introduce a novel algorithm that uses a multilevel type approach to compute the correlation function of observables smoothed with the Yang-Mills gradient flow. When applied to the topological charge density and the Yang-Mills energy density, our results agree with a scaling of the error proportional to 1/n, instead of the 1/ √ n scaling from traditional Monte-Carlo simulations, where n is the number of independent measurements. We compute the topological susceptibility in the pure Yang-Mills gauge theory for the gauge groups with N = 4, 5, 6 and three different lattice spacings. In order to deal with the freezing of topology, we use open boundary conditions, which allows us to go to finer lattice spacings when compared to previous works in the literature. In addition, we employ the theoretically sound definition of the topological charge density through the gradient flow. Our final result for the dimensionless quantity t 2 0 χ YM = 7.03(13) × 10 −4 in the limit N → ∞, represents a new quality in the verification of the Witten-Veneziano formula. Lastly, we use the lattice formulation to verify the factorization of the expectation value of the product of gauge invariant operators in the large N limit. We work with Wilson loops smoothed with the Yang-Mills gradient flow and simulations up to the gauge group SU(8). Loops at different N are matched using the scale t 0, and thanks to the favourable renormalizati
Many real-world engineering problems can be expressed in terms of partial differential equations an solved by using the finite-element method, which is usually arallelized, i.e. the mesh is divided among several proce...
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Many real-world engineering problems can be expressed in terms of partial differential equations an solved by using the finite-element method, which is usually arallelized, i.e. the mesh is divided among several processors. To achieve high parallel efficiency it is important that the mesh is partitioned in such a way that workloads are well balanced and interprocessor communication is minimized In this paper we present an enhancement of a technique that uses a nature-inspired metaheuristir approach to achieve higher-quality partitions. We present two heuristic mesh-partitioning methods, both of which build on the multiple ant-colony algorithm in order to improve the quality of the mesh partitions. The first method augments the multiple ant-colony algorithm with a multilevel paradigm, whereas the second uses the multiple ant colony algorithm as a refinement to the initial partition obtained by vector quantization. The two methods are experimentally compared with the well-known mesh-partitioning programs, p-METIS and Chaco.
We study the convergence rate of multilevel algorithms from an algebraic point of view. This requires a detailed analysis of the constant in the strengthened Cauchy–Schwarz inequality between the coarse-grid space an...
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