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We present algorithms for solving symmetric, diagonally-dominant linear systems to accuracy ɛ in time linear in their number of non-zeros and log(κf (A)/ɛ), where κf (A) isthe condition number of the matrix defining the linear system. Our algorithm applies the preconditioned Chebyshev iteration with preconditioners designed using nearly-linear time algorithms for graph sparsification and graph partitioning.

Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity, and provide excellent graph partitions. However, there exists little theoretical analysis that could explain the ability of multilevel algorithms to produce good partitions. In this paper we present such an analysis. We show under certain reasonable assumptions that even if no refinement is used in the uncoarsening phase, a good bisection of the coarser graph is worse than a good bisection of the finer graph by at most a small factor. We also show that the size of a good vertex-separator of the coarse graph projected to the finer graph (without performing refinement in the uncoarsening phase) is higher than the size of a good vertexseparator of the finer graph by at most a small factor.

In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

For a large class of irregular mesh applications, the structure of the mesh changes from one phase of the computation to the next. Eventually, as the mesh evolves, the adapted mesh has to be repartitioned to ensure good load balance. If this new graph is partitioned from scratch, it may lead to an excessive migration of data among processors. In this paper, we present schemes for computing repartitionings of adaptively refined meshes that perform diffusion of

Graph partitioning is a fundamental problem in several scientific and engineering applications. In this paper, we describe heuristics that improve the state-of-the-art practical algorithms used in graph-partitioning software in terms of both partitioning speed and quality. An important use of graph-partitioning is in ordering sparse matrices for obtaining direct solutions to sparse systems of linear equations arising in engineering and optimization applications. The experiments reported in this paper show that the use of these heuristics results in a considerable improvement in the quality of sparse-matrix orderings over conventional ordering methods, especially for sparse matrices arising in linear programming problems. In addition, our graph-partitioning-based ordering algorithm is more parallelizable than minimum-degree-based ordering algorithms, and it renders the ordered matrix more amenable to parallel factorization.

In this paper we provide a robust reordering scheme for sparse matrices. The scheme relies on the notion of multisection, a generalization of bisection. The reordering strategy is demonstrated to have consistently good performance in terms of fill reduction when compared with multiple minimum degree and generalized nested dissection. Experimental results show that by using multisection, we obtain an ordering which is consistently as good as or better than both for a wide spectrum of sparse problems. 1 Introduction It is well recognized that finding a fill-reducing ordering is crucial in the success of the numerical solution of sparse linear systems. For symmetric positive-definite systems, the minimum degree [38] and the nested dissection [11] orderings are perhaps the most popular ordering schemes. They represent two opposite approaches to the ordering problem. However, they share a common undesirable characteristic. Both schemes produce generally good orderings, but the ordering qua...

This paper describes heuristics for partitioning a general M x N matrix into arrowhead form. Such heuristics are useful for decomposing large, constrained, optimization problems into forms that are amenable to parallel processing. The heuristics presented can be easily implemented using publicly available graph partitioning algorithms. The application of such techniques for solving large linear programs is described. Extensive computational results on the effectiveness of our partitioning procedures and their usefulness for parallel optimization are presented. @ 1998 The

We analyze single-node performance of sparse matrix-vector multiplication by investigating issues of data locality and fine-grained parallelism. We examine the data-locality characteristics of the compressedsparse -row representation and consider improvements in locality through matrix permutation. Motivated by potential improvements in fine-grained parallelism, we evaluate modified sparse-matrix representations. The results lead to general conclusions about improving single-node performance of sparse matrix-vector multiplication in parallel libraries of sparse iterative solvers. 1 Introduction One of the core operations of iterative sparse solvers is sparse matrix-vector multiplication. In order to achieve high performance, a parallel implementation of sparse matrix-vector multiplication must maintain scalability. This scalability comes from a balanced mapping of the matrix and vectors among the distributed processors, a mapping that minimizes interprocessor communication. Load balan...

In this paper we introduce a three-step approach to find a vertex bisector of a graph. The first step finds a domain decomposition of the graph, a set of connected subgraphs, the domains, and a multisector, the remaining vertices that separate the domains from each other. The second step uses a block variant of the Kernighan-Lin scheme to find a bisector that is a subset of the multisector. The third step improves the bisector by bipartite graph matching. Experimental results show this domain decomposition method finds graph partitions that compare favorably with a state-of-the-art multilevel partitioning scheme in both quality and execution time. 1 Introduction Graph partitioning is a well-known practical problem that has many important applications, such as task allocation for parallel computations [13] and circuit partitioning for VLSI design [22]. Our driving interest is to find low-fill orderings for sparse matrix computation [4], [6], [15], [19]. An effective approach to find fi...