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Nested dissection: A survey and comparison of various nested dissection algorithms
, 1992
"... Methods for solving sparse linear systems of equations can be categorized under two broad classes direct and iterative. Direct methods are methods based on gaussian elimination. This report discusses one such direct method namely Nested dissection. Nested Dissection, originally proposed by Alan Geo ..."
Abstract

Cited by 8 (1 self)
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Methods for solving sparse linear systems of equations can be categorized under two broad classes direct and iterative. Direct methods are methods based on gaussian elimination. This report discusses one such direct method namely Nested dissection. Nested Dissection, originally proposed by Alan George, is a technique for solving sparse linear systems efficiently. This report is a survey of some of the work in the area of nested dissection and attempts to put it together using a common framework.
On the stabbing number of a random Delaunay triangulation
"... We consider a Delaunay triangulation defined on n points distributed independently and uniformly on a planar compact convex set of positive volume. Let the stabbing number be the maximal number of intersections between a line and edges of the triangulation. We show that the stabbing number Sn is Θ ..."
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Cited by 6 (0 self)
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We consider a Delaunay triangulation defined on n points distributed independently and uniformly on a planar compact convex set of positive volume. Let the stabbing number be the maximal number of intersections between a line and edges of the triangulation. We show that the stabbing number Sn is Θ ( √ n) in the mean, and provide tail bounds for P{Sn ≥ t √ n}. Applications to planar point location, nearest neighbor searching, range queries, planar separator determination, approximate shortest paths, and the diameter of the Delaunay triangulation are discussed.
MinMaxBoundary Domain Decomposition
 Theor. Comput. Sci
, 1998
"... Domain decomposition is one of the most effective and popular parallel computing techniques for solving large scale numerical systems. In the special case when the amount of computation in a subdomain is proportional to the volume of the subdomain, domain decomposition amounts to minimizing the surf ..."
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Cited by 5 (1 self)
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Domain decomposition is one of the most effective and popular parallel computing techniques for solving large scale numerical systems. In the special case when the amount of computation in a subdomain is proportional to the volume of the subdomain, domain decomposition amounts to minimizing the surface area of each subdomain while dividing the volume evenly. Motivated by this fact, we study the following minmax boundary multiway partitioning problem: Given a graph G and an integer k ? 1, we would like to divide G into k subgraphs G 1 ; : : : ; G k (by removing edges) such that (i) jG i j = \Theta(jGj=k) for all i 2 f1; : : : ; kg; and (ii) the maximum boundary size of any subgraph (the set of edges connecting it with other subgraphs) is minimized. We provide an algorithm that given G, a wellshaped mesh in d dimensions, finds a partition of G into k subgraphs G 1 ; : : : ; G k , such that for all i, G i has \Theta(jGj=k) vertices and the number of edges connecting G i with the ot...
PARALLEL UNSYMMETRICPATTEN MULTIFRONTAL SPARSE LU WITH COLUMN PREORDERING
"... Abstract. We present a new parallel sparse LU factorization algorithm and code. The algorithm uses a columnpreordering partialpivoting unsymmetricpattern multifrontal approach. Our baseline sequential algorithm is based on umfpack 4 but is somewhat simpler and is often somewhat faster than umfpac ..."
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Abstract. We present a new parallel sparse LU factorization algorithm and code. The algorithm uses a columnpreordering partialpivoting unsymmetricpattern multifrontal approach. Our baseline sequential algorithm is based on umfpack 4 but is somewhat simpler and is often somewhat faster than umfpack version 4.0. Our parallel algorithm is designed for sharedmemory machines with a small or moderate number of processors (we tested it on up to 32 processors). We experimentally compare our algorithm with SuperLU MT, an existing sharedmemory sparse LU factorization with partial pivoting. SuperLU MT scales better than our new algorithm, but our algorithm is more reliable and is usually faster in absolute (on up to 16 processors; we were not able to run SuperLU MT on 32). More specifically, on large matrices our algorithm is always faster on up to 4 processors, and is usually faster on 8 and 16. The main contribution of this paper is showing that the columnpreordering partialpivoting unsymmetricpattern multifrontal approach, developed as a sequential algorithm by Davis in several recent versions of umfpack, can be effectively parallelized. 1.
NestedDissection Orderings for Sparse LU with Partial Pivoting
 SIAM J. Matrix Anal. Appl
, 2000
"... this paper does not apply to them.) ..."