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In applications ranging from DNA sequencing through archeological dating to sparse matrix reordering, a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other. That is, given a correlation function f reflecting the desire for each pair of elements to be near each other, find all… (More)

We present support theory, a set of techniques for bounding extreme eigenvalues and condition numbers for matrix pencils. Our intended application of support theory is to enable proving condition number bounds for preconditioners for symmetric, positive definite systems. One key feature sets our approach apart from most other works: We use support numbers… (More)

1. Introduction. Partitioning and load-balancing are important issues in parallel scientific computing. The goal is to distribute data (and work) evenly among processors such as to reduce communication cost and achieve maximal performance. Graph partitioning has long served as a useful model for load balancing in parallel computing. Data are represented as… (More)

We show in this note how support preconditioners can be applied to a class of linear systems arising from use of the finite element method to solve linear elliptic problems. Our technique reduces the problem, which is symmetric and positive definite, to a symmetric positive definite diagonally dominant problem. Significant theory has already been developed… (More)

In parallel adaptive applications, the computational structure of the applications changes over time, leading to load imbalances even though the initial load distributions were balanced. To restore balance and to keep communication volume low in further iterations of the applications, dynamic load balancing (repartitioning) of the changed computational… (More)

This paper analyzes a novel method for constructing preconditioners for diagonally-dominant symmetric positive-definite matrices. The method discussed here is based on a simple idea: we construct M by simply dropping offdiagonal nonzeros from A and modifying the diagonal elements to maintain a certain row-sum property. The preconditioners are extensions of… (More)

In this paper we present HUND, a hypergraph-based unsymmetric nested dissection ordering algorithm for reducing the fill-in incurred during Gaussian elimination. HUND has several important properties. It takes a global perspective of the entire matrix, as opposed to local heuristics. It takes into account the assymetry of the input matrix by using a… (More)

Partitioning and load balancing are important problems in scientific computing that can be modeled as combinatorial problems using graphs or hypergraphs. The Zoltan toolkit was developed primarily for partitioning and load balancing to support dynamic parallel applications, but has expanded to support other problems in combinatorial scientific computing,… (More)

Adaptive scientific computations require that periodic repartitioning (load balancing) occur dynamically to maintain load balance. Hypergraph partitioning is a successful model for minimizing communication volume in scientific computations, and partitioning software for the static case is widely available. In this paper, we present a new hy-pergraph model… (More)