5,565 research outputs found
A Direct Elliptic Solver Based on Hierarchically Low-rank Schur Complements
A parallel fast direct solver for rank-compressible block tridiagonal linear
systems is presented. Algorithmic synergies between Cyclic Reduction and
Hierarchical matrix arithmetic operations result in a solver with arithmetic complexity and memory footprint. We provide a
baseline for performance and applicability by comparing with well known
implementations of the -LU factorization and algebraic multigrid
with a parallel implementation that leverages the concurrency features of the
method. Numerical experiments reveal that this method is comparable with other
fast direct solvers based on Hierarchical Matrices such as -LU and
that it can tackle problems where algebraic multigrid fails to converge
On sparse representations of linear operators and the approximation of matrix products
Thus far, sparse representations have been exploited largely in the context
of robustly estimating functions in a noisy environment from a few
measurements. In this context, the existence of a basis in which the signal
class under consideration is sparse is used to decrease the number of necessary
measurements while controlling the approximation error. In this paper, we
instead focus on applications in numerical analysis, by way of sparse
representations of linear operators with the objective of minimizing the number
of operations needed to perform basic operations (here, multiplication) on
these operators. We represent a linear operator by a sum of rank-one operators,
and show how a sparse representation that guarantees a low approximation error
for the product can be obtained from analyzing an induced quadratic form. This
construction in turn yields new algorithms for computing approximate matrix
products.Comment: 6 pages, 3 figures; presented at the 42nd Annual Conference on
Information Sciences and Systems (CISS 2008
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