7,144 research outputs found
A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization
We present a distributed-memory library for computations with dense
structured matrices. A matrix is considered structured if its off-diagonal
blocks can be approximated by a rank-deficient matrix with low numerical rank.
Here, we use Hierarchically Semi-Separable representations (HSS). Such matrices
appear in many applications, e.g., finite element methods, boundary element
methods, etc. Exploiting this structure allows for fast solution of linear
systems and/or fast computation of matrix-vector products, which are the two
main building blocks of matrix computations. The compression algorithm that we
use, that computes the HSS form of an input dense matrix, relies on randomized
sampling with a novel adaptive sampling mechanism. We discuss the
parallelization of this algorithm and also present the parallelization of
structured matrix-vector product, structured factorization and solution
routines. The efficiency of the approach is demonstrated on large problems from
different academic and industrial applications, on up to 8,000 cores.
This work is part of a more global effort, the STRUMPACK (STRUctured Matrices
PACKage) software package for computations with sparse and dense structured
matrices. Hence, although useful on their own right, the routines also
represent a step in the direction of a distributed-memory sparse solver
Fast linear algebra is stable
In an earlier paper, we showed that a large class of fast recursive matrix
multiplication algorithms is stable in a normwise sense, and that in fact if
multiplication of -by- matrices can be done by any algorithm in
operations for any , then it can be done
stably in operations for any . Here we extend
this result to show that essentially all standard linear algebra operations,
including LU decomposition, QR decomposition, linear equation solving, matrix
inversion, solving least squares problems, (generalized) eigenvalue problems
and the singular value decomposition can also be done stably (in a normwise
sense) in operations.Comment: 26 pages; final version; to appear in Numerische Mathemati
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
Acceleration of Randomized Kaczmarz Method via the Johnson-Lindenstrauss Lemma
The Kaczmarz method is an algorithm for finding the solution to an
overdetermined consistent system of linear equations Ax=b by iteratively
projecting onto the solution spaces. The randomized version put forth by
Strohmer and Vershynin yields provably exponential convergence in expectation,
which for highly overdetermined systems even outperforms the conjugate gradient
method. In this article we present a modified version of the randomized
Kaczmarz method which at each iteration selects the optimal projection from a
randomly chosen set, which in most cases significantly improves the convergence
rate. We utilize a Johnson-Lindenstrauss dimension reduction technique to keep
the runtime on the same order as the original randomized version, adding only
extra preprocessing time. We present a series of empirical studies which
demonstrate the remarkable acceleration in convergence to the solution using
this modified approach
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