770 research outputs found
Analysis of A Splitting Approach for the Parallel Solution of Linear Systems on GPU Cards
We discuss an approach for solving sparse or dense banded linear systems
on a Graphics Processing Unit (GPU) card. The
matrix is possibly nonsymmetric and
moderately large; i.e., . The ${\it split\ and\
parallelize}{\tt SaP}{\bf A}{\bf A}_ii=1,\ldots,P{\bf A}_i{\tt SaP::GPU}{\tt PARDISO}{\tt SuperLU}{\tt MUMPS}{\tt SaP::GPU}{\tt MKL}{\tt SaP::GPU}{\tt SaP::GPU}$ is publicly available and distributed as
open source under a permissive BSD3 license.Comment: 38 page
Dendrogram seriation in data visualisation: algorithms and applications
Seriation is a data analytic tool for obtaining a permutation of a set of objects
with the goal of revealing structural information within the set of objects. The
purpose of this thesis is to investigate and develop tools for seriation with the
goal of using these tools to enhance data visualisation.
The particular focus of this thesis is on dendrogram seriation algorithms.
A dendrogram is a tree-like structure used for visualising the results of a hierarchical
clustering and the order of the leaves in a dendrogram provides a
permutation of a set of objects. Dendrogram seriation algorithms rearrange
the leaves of a dendrogram in order to find a permutation that optimises a
given criterion.
Dendrogram seriation algorithms are widely used, however, the research in
this area is often confusing because of inconsistent or inadequate terminology.
This thesis proposes new notation and terminology with the goal of better
understanding and comparing dendrogram seriation algorithms.
Seriation criteria measure the goodness of a permutation of a set of objects.
Popular seriation criteria include the path length of a permutation and measuring
anti-Robinson form in a symmetric matrix. This thesis proposes two
new seriation criteria, lazy path length and banded anti-Robinson form,
and demonstrates their effectiveness in improving a variety of visualisations.
The main contribution of this thesis is a new dendrogram seriation algorithm.
This algorithm improves on other dendrogram seriation algorithms and
is also flexible because it allows the user to either choose from a variety of seriation
criteria, including the new criteria mentioned above, or to input their
own criteria.
Finally, this thesis performs a comparison of several seriation algorithms,
the results of which show that the proposed algorithm performs competitively
against other algorithms. This leads to a set of general guidelines for choosing
the most appropriate seriation algorithm for different seriation interests and
visualisation settings
A New Flexible Dendrogram Seriation Algorithm for Data Visualisation
Seriation is a data analytic tool for obtaining a permutation of a set of objects with the goal
of revealing structural information within the set of objects. Seriating variables, cases or categories
generally improves visualisations of statistical data, for example, by revealing hidden patterns in data
or by making large datasets easier to understand. In this paper we present a new algorithm for seriation
based on dendrograms. Dendrogram seriation algorithms rearrange the nodes in a dendrogram in order
to obtain a permutation of the leaves (i.e. objects) that optimises a given criterion. Our algorithm is
more flexible than currently available seriation algorithms because it allows the user to either choose
from a variety of seriation criteria or to input their own criteria. This choice of seriation criteria is
an important feature because different criteria are suitable for different visualisation settings. Common
seriation criteria include measurements of the path length through a set of objects and measurements
of anti-Robinson form in a symmetric matrix. We propose new seriation criteria called lazy path
length and banded anti-Robinson form, and demonstrate their effectiveness in a variety of visualisation
settings
A New Flexible Dendrogram Seriation Algorithm for Data Visualisation
Seriation is a data analytic tool for obtaining a permutation of a set of objects with the goal
of revealing structural information within the set of objects. Seriating variables, cases or categories
generally improves visualisations of statistical data, for example, by revealing hidden patterns in data
or by making large datasets easier to understand. In this paper we present a new algorithm for seriation
based on dendrograms. Dendrogram seriation algorithms rearrange the nodes in a dendrogram in order
to obtain a permutation of the leaves (i.e. objects) that optimises a given criterion. Our algorithm is
more flexible than currently available seriation algorithms because it allows the user to either choose
from a variety of seriation criteria or to input their own criteria. This choice of seriation criteria is
an important feature because different criteria are suitable for different visualisation settings. Common
seriation criteria include measurements of the path length through a set of objects and measurements
of anti-Robinson form in a symmetric matrix. We propose new seriation criteria called lazy path
length and banded anti-Robinson form, and demonstrate their effectiveness in a variety of visualisation
settings
Communication aspects of parallel processing
Cover title.Includes bibliographical references.Supported in part by the Air Force Office of Scientific Research. AFOSR-88-0032Cüneyt Özveren
Algebraic, Block and Multiplicative Preconditioners based on Fast Tridiagonal Solves on GPUs
This thesis contributes to the field of sparse linear algebra, graph applications, and preconditioners for Krylov iterative solvers of sparse linear equation systems, by providing a (block) tridiagonal solver library, a generalized sparse matrix-vector implementation, a linear forest extraction, and a multiplicative preconditioner based on tridiagonal solves. The tridiagonal library, which supports (scaled) partial pivoting, outperforms cuSPARSE's tridiagonal solver by factor five while completely utilizing the available GPU memory bandwidth. For the performance optimized solving of multiple right-hand sides, the explicit factorization of the tridiagonal matrix can be computed. The extraction of a weighted linear forest (union of disjoint paths) from a general graph is used to build algebraic (block) tridiagonal preconditioners and deploys the generalized sparse-matrix vector implementation of this thesis for preconditioner construction. During linear forest extraction, a new parallel bidirectional scan pattern, which can operate on double-linked list structures, identifies the path ID and the position of a vertex. The algebraic preconditioner construction is also used to build more advanced preconditioners, which contain multiple tridiagonal factors, based on generalized ILU factorizations. Additionally, other preconditioners based on tridiagonal factors are presented and evaluated in comparison to ILU and ILU incomplete sparse approximate inverse preconditioners (ILU-ISAI) for the solution of large sparse linear equation systems from the Sparse Matrix Collection. For all presented problems of this thesis, an efficient parallel algorithm and its CUDA implementation for single GPU systems is provided
Decay properties of spectral projectors with applications to electronic structure
Motivated by applications in quantum chemistry and solid state physics, we
apply general results from approximation theory and matrix analysis to the
study of the decay properties of spectral projectors associated with large and
sparse Hermitian matrices. Our theory leads to a rigorous proof of the
exponential off-diagonal decay ("nearsightedness") for the density matrix of
gapped systems at zero electronic temperature in both orthogonal and
non-orthogonal representations, thus providing a firm theoretical basis for the
possibility of linear scaling methods in electronic structure calculations for
non-metallic systems. We further discuss the case of density matrices for
metallic systems at positive electronic temperature. A few other possible
applications are also discussed.Comment: 63 pages, 13 figure
Sparse Matrix Multiplication on a Field-Programmable Gate Array
To extract data from highly sophisticated sensor networks, algorithms derived from graph theory are often applied to raw sensor data. Embedded digital systems are used to apply these algorithms. A common computation performed in these algorithms is finding the product of two sparsely populated matrices. When processing a sparse matrix, certain optimizations can be made by taking advantage of the large percentage of zero entries. This project proposes an optimized algorithm for performing sparse matrix multiplications in an embedded system and investigates how a parallel architecture constructed of multiple processors on a single Field-Programmable Gate Array (FPGA) can be used to speed up computations
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