29,421 research outputs found
The Complexity Of The NP-Class
This paper presents a novel and straight formulation, and gives a complete
insight towards the understanding of the complexity of the problems of the so
called NP-Class. In particular, this paper focuses in the Searching of the
Optimal Geometrical Structures and the Travelling Salesman Problems. The main
results are the polynomial reduction procedure and the solution to the Noted
Conjecture of the NP-Class
A Library for Pattern-based Sparse Matrix Vector Multiply
Pattern-based Representation (PBR) is a novel approach to improving the performance of Sparse Matrix-Vector Multiply (SMVM) numerical kernels. Motivated by our observation that many matrices can be divided into blocks that share a small number of distinct patterns, we generate custom multiplication kernels for frequently recurring block patterns.
The resulting reduction in index overhead significantly reduces memory bandwidth requirements and improves performance. Unlike existing methods, PBR requires neither detection of dense blocks nor zero filling, making it particularly advantageous for matrices that lack dense nonzero concentrations. SMVM kernels for PBR can benefit from explicit prefetching and vectorization, and are amenable to parallelization. The analysis and format conversion to PBR is implemented as a library, making it suitable for applications that generate matrices dynamically at runtime. We present sequential and parallel performance results for PBR on two current multicore architectures, which show that PBR outperforms available alternatives for the matrices to which it is applicable,
and that the analysis and conversion overhead is amortized in realistic application scenarios
Visualizing dimensionality reduction of systems biology data
One of the challenges in analyzing high-dimensional expression data is the
detection of important biological signals. A common approach is to apply a
dimension reduction method, such as principal component analysis. Typically,
after application of such a method the data is projected and visualized in the
new coordinate system, using scatter plots or profile plots. These methods
provide good results if the data have certain properties which become visible
in the new coordinate system and which were hard to detect in the original
coordinate system. Often however, the application of only one method does not
suffice to capture all important signals. Therefore several methods addressing
different aspects of the data need to be applied. We have developed a framework
for linear and non-linear dimension reduction methods within our visual
analytics pipeline SpRay. This includes measures that assist the interpretation
of the factorization result. Different visualizations of these measures can be
combined with functional annotations that support the interpretation of the
results. We show an application to high-resolution time series microarray data
in the antibiotic-producing organism Streptomyces coelicolor as well as to
microarray data measuring expression of cells with normal karyotype and cells
with trisomies of human chromosomes 13 and 21
A robust and efficient implementation of LOBPCG
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely
used to compute eigenvalues of large sparse symmetric matrices. The algorithm
can suffer from numerical instability if it is not implemented with care. This
is especially problematic when the number of eigenpairs to be computed is
relatively large. In this paper we propose an improved basis selection strategy
based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence
criterion which is backward stable to enhance the robustness. We also suggest
several algorithmic optimizations that improve performance of practical LOBPCG
implementations. Numerical examples confirm that our approach consistently and
significantly outperforms previous competing approaches in both stability and
speed
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