82 research outputs found
Differential qd algorithm with shifts for rank-structured matrices
Although QR iterations dominate in eigenvalue computations, there are several
important cases when alternative LR-type algorithms may be preferable. In
particular, in the symmetric tridiagonal case where differential qd algorithm
with shifts (dqds) proposed by Fernando and Parlett enjoys often faster
convergence while preserving high relative accuracy (that is not guaranteed in
QR algorithm). In eigenvalue computations for rank-structured matrices QR
algorithm is also a popular choice since, in the symmetric case, the rank
structure is preserved. In the unsymmetric case, however, QR algorithm destroys
the rank structure and, hence, LR-type algorithms come to play once again. In
the current paper we discover several variants of qd algorithms for
quasiseparable matrices. Remarkably, one of them, when applied to Hessenberg
matrices becomes a direct generalization of dqds algorithm for tridiagonal
matrices. Therefore, it can be applied to such important matrices as companion
and confederate, and provides an alternative algorithm for finding roots of a
polynomial represented in the basis of orthogonal polynomials. Results of
preliminary numerical experiments are presented
Fast and scalable Gaussian process modeling with applications to astronomical time series
The growing field of large-scale time domain astronomy requires methods for
probabilistic data analysis that are computationally tractable, even with large
datasets. Gaussian Processes are a popular class of models used for this
purpose but, since the computational cost scales, in general, as the cube of
the number of data points, their application has been limited to small
datasets. In this paper, we present a novel method for Gaussian Process
modeling in one-dimension where the computational requirements scale linearly
with the size of the dataset. We demonstrate the method by applying it to
simulated and real astronomical time series datasets. These demonstrations are
examples of probabilistic inference of stellar rotation periods, asteroseismic
oscillation spectra, and transiting planet parameters. The method exploits
structure in the problem when the covariance function is expressed as a mixture
of complex exponentials, without requiring evenly spaced observations or
uniform noise. This form of covariance arises naturally when the process is a
mixture of stochastically-driven damped harmonic oscillators -- providing a
physical motivation for and interpretation of this choice -- but we also
demonstrate that it can be a useful effective model in some other cases. We
present a mathematical description of the method and compare it to existing
scalable Gaussian Process methods. The method is fast and interpretable, with a
range of potential applications within astronomical data analysis and beyond.
We provide well-tested and documented open-source implementations of this
method in C++, Python, and Julia.Comment: Updated in response to referee. Submitted to the AAS Journals.
Comments (still) welcome. Code available: https://github.com/dfm/celerit
An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling
We present a sparse linear system solver that is based on a multifrontal
variant of Gaussian elimination, and exploits low-rank approximation of the
resulting dense frontal matrices. We use hierarchically semiseparable (HSS)
matrices, which have low-rank off-diagonal blocks, to approximate the frontal
matrices. For HSS matrix construction, a randomized sampling algorithm is used
together with interpolative decompositions. The combination of the randomized
compression with a fast ULV HSS factorization leads to a solver with lower
computational complexity than the standard multifrontal method for many
applications, resulting in speedups up to 7 fold for problems in our test
suite. The implementation targets many-core systems by using task parallelism
with dynamic runtime scheduling. Numerical experiments show performance
improvements over state-of-the-art sparse direct solvers. The implementation
achieves high performance and good scalability on a range of modern shared
memory parallel systems, including the Intel Xeon Phi (MIC). The code is part
of a software package called STRUMPACK -- STRUctured Matrices PACKage, which
also has a distributed memory component for dense rank-structured matrices
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