74 research outputs found
Large-scale computation of pseudospectra using ARPACK and eigs
ARPACK and its MATLAB counterpart, eigs, are software packages that calculate some eigenvalues of a large non-symmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in large-scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations, but also information as to whether or not the eigenvalues are likely to be physically significant. Examples are presented for matrices with dimension up to 200,000
Theoretical error estimates for computing the matrix logarithm by Pad\'e-type approximants
In this article, we focus on the error that is committed when computing the
matrix logarithm using the Gauss--Legendre quadrature rules. These formulas can
be interpreted as Pad\'e approximants of a suitable Gauss hypergeometric
function. Empirical observation tells us that the convergence of these
quadratures becomes slow when the matrix is not close to the identity matrix,
thus suggesting the usage of an inverse scaling and squaring approach for
obtaining a matrix with this property. The novelty of this work is the
introduction of error estimates that can be used to select a priori both the
number of Legendre points needed to obtain a given accuracy and the number of
inverse scaling and squaring to be performed. We include some numerical
experiments to show the reliability of the estimates introduced
Structured pseudospectra and random eigenvalues problems in vibrating systems
This paper introduces the concept of pseudospectra as a generalized tool for uncertainty quantification and
propagation in structural dynamics. Different types of pseudospectra of matrices and matrix polynomials are
explained. Particular emphasis is given to structured pseudospectra for matrix polynomials, which offer a
deterministic way of dealing with uncertainties for structural dynamic systems. The pseudospectra analysis is
compared with the results from Monte Carlo simulations of uncertain discrete systems. Two illustrative example
problems, one with probabilistic uncertainty with various types of statistical distributions and the other with interval
type of uncertainty, are studied in detail. Excellent agreement is found between the pseudospectra results and Monte
Carlo simulation results
High Performance Computing for Stability Problems - Applications to Hydrodynamic Stability and Neutron Transport Criticality
In this work we examine two kinds of applications in terms of stability and perform numerical evaluations and benchmarks on parallel platforms. We consider the applicability
of pseudospectra in the field of hydrodynamic stability to obtain more information than a
traditional linear stability analysis can provide. Furthermore, we treat the neutron transport criticality problem and highlight the Davidson method as an attractive alternative to the so far widely used power method in that context
A pseudospectral matrix method for time-dependent tensor fields on a spherical shell
We construct a pseudospectral method for the solution of time-dependent,
non-linear partial differential equations on a three-dimensional spherical
shell. The problem we address is the treatment of tensor fields on the sphere.
As a test case we consider the evolution of a single black hole in numerical
general relativity. A natural strategy would be the expansion in tensor
spherical harmonics in spherical coordinates. Instead, we consider the simpler
and potentially more efficient possibility of a double Fourier expansion on the
sphere for tensors in Cartesian coordinates. As usual for the double Fourier
method, we employ a filter to address time-step limitations and certain
stability issues. We find that a tensor filter based on spin-weighted spherical
harmonics is successful, while two simplified, non-spin-weighted filters do not
lead to stable evolutions. The derivatives and the filter are implemented by
matrix multiplication for efficiency. A key technical point is the construction
of a matrix multiplication method for the spin-weighted spherical harmonic
filter. As example for the efficient parallelization of the double Fourier,
spin-weighted filter method we discuss an implementation on a GPU, which
achieves a speed-up of up to a factor of 20 compared to a single core CPU
implementation.Comment: 33 pages, 9 figure
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