401 research outputs found
Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations
This paper is concerned with polynomial approximations of the spectral
abscissa function (the supremum of the real parts of the eigenvalues) of a
parameterized eigenvalue problem, which are closely related to polynomial chaos
approximations if the parameters correspond to realizations of random
variables.
Unlike in existing works, we highlight the major role of the smoothness
properties of the spectral abscissa function. Even if the matrices of the
eigenvalue problem are analytic functions of the parameters, the spectral
abscissa function may not be everywhere differentiable, even not everywhere
Lipschitz continuous, which is related to multiple rightmost eigenvalues or
rightmost eigenvalues with multiplicity higher than one.
The presented analysis demonstrates that the smoothness properties heavily
affect the approximation errors of the Galerkin and collocation-based
polynomial approximations, and the numerical errors of the evaluation of
coefficients with integration methods. A documentation of the experiments,
conducted on the benchmark problems through the software Chebfun, is publicly
available.Comment: This is a pre-print of an article published in Numerical Algorithms.
The final authenticated version is available online at:
https://doi.org/10.1007/s11075-018-00648-
Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using The Matrix Lambert W function
This paper revisits a recently developed methodology based on the matrix
Lambert W function for the stability analysis of linear time invariant, time
delay systems. By studying a particular, yet common, second order system, we
show that in general there is no one to one correspondence between the branches
of the matrix Lambert W function and the characteristic roots of the system.
Furthermore, it is shown that under mild conditions only two branches suffice
to find the complete spectrum of the system, and that the principal branch can
be used to find several roots, and not the dominant root only, as stated in
previous works. The results are first presented analytically, and then verified
by numerical experiments
Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method
The partial Schur factorization can be used to represent several eigenpairs
of a matrix in a numerically robust way. Different adaptions of the Arnoldi
method are often used to compute partial Schur factorizations. We propose here
a technique to compute a partial Schur factorization of a nonlinear eigenvalue
problem (NEP). The technique is inspired by the algorithm in [8], now called
the infinite Arnoldi method. The infinite Arnoldi method is a method designed
for NEPs, and can be interpreted as Arnoldi's method applied to a linear
infinite-dimensional operator, whose reciprocal eigenvalues are the solutions
to the NEP. As a first result we show that the invariant pairs of the operator
are equivalent to invariant pairs of the NEP. We characterize the structure of
the invariant pairs of the operator and show how one can carry out a
modification of the infinite Arnoldi method by respecting the structure. This
also allows us to naturally add the feature known as locking. We nest this
algorithm with an outer iteration, where the infinite Arnoldi method for a
particular type of structured functions is appropriately restarted. The
restarting exploits the structure and is inspired by the well-known implicitly
restarted Arnoldi method for standard eigenvalue problems. The final algorithm
is applied to examples from a benchmark collection, showing that both
processing time and memory consumption can be considerably reduced with the
restarting technique
An Inequality Constrained SL/QP Method for Minimizing the Spectral Abscissa
We consider a problem in eigenvalue optimization, in particular finding a
local minimizer of the spectral abscissa - the value of a parameter that
results in the smallest value of the largest real part of the spectrum of a
matrix system. This is an important problem for the stabilization of control
systems. Many systems require the spectra to lie in the left half plane in
order for them to be stable. The optimization problem, however, is difficult to
solve because the underlying objective function is nonconvex, nonsmooth, and
non-Lipschitz. In addition, local minima tend to correspond to points of
non-differentiability and locally non-Lipschitz behavior. We present a
sequential linear and quadratic programming algorithm that solves a series of
linear or quadratic subproblems formed by linearizing the surfaces
corresponding to the largest eigenvalues. We present numerical results
comparing the algorithms to the state of the art
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