21 research outputs found
The greedy strategy in optimizing the Perron eigenvalue
We address the problems of minimizing and of maximizing the spectral radius
overa compact family of non-negative matrices. Those problems being hard in
generalcan be efficiently solved for some special families. We consider the
so-called prod-uct families, where each matrix is composed of rows chosen
independently from givensets. A recently introduced greedy method works very
fast. However, it is applicablemostly for strictly positive matrices. For
sparse matrices, it often diverges and gives awrong answer. We present the
"selective greedy method" thatworks equally well forall non-negative product
families, including sparse ones.For this method, we provea quadratic rate of
convergence and demonstrate its efficiency in numerical examples.The numerical
examples are realised for two cases: finite uncertainty sets and poly-hedral
uncertainty sets given by systems of linear inequalities. In dimensions up to
2000, the matrices with minimal/maximal spectral radii in product families are
foundwithin a few iterations. Applications to dynamical systemsand to the graph
theoryare considere
Invariant polytopes of linear operators with applications to regularity of wavelets and of subdivisions
We generalize the recent invariant polytope algorithm for computing the joint
spectral radius and extend it to a wider class of matrix sets. This, in
particular, makes the algorithm applicable to sets of matrices that have
finitely many spectrum maximizing products. A criterion of convergence of the
algorithm is proved.
As an application we solve two challenging computational open problems. First
we find the regularity of the Butterfly subdivision scheme for various
parameters . In the "most regular" case , we
prove that the limit function has H\"older exponent and its derivative is
"almost Lipschitz" with logarithmic factor . Second we compute the H\"older
exponent of Daubechies wavelets of high order.Comment: 36 page
Maximal acyclic subgraphs and closest stable matrices
We develop a matrix approach to the Maximal Acyclic Subgraph (MAS) problem by
reducing it to finding the closest nilpotent matrix to the matrix of the graph.
Using recent results on the closest Schur stable systems and on minimising the
spectral radius over special sets of non-negative matrices we obtain an
algorithm for finding an approximate solution of MAS. Numerical results for
graphs from 50 to 1500 vertices demonstrate its fast convergence and give the
rate of approximation in most cases larger than 0.6. The same method gives the
precise solution for the following weakened version of MAS: and the minimal
such that the graph can be made acyclic by cutting at most incoming edges
from each vertex. Several modifications, when each vertex is assigned with its
own maximal number of cut edges, when some of edges are "untouchable",
are also considered. Some applications are discussed
Stability of linear switching systems and Markov-Bernstein inequalities for exponents
We analyse the problem of stability of a continuous time linear switching
system (LSS) versus the stability of its Euler discretization. It is well-known
that the existence of a positive {\tau} for which the corresponding discrete
time system with step size {\tau} is stable implies the stability of LSS. Our
main goal is to obtain a converse statement, that is, to estimate the
discretization step size {\tau} > 0 up to a given accuracy {\epsilon} > 0. This
leads to a method of deciding the stability of continuous time LSS with a
guaranteed accuracy. As the first step, we solve this problem for matrices with
real spectrum and conjecture that our method stays valid for the general case.
Our approach is based on Markov-Bernstein type inequalities for systems of
exponents. We obtain universal estimates for sharp constants in those
inequalities. Our work provides the first estimate of the computational cost of
the stability problem for continuous-time LSS (though restricted to the
real-spectrum case)
Tiling of polyhedral sets
A self-affine tiling of a compact set G of positive Lebesgue measure is its
partition to parallel shifts of a compact set which is affinely similar to G.
We find all polyhedral sets (unions of finitely many convex polyhedra) that
admit self-affine tilings. It is shown that in R^d there exist an infinite
family of such polyhedral sets, not affinely equivalent to each other. A
special attention is paid to an important particular case when the matrix of
affine similarity and the translation vectors are integer. Applications to the
approximation theory and to the functional analysis are discussed
On the closest stable/unstable nonnegative matrix and related stability radii
We consider the problem of computing the closest stable/unstable non-negative
matrix to a given real matrix. This problem is important in the study of linear
dynamical systems, numerical methods, etc. The distance between matrices is
measured in the Frobenius norm. The problem is addressed for two types of
stability: the Schur stability (the matrix is stable if its spectral radius is
smaller than one) and the Hurwitz stability (the matrix is stable if its
spectral abscissa is negative). We show that the closest unstable matrix can
always be explicitly found. For the closest stable matrix, we present an
iterative algorithm which converges to a local minimum with a linear rate. It
is shown that the total number of local minima can be exponential in the
dimension. Numerical results and the complexity estimates are presented
Self-affine 2-attractors and tiles
We study two-digit attractors (2-attractors) in which are
self-affine compact sets defined by two contraction affine mappings with the
same linear part. They are widely studied in the literature under various
names: twindragons, two-digit tiles, 2-reptiles, etc., due to many applications
in approximation theory, in the construction of multivariate Haar systems and
other wavelet bases, in the discrete geometry, and in the number theory. We
obtain a complete classification of isotropic 2-attractors in
and show that they are all homeomorphic but not diffeomorphic. In the general,
non-isotropic, case it is proved that a 2-attractor is uniquely defined, up to
an affine similarity, by the spectrum of the dilation matrix. We estimate the
number of different 2-attractors in by analysing integer unitary
expanding polynomials with the free coefficient . The total number of
such polynomials is estimated by the Mahler measure. We present several
infinite series of such polynomials. For some of the 2-attractors, their
H\"older exponents are found. Some of our results are extended to attractors
with an arbitrary number of digits.Comment: 40 pages, 7 figure
Analytic functions in shift-invariant spaces and analytic limits of level dependent subdivision
The structure of exponential subspaces of finitely generated shift-invariant
spaces is well understood and the role of such subspaces for the approximation
power of refinable function vectors and related multi-wavelets is well studied.
In this paper, in the univariate setting, we characterize all analytic
subspaces of finitely generated shift-invariant spaces and provide explicit
descriptions of elements of such subspaces. Consequently, we depict the
analytic functions generated by level dependent (non-stationary) subdivision
schemes with masks of unbounded support. And we confirm the belief that the
exponential polynomials are indeed the only analytic functions generated by
such subdivision schemes with finitely supported masks
Elliptic polytopes and invariant norms of linear operators
We address the problem of constructing elliptic polytopes in R^d, which are
convex hulls of finitely many two-dimensional ellipses with a common center.
Such sets arise in the study of spectral properties of matrices, asymptotics of
long matrix products, in the Lyapunov stability, etc.. The main issue in the
construction is to decide whether a given ellipse is in the convex hull of
others. The computational complexity of this problem is analysed by considering
an equivalent optimisation problem. We show that the number of local extrema of
that problem may grow exponentially in d. For d=2,3, it admits an explicit
solution for an arbitrary number of ellipses; for higher dimensions, several
geometric methods for approximate solutions are derived. Those methods are
analysed numerically and their efficiency is demonstrated in applications
Convex Optimization methods for computing the Lyapunov Exponent of matrices
We introduce a new approach to evaluate the largest Lyapunov exponent of a
family of nonnegative matrices. The method is based on using special positive
homogeneous functionals on which gives iterative lower and upper
bounds for the Lyapunov exponent. They improve previously known bounds and
converge to the real value. The rate of convergence is estimated and the
efficiency of the algorithm is demonstrated on several problems from
applications (in functional analysis, combinatorics, and lan- guage theory) and
on numerical examples with randomly generated matrices. The method computes the
Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up
to 60). We generalize this approach to all matrices, not necessar- ily
nonnegative, derive a new universal upper bound for the Lyapunov exponent, and
show that such a lower bound, in general, does not exist