arXiv subjects

Vladimir Yu. Protasov

Publications and source records attributed to Vladimir Yu. Protasov.

27 records · Page 2Linked to original sources

arXiv2020

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 $r$ such that the graph can be made acyclic by cutting at most $r$ incoming edges from each vertex. Several modifications, when each vertex is assigned with its own maximal number $r_i$ of cut edges, when some of edges are "untouchable", are also considered. Some applications are discussed.

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arXiv2020

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 considered

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arXiv2019

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.

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arXiv2019

Regularity of anisotropic refinable functions

This paper presents a detailed regularity analysis of anisotropic wavelet frames and subdivision. In the univariate setting, the smoothness of wavelet frames and subdivision is well understood by means of the matrix approach. In the multivariate setting, this approach has been extended only to the special case of isotropic refinement with the dilation matrix all of whose eigenvalues are equal in the absolute value. The general anisotropic case has resisted to be fully understood: the matrix approach can determine whether a refinable function belongs to $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$, but its Hölder regularity remained mysteriously unattainable. It this paper we show how to compute the Hölder regularity in $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$. In the anisotropic case, our expression for the exact Hölder exponent of a refinable function reflects the impact of the variable moduli of the eigenvalues of the corresponding dilation matrix. In the isotropic case, our results reduce to the well-known facts from the literature. We provide an efficient algorithm for determining the finite set of the restricted transition matrices whose spectral properties characterize the Hölder exponent of the corresponding refinable function. We also analyze the higher regularity, the local regularity, the corresponding moduli of continuity, and the rate of convergence of the corresponding subdivision schemes. We illustrate our results with several examples.

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arXiv2018

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.

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arXiv2016

Universal measure for Poncelet-type theorems

We give a simple proof of the Emch closing theorem by introducing a new invariant measure on the circle. Special cases of that measures are well-known and have been used in the literature to prove Poncelet's and Zigzag theorems. Some further generalizations are also obtained by applying the new measure.

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arXiv2015

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 $ω$. In the "most regular" case $ω= \frac{1}{16}$, we prove that the limit function has Hölder exponent $2$ and its derivative is "almost Lipschitz" with logarithmic factor $2$. Second we compute the Hölder exponent of Daubechies wavelets of high order.

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arXiv2014

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 τ for which the corresponding discrete time system with step size τ is stable implies the stability of LSS. Our main goal is to obtain a converse statement, that is, to estimate the discretization step size τ > 0 up to a given accuracy ε > 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).

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arXiv2012

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 $R^{d}_+,$ 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.

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