Abscissa

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Zhendong Sun - One of the best experts on this subject based on the ideXlab platform.

Wim Michiels - One of the best experts on this subject based on the ideXlab platform.

  • Polynomial (chaos) approximation of maximum eigenvalue functions
    Numerical Algorithms, 2019
    Co-Authors: Luca Fenzi, Wim Michiels
    Abstract:

    This paper is concerned with polynomial approximations of the spectral Abscissa function (defined by 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 previous work, we highlight the major role of this function smoothness properties. Even if the eigenvalue problem matrices are analytic functions of the parameters, the spectral Abscissa function may not be differentiable, and even non-Lipschitz continuous, due to multiple rightmost eigenvalues counted with multiplicity. This analysis demonstrates smoothness properties not only heavily affect the approximation errors of the Galerkin and collocation based polynomial approximations, but also the numerical errors in the evaluation of coefficients in the Galerkin approach with integration methods. A documentation of the experiments, conducted on the benchmark problems through the software Chebfun, is publicly available.

  • A predictor-corrector type algorithm for the pseudospectral Abscissa computation of time-delay systems
    Automatica, 2010
    Co-Authors: Suat Gumussoy, Wim Michiels
    Abstract:

    The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral Abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important from a robust stability point of view. In this paper we present an accurate method for the computation of the pseudospectral Abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectrum and the level sets of an appropriately defined complex function. The computation is done in two steps. In the prediction step, an approximation of the pseudospectral is obtained based on a rational approximation of the characteristic matrix and the application of a bisection algorithm. Each step in this bisection algorithm relies on checking the presence of the imaginary axis eigenvalues of a complex matrix, similar to the delay free case. In the corrector step, the approximate pseudospectral Abscissa is corrected to any given accuracy, by solving a set of nonlinear equations that characterizes the extreme points in the pseudospectrum contours.

  • Computing the pseudospectral Abscissa of time-delay systems
    IFAC Proceedings Volumes, 2009
    Co-Authors: Suat Gumussoy, Wim Michiels
    Abstract:

    Abstract The pseudospectra of a linear time-invariant system are the sets in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral Abscissa are defined as the maximum real part of the characteristic roots in the pseudospectra and, therefore, they are for instance important from a robust stability point of view. In this paper we present a numerical method for the computation of the pseudospectral Abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectra and the level sets of an appropriately defined complex function. These connections lead us to a bisection algorithm for the computation of the pseudospectral Abscissa, where each step relies on checking the presence of imaginary axis eigenvalues of an appropriately defined operator. Because this operator is infinite-dimensional a predictor-corrector approach is taken. In the predictor step the bisection algorithm is applied where the operator is discretized into a matrix, yielding approximations for the pseudospectral Abscissa. The effect of the discretization is fully characterized in the paper. In the corrector step, the approximate pseudospectral Abscissa are corrected to any given accuracy, by solving a set of nonlinear equations that characterize extreme points in the pseudospectra contours.

  • The Smoothed Spectral Abscissa for Robust Stability Optimization
    SIAM Journal on Optimization, 2009
    Co-Authors: Joris Vanbiervliet, Bart Vandereycken, Wim Michiels, Stefan Vandewalle, Moritz Diehl
    Abstract:

    This paper concerns the stability optimization of (parameterized) matrices $A(x)$, a problem typically arising in the design of fixed-order or fixed-structured feedback controllers. It is well known that the minimization of the spectral Abscissa function $\alpha(A)$ gives rise to very difficult optimization problems, since $\alpha(A)$ is not everywhere differentiable and even not everywhere Lipschitz. We therefore propose a new stability measure, namely, the smoothed spectral Abscissa $\tilde\alpha_{\epsilon}(A)$, which is based on the inversion of a relaxed $H_2$-type cost function. The regularization parameter $\epsilon$ allows tuning the degree of smoothness. For $\epsilon$ approaching zero, the smoothed spectral Abscissa converges towards the nonsmooth spectral Abscissa from above so that $\tilde\alpha_{\epsilon}(A)\leq0$ guarantees asymptotic stability. Evaluation of the smoothed spectral Abscissa and its derivatives w.r.t. matrix parameters $x$ can be performed at the cost of solving a primal-dual Lyapunov equation pair, allowing for an efficient integration into a derivative-based optimization framework. Two optimization problems are considered: On the one hand, the minimization of the smoothed spectral Abscissa $\tilde\alpha_{\epsilon}(A(x))$ as a function of the matrix parameters for a fixed value of $\epsilon$, and, on the other hand, the maximization of $\epsilon$ such that the stability requirement $\tilde\alpha_{\epsilon}(A(x))\leq0$ is still satisfied. The latter problem can be interpreted as an $H_2$-norm minimization problem, and its solution additionally implies an upper bound on the corresponding $H_\infty$-norm or a lower bound on the distance to instability. In both cases, additional equality and inequality constraints on the variables can be naturally taken into account in the optimization problem.

  • Stability optimization with the smoothed spectral Abscissa
    2007
    Co-Authors: Joris Vanbiervliet, Bart Vandereycken, Wim Michiels, Stefan Vandewalle, Moritz Diehl
    Abstract:

    The problem of finding fixed-order stabilizing feedback controllers can be transformed into an optimization problem, where the controller parameters are the optimization variables. The most straightforward choice of objective function is related to the eigenvalues of the system matrix A. Specifically, minimization of the spectral Abscissa α(A), i.e. the rightmost eigenvalue of the spectrum σ(A), stabilizes the system as soon as α(A) < 0. However, the spectral Abscissa is in general nonsmooth and nonconvex and therefore typically a very hard function to optimize. Moreover, it is known to be a lousy measure of stability when robustness against parameter uncertainties is regarded. For this reason, more robust criteria have been proposed, most prominently the minimization of the pseudospectral Abscissa [1, 2]. While this approach, being closely related to maximizing the distance to instability (or to minimizing the H∞-norm), indeed yields much more robust solutions, the pseudospectral Abscissa still suffers from nonsmoothness and associated high computational costs in optimization. In this talk we propose a new type of “pseudospectral Abscissa αǫ(A)”, namely the smooth spectral Abscissa, which is based on the inversion of a relaxed H2-type cost function. For ǫ → 0, the smooth spectral Abscissa αǫ(A) is shown to converge from above towards the exact spectral Abscissa α(A), so that stability is guaranteed if αǫ(A) ≤ 0. In addition, the smooth spectral Abscissa and its derivatives can be evaluated at the cost of solving a few relaxed Lyapunov equations, allowing for an efficient integration in a derivative based optimization framework. Similarly to the pseuodspectral Abscissa, the problem of maximizing ǫ while still satisfying the stability condition αǫ(A) ≤ 0 is shown to be equivalent to the H2-norm optimization problem.

Jokke Hasa - One of the best experts on this subject based on the ideXlab platform.

  • representation growth of compact linear groups
    Transactions of the American Mathematical Society, 2019
    Co-Authors: Jokke Hasa, Alexander Stasinski
    Abstract:

    We study the representation growth of simple compact Lie groups and of SLn(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GLn(O). This amounts to a study of the Abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the Abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the Abscissa of Witten zeta functions is r/κ, where r is the rank and κ the number of positive roots. We then show that the twist zeta function of GLn(O) exists and has the same Abscissa of convergence as the zeta function of SLn(O), provided n does not divide char O. We compute the twist zeta function of GL2(O) when the residue characteristic p of O is odd and approximate the zeta function when p = 2 to deduce that the Abscissa is 1. Finally, we construct a large part of the representations of SL2(Fq[[t]]), q even, and deduce that its Abscissa lies in the interval [1, 5/2].

  • representation growth of compact linear groups
    arXiv: Representation Theory, 2017
    Co-Authors: Jokke Hasa, Alexander Stasinski
    Abstract:

    We study the representation growth of simple compact Lie groups and of $\mathrm{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is a compact discrete valuation ring, as well as the twist representation growth of $\mathrm{GL}_n(\mathcal{O})$. This amounts to a study of the Abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the Abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the Abscissa of Witten zeta functions is $r/\kappa$, where $r$ is the rank and $\kappa$ the number of positive roots. We then show that the twist zeta function of $\mathrm{GL}_n(\mathcal{O})$ exists and has the same Abscissa of convergence as the zeta function of $\mathrm{SL}_n(\mathcal{O})$, provided $n$ does not divide $\text{char}\,{\mathcal{O}}$. We compute the twist zeta function of $\mathrm{GL}_2(\mathcal{O})$ when the residue characteristic $p$ of $\mathcal{O}$ is odd, and approximate the zeta function when $p=2$ to deduce that the Abscissa is $1$. Finally, we construct a large part of the representations of $\mathrm{SL}_2(\mathbb{F}_q[[t]])$, $q$ even, and deduce that its Abscissa lies in the interval $[1,\,5/2]$.

Meili Lin - One of the best experts on this subject based on the ideXlab platform.

Alexander Stasinski - One of the best experts on this subject based on the ideXlab platform.

  • representation growth of compact linear groups
    Transactions of the American Mathematical Society, 2019
    Co-Authors: Jokke Hasa, Alexander Stasinski
    Abstract:

    We study the representation growth of simple compact Lie groups and of SLn(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GLn(O). This amounts to a study of the Abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the Abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the Abscissa of Witten zeta functions is r/κ, where r is the rank and κ the number of positive roots. We then show that the twist zeta function of GLn(O) exists and has the same Abscissa of convergence as the zeta function of SLn(O), provided n does not divide char O. We compute the twist zeta function of GL2(O) when the residue characteristic p of O is odd and approximate the zeta function when p = 2 to deduce that the Abscissa is 1. Finally, we construct a large part of the representations of SL2(Fq[[t]]), q even, and deduce that its Abscissa lies in the interval [1, 5/2].

  • representation growth of compact linear groups
    arXiv: Representation Theory, 2017
    Co-Authors: Jokke Hasa, Alexander Stasinski
    Abstract:

    We study the representation growth of simple compact Lie groups and of $\mathrm{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is a compact discrete valuation ring, as well as the twist representation growth of $\mathrm{GL}_n(\mathcal{O})$. This amounts to a study of the Abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the Abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the Abscissa of Witten zeta functions is $r/\kappa$, where $r$ is the rank and $\kappa$ the number of positive roots. We then show that the twist zeta function of $\mathrm{GL}_n(\mathcal{O})$ exists and has the same Abscissa of convergence as the zeta function of $\mathrm{SL}_n(\mathcal{O})$, provided $n$ does not divide $\text{char}\,{\mathcal{O}}$. We compute the twist zeta function of $\mathrm{GL}_2(\mathcal{O})$ when the residue characteristic $p$ of $\mathcal{O}$ is odd, and approximate the zeta function when $p=2$ to deduce that the Abscissa is $1$. Finally, we construct a large part of the representations of $\mathrm{SL}_2(\mathbb{F}_q[[t]])$, $q$ even, and deduce that its Abscissa lies in the interval $[1,\,5/2]$.

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