Browsing by Author "Guglielmi, Nicola"
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- ItemLow rank differential equations for Hamiltonian matrix nearness problems(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Guglielmi, Nicola; Kressner, Daniel; Lubich, ChristianFor a Hamiltonian matrix with purely imaginary eigenvalues, we aim to determine the nearest Hamiltonian matrix such that some or all eigenvalues leave the imaginary axis. Conversely, for a Hamiltonian matrix with all eigenvalues lying off the imaginary axis, we look for a nearest Hamiltonian matrix that has a pair of imaginary eigenvalues. The Hamiltonian matrices can be allowed to be complex or restricted to be real. Such Hamiltonian matrix nearness problems are motivated by applications such as the analysis of passive control systems. They are closely related to the problem of determining extremal points of Hamiltonian pseudospectra. We obtain a characterization of optimal perturbations, which turn out to be of low rank and are attractive stationary points of low-rank differential equations that we derive. This permits us to give fast algorithms - which show quadratic convergence - for solving the considered Hamiltonian matrix nearness problems.
- ItemStationary multivariate subdivision: Joint spectral radius and asymptotic similarity(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Charina, Maria; Conti, Costanza; Guglielmi, Nicola; Protasov, VladimirIn this paper we study scalar multivariate non-stationary subdivision schemes with a general integer dilation matrix. We present a new numerically efficient method for checking convergence and H ̈older regularity of such schemes. This method relies on the concepts of approximate sum rules, asymptotic similarity and the so-called joint spectral radius of a finite set of square matrices. The combination of these concepts allows us to employ recent advances in linear algebra for exact computation of the joint spectral radius that have had already a great impact on studies of stationary subdivision schemes. We also expose the limitations of non-stationary schemes in their capability to reproduce and generate certain function spaces. We illustrate our results with several examples.