April 2015 - NumPI - Mailing list — Dipartimento di Matematica, UniPI

Reminder: Optimization & Numerical Analysis Seminar: M.G. Scutellà
by Federico Poloni 29 Apr '15

29 Apr '15

Our next seminar is tomorrow morning. Date: Thursday April 30, 2015 Time: 10:00 Place: Sala Seminari Ovest, dipartimento di informatica Speaker: Maria Grazia Scutellà, dipartimento di informatica, U Pisa. Title: *Optimizing Green Wireless LANs* Abstract: We consider problems arising in the design of Green (or energy-saving) Wireless Local Area Networks (GWLANs). In this context, decisions on powering-on a set of access points, and decisions on the assignment of the user terminals to the opened access points, have to be taken simultaneously. In particular, the level of power assigned to each access point may affect, in a nonlinear way, the capacity of the connections between the access points and the user terminals that are assigned to them. The aim is to minimize the overall power consumption of the access points, which is given by a linear dependency between the power consumed and the total demand assigned to the access points, via assignment costs. In the first part of the talk we present a Mixed Integer formulation with NonLinear constraints (MINLP) and an optimization approach, belonging to the framework of Branch and Benders Cut methods, which is based on the proposed MINLP formulation. In a non-standard fashion, the master problem of the approach includes the variables of the Benders subproblem, but relaxes their integrality. The approach has been tested on a large set of instances, and compared to a more traditional Benders decomposition algorithm on a subset of instances without the assignment costs, where the two approaches can be compared. The main achievements of the computational experimentation are shortly discussed. Then, in the second part of the talk, we address the optimization of GWLANs in a more realistic scenario, which takes into account also some uncertainty factors. Specifically, we present a robust optimization approach that incorporates both link capacity fluctuations and user mobility under Bertsimas and Sim's robust optimization paradigm. Again, preliminary computational results are shortly discussed. === Everyone is welcome! -- --federico poloni Dipartimento di Informatica, Università di Pisa http://www.di.unipi.it/~fpoloni/ tel. +39 050 2213143

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Optimization & Numerical Analysis Seminar: M.G. Scutellà
by Federico Poloni 23 Apr '15

23 Apr '15

Date: Thursday April 30, 2015 Time: 10:00 Place: Sala Seminari Ovest, dipartimento di informatica Speaker: Maria Grazia Scutellà, dipartimento di informatica, U Pisa. Title: *Optimizing Green Wireless LANs* Abstract: We consider problems arising in the design of Green (or energy-saving) Wireless Local Area Networks (GWLANs). In this context, decisions on powering-on a set of access points, and decisions on the assignment of the user terminals to the opened access points, have to be taken simultaneously. In particular, the level of power assigned to each access point may affect, in a nonlinear way, the capacity of the connections between the access points and the user terminals that are assigned to them. The aim is to minimize the overall power consumption of the access points, which is given by a linear dependency between the power consumed and the total demand assigned to the access points, via assignment costs. In the first part of the talk we present a Mixed Integer formulation with NonLinear constraints (MINLP) and an optimization approach, belonging to the framework of Branch and Benders Cut methods, which is based on the proposed MINLP formulation. In a non-standard fashion, the master problem of the approach includes the variables of the Benders subproblem, but relaxes their integrality. The approach has been tested on a large set of instances, and compared to a more traditional Benders decomposition algorithm on a subset of instances without the assignment costs, where the two approaches can be compared. The main achievements of the computational experimentation are shortly discussed. Then, in the second part of the talk, we address the optimization of GWLANs in a more realistic scenario, which takes into account also some uncertainty factors. Specifically, we present a robust optimization approach that incorporates both link capacity fluctuations and user mobility under Bertsimas and Sim's robust optimization paradigm. Again, preliminary computational results are shortly discussed. === Everyone is welcome! -- --federico poloni Dipartimento di Informatica, Università di Pisa http://www.di.unipi.it/~fpoloni/ tel. +39 050 2213143

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Reminder: Optimization & Numerical Analysis Seminar: F. Poloni
by Federico Poloni 16 Apr '15

16 Apr '15

Date: Thursday April 16, 2015 Time: 11:00 Place: Sala Riunioni Est, dipartimento di informatica Speaker: Federico Poloni, dipartimento di informatica, U Pisa. Title: *Methods for accurate invariant subspace computations.* Abstract: We consider two modelling problems, one arising in control theory (linear-quadratic regulator) and one arising in the modelling of queues (or buffers) under a continuous-time Markov model. The solution of both can be reduced to finding the invariant subspace of a suitable structured matrix, or equivalently solving a Riccati-like matrix equation. I will present linear algebra tools that can be used to solve them accurately; in particular: * a QR-like matrix factorization in which a tall skinny n x m matrix is factored as U*R, where R is m x m square invertible and U has a m x m identity submatrix and all its other entries bounded in modulus by 1. This factorization can be formulated as an optimization problem (maximum volume submatrix) or as finding a suitable set of basic variables in a tableau. * (if time allows) a numerical method for the Markov problem based on a technique (GTH method) to solve linear systems with a (possibly ill-conditioned) M-matrix without subtractions, obtaining high accuracy in both large and small entries of the solution. === Everyone is welcome! -- --federico poloni Dipartimento di Informatica, Università di Pisa http://www.di.unipi.it/~fpoloni/ tel. +39 050 2213143

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Optimization & Numerical Analysis Seminar: F. Poloni
by Federico Poloni 10 Apr '15

10 Apr '15

Date: Thursday April 16, 2015 Time: 11:00 Place: Sala Riunioni Est, dipartimento di informatica Speaker: Federico Poloni, dipartimento di informatica, U Pisa. Title: *Methods for accurate invariant subspace computations.* Abstract: We consider two modelling problems, one arising in control theory (linear-quadratic regulator) and one arising in the modelling of queues (or buffers) under a continuous-time Markov model. The solution of both can be reduced to finding the invariant subspace of a suitable structured matrix, or equivalently solving a Riccati-like matrix equation. I will present linear algebra tools that can be used to solve them accurately; in particular: * a QR-like matrix factorization in which a tall skinny n x m matrix is factored as U*R, where R is m x m square invertible and U has a m x m identity submatrix and all its other entries bounded in modulus by 1. This factorization can be formulated as an optimization problem (maximum volume submatrix) or as finding a suitable set of basic variables in a tableau. * (if time allows) a numerical method for the Markov problem based on a technique (GTH method) to solve linear systems with a (possibly ill-conditioned) M-matrix without subtractions, obtaining high accuracy in both large and small entries of the solution. === Everyone is welcome! -- --federico poloni Dipartimento di Informatica, Università di Pisa http://www.di.unipi.it/~fpoloni/ tel. +39 050 2213143

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Reminder: Optimization & Numerical Analysis seminars: R. Vandebril
by Federico Poloni 08 Apr '15

08 Apr '15

Date: Giovedi' 9 Aprile Time: 11:00 Place: Aula Seminari, dipartimento di matematica Speaker: Raf Vandebril, Department of Computer Science, KU Leuven. Title: *The Companion QR* Abstract: In this lecture we will propose a new fast and stable manner of computing roots of polynomials. Roots of polynomials are typically computed by putting the coefficients of the polynomial in the companion matrix and then computing the eigenvalues of this matrix. Exploiting the available low-rank structure leads to an algorithm of quadratic instead of cubic complexity. Several low-cost algorithms have already been proposed. Either they fully exploit the low-rank properties of the involved matrices by representing them for instance via quasiseparable factors, or they write the companion matrix as the sum of a unitary plus low rank matrix and exploit this structure. In this lecture we will use the second manner. However, only a single QR step requires the use of the low rank part. After that we can put the low rank term aside and continue only with the unitary matrix, translating the problem thereby to a unitary eigenvalue problem. Only in the end the low rank matrix is reconstructed to retrieve the eigenvalues. Numerical experiments validate the approach, illustrate its reliability and speed. The algorithm is compared against other available methods and a proof of backward stability is provided. This research is joint work with David S.\ Watkins and Jared L.\ Aurentz from Washington State University, USA and Thomas Mach from the KU Leuven, Belgium. === Everyone is welcome! -- --federico poloni Dipartimento di Informatica, Università di Pisa http://www.di.unipi.it/~fpoloni/ tel:+39-050-2213143

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Optimization & Numerical Analysis seminars: R. Vandebril
by Federico Poloni 01 Apr '15

01 Apr '15

Date: Giovedi' 9 Aprile Time: 11:00 Place: Aula Seminari, dipartimento di matematica Speaker: Raf Vandebril, Department of Computer Science, KU Leuven. Title: *The Companion QR* Abstract: In this lecture we will propose a new fast and stable manner of computing roots of polynomials. Roots of polynomials are typically computed by putting the coefficients of the polynomial in the companion matrix and then computing the eigenvalues of this matrix. Exploiting the available low-rank structure leads to an algorithm of quadratic instead of cubic complexity. Several low-cost algorithms have already been proposed. Either they fully exploit the low-rank properties of the involved matrices by representing them for instance via quasiseparable factors, or they write the companion matrix as the sum of a unitary plus low rank matrix and exploit this structure. In this lecture we will use the second manner. However, only a single QR step requires the use of the low rank part. After that we can put the low rank term aside and continue only with the unitary matrix, translating the problem thereby to a unitary eigenvalue problem. Only in the end the low rank matrix is reconstructed to retrieve the eigenvalues. Numerical experiments validate the approach, illustrate its reliability and speed. The algorithm is compared against other available methods and a proof of backward stability is provided. This research is joint work with David S.\ Watkins and Jared L.\ Aurentz from Washington State University, USA and Thomas Mach from the KU Leuven, Belgium. === Everyone is welcome! -- --federico poloni Dipartimento di Informatica, Università di Pisa http://www.di.unipi.it/~fpoloni/ tel:+39-050-2213143

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