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265 discussions

Seminar on 04/07 (Federico Poloni)
by Leonardo Robol 04 Jul '19

04 Jul '19

Speaker: Federico Poloni Affiliation: Department of Computer Science, University of Pisa Time: Thursday, 04/07/2019 Place: Aula Magna, Dipartimento di Matematica Title: Principal pivot transforms, quasidefinite matrices, and matrix equations Principal pivot transforms are maps between matrices that generalize, in some sense, Schur complementation and inversion. They are a somewhat counterintuitive but powerful tool that allows one to reinterpret Gaussian elimination-type algorithms in a different framework. We will show how they can deal successfully with various kinds of matrices in a structure-preserving way; this includes (weakly) quasi-definite matrices, i.e., symmetric matrices with two complementary diagonal blocks of opposite definiteness, as well as Hamiltonian/symplectic matrices. We will then present an algorithm to solve dense algebraic Riccati equations in a factored form, which relies on these transforms as a basic building block.

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Seminar on 25/06 (Michael Overton)
by Leonardo Robol 24 Jun '19

24 Jun '19

Speaker: Michael Overton Affiliation: New York University Time: Wednesday, 25/06/2019, 11:00 Place: Aula Mancini, SNS Title: Crouzeix's conjecture Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years. Made in 2004 by Michel Crouzeix, it postulates that, for any polynomial p and any matrix A, ||p(A)|| <= 2 max(|p(z)|: z in W(A)), where the norm is the 2-norm and W(A) is the field of values (numerical range) of A, that is the set of points attained by v*Av for some vector v of unit length. Crouzeix proved in 2007 that the inequality above holds if 2 is replaced by 11.08, and recently this was greatly improved by Palencia, replacing 2 by 1+sqrt(2). Furthermore, it is known that the conjecture holds in a number of special cases, including n=2. We use nonsmooth optimization to investigate the conjecture numerically by locally minimizing the “Crouzeix ratio”, defined as the quotient with numerator the right-hand side and denominator the left-hand side of the conjectured inequality. We also present local nonsmooth variational analysis of the Crouzeix ratio at conjectured global minimizers. All our results strongly support the truth of Crouzeix’s conjecture.

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Seminar on 05/06 (Leonardo Robol)
by Leonardo Robol 03 Jun '19

03 Jun '19

Dear all, I am sorry for the delay in this announcement, I was planning to send it before, but it slipped off my mind. Everybody is welcome to attend, as usual. Speaker: Leonardo Robol Affiliation: Department of Mathematics, University of Pisa Time: Wednesday, 05/06/2019 Place: Aula Riunioni, Dipartimento di Matematica Title: Rank structures in matrix equations and matrix functions Low-rank matrices (and tensors) appear often in applications, and this can be exploited in several ways to treat large scale problems that would be otherwise unfeasible. It has been known since a long time that in some applications, such as the solution of the Sylvester equation AX + XB + C = 0, working with low-rank C often leads to a solution X which is full rank, but can be efficiently approximated by a low-rank matrix. This can be used, among other things, in solving PDEs defined by separable operators, such as the 2D Laplace operator on rectangular domains. We briefly review some of these results, and the connection with some rational approximation problems. We show that these structures appear in a multitude of interesting cases. For instance, the same properties can be found when modifying the coefficients of linear matrix equation by performing low-rank updates to their coefficients, and the theory can be extended to more general matrices which have off-diagonal blocks of low-rank. Time permitting, we discuss some recent developments that allow to prove the existence of a low-rank structures in the action of functions of matrices that have a Kronecker sum structure, such as x = f (A⊗I + B^T⊗I)v, for particularly structured vector v. An efficient algorithm for the approximation of the vector x, reshaped in matrix form, is given. This founds applications in extending the ideas used for the Laplace operator and 2D PDEs to more general nonlocal operators, such as the fractional Laplacian.

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Seminar on 11/04 (Fabio Carrara)
by Leonardo Robol 11 Apr '19

11 Apr '19

Speaker: Fabio Carrara Affiliation: ISTI-CNR, Pisa Time: Thursday, 11 April 2019, h. 14:00 Place: Aula Riunioni, Dipartimento di Matematica Title: Neural networks for image understanding: methods, limitations, and new frontiers Vision is natural and easy for living beings, while it is incredibly complex to be replicated in machines. In the last 40 years, a lot of work in Computer Vision has been spent trying to model and implement visual perception manually, with slow improvements over time. However, in the last decade, deep neural networks revolutionized this field by automatically learning complex models for vision from huge amounts of data, thus relieving the practitioners from defining and extracting complex visual features that would require a lot of manual effort to find. In this talk, we will make a journey into neural networks for computer vision focusing on tasks in which they shine, that is image recognition and understanding. We will cover the most successful neural network formulation for visual data, that is the convolutional neural network, from the basics to recent architectures, while pinpointing current limitations in the field. In conclusion, we will discuss novel neural network architectures, such as continuous ODE-defined models, that have been recently proposed and provide new research directions.

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Seminar on 08/04 (Raf Vandebril)
by Leonardo Robol 08 Apr '19

08 Apr '19

Speaker: Raf Vandebril Affiliation: Department of Computer Science, KU Leuven Time: Monday, 08 April 2019, h. 14:00 Place: Aula Seminari, Dipartimento di Matematica Title: Structures, Gramians, and Krylov Krylov methods are typically used for transforming large scale prolems and matrices to matrices of a manageable size. Krylov inspired algorithms are based on the principle: project - solve - lift. In this talk we will focus on the projection step, the orthogonal bases involved, and the structured of the resulting projection. Two bases are needed for the projection: a basis for a search Krylov subspace, say V, and another basis for the constraint Krylov subspace, name it W. For a matrix A, the projected small to medium sized matrix will be W^*AV. There is now a wide variety of possibilities for constructing the matrices W and V. We could go from classical Krylov to extended and rational Krylov subspaces. But we could also attempt to construct subspaces with variants of the matrix A, such as for instance its inverse, its conjugate transpose, or the inverse of its conjugate transpose. In this lecture we will examine some of these possibilities. A general framework is proposed in which we will touch upon the following aspects: biorthogonal (rational) Krylov, CMV factorizations, non-unitary CMV factorizations, quasiseparable Hessenberg matrices, Hankel and Toeplitz Gramians.

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Seminar on 12/04 (James Nagy)
by Leonardo Robol 04 Apr '19

04 Apr '19

Speaker: James Nagy Affiliation: Department of Mathematics, Emory University Time: Friday, 12 April 2019, h. 11:00 Place: Aula Mancini, SNS Title: MATLAB Tools for Large-Scale Linear Inverse Problems Inverse problems arise in a variety of applications: image processing, finance, mathematical biology, and more. Mathematical models for these applications may involve integral equations, partial differential equations, and dynamical systems, and solution schemes are formulated by applying algorithms that incorporate regularization techniques and/or statistical approaches. In most cases these solutions schemes involve the need to solve a large-scale ill-conditioned linear system that is corrupted by noise and other errors. In this talk we describe and demonstrate capabilities of a new MATLAB software package that consists of state-of-the-art iterative methods for solving such systems, which includes approaches that can automatically estimate regularization parameters, stopping iterations, etc., making them very simple to use. Thus, the package allows users to easily incorporate into their own applications (or simply experiment with) different iterative methods and regularization strategies with very little programming effort. On the other hand, sophisticated users can also easily access various options to tune the algorithms for certain applications. Moreover, the package includes several test problems and examples to illustrate how the iterative methods can be used on a variety of large-scale inverse problems. The talk will begin with a brief introduction to inverse problems, discuss considerations that are needed to compute an approximate solution, and describe some details about new efficient hybrid Krylov subspace methods that are implemented in our package. These methods can guide users in automatically choosing regularization parameters, and can be used to enforce various regularization schemes, such as sparsity. We will use imaging examples that arise in medicine and astronomy to illustrate the performance of the methods. This is joint work with Silvia Gazzola (University of Bath) and Per Christian Hansen (Technical University of Denmark).

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Seminar on 04/04 (Peter Benner)
by Leonardo Robol 22 Mar '19

22 Mar '19

Speaker: Peter Benner Affiliation: Max Planck Institute for Dynamics of Complex Technical Systems Time: Thursday, 04 April 2019, h. 15:00 Place: Sala Seminari Ovest, Dipartimento di Informatica Title: Low-rank tensor methods for PDE-constrained optimization under uncertainty We discuss optimization and control of unsteady partial differential equations (PDEs), where some coefficient of the PDE as well as the control may be uncertain. This may be due to the lack of knowledge about the exact physical parameters, like material properties describing a real-world problem (''epistemic uncertainty'') or the inability to apply a computed optimal control exactly in practice. Using a stochastic Galerkin space-time discretization of the optimality system resulting from such PDE-constrained optimization problems under uncertainty leads to large-scale linear or nonlinear systems of equations in saddle point form. Nonlinearity is treated with a Picard-type iteration in which linear saddle point systems have to be solved in each iteration step. Using data compression based on separation of variables and the tensor train (TT) format, we show how these large-scale indefinite and (non)symmetric systems that typically have 10⁸ to 10¹⁵ unknowns can be solved without the use of HPC technology. The key observation is that the unknown and the data can be well approximated in a new block TT format that reduces complexity by several orders of magnitude. As examples, we consider control and optimization problems for the linear heat equation, the unsteady Stokes and Stokes-Brinkman equations, as well as the incompressible unsteady Navier-Stokes equations. The talk surveys the results published in [1,2] and provides new results for the Navier-Stokes case. [1] P. Benner, A. Onwunta, M. Stoll, Block-diagonal preconditioning for optimal control problems constrained by PDEs with uncertain inputs, SIAM Journal on Matrix Analysis and Applications, 37(2):491--518, 2016. [2] P. Benner, S. Dolgov, A. Onwunta, M. Stoll, Low-rank solvers for unsteady Stokes-Brinkman optimal control problem with random data, Computer Methods in Applied Mechanics and Engineering, 304:26--54, 2016.

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Fwd: [Dipartimento.di] Seminar: Machine Learning & Optimization: A Fruitful Interplay - M. Sanguineti
by Leonardo Robol 22 Mar '19

22 Mar '19

Dear all, I would like to advertise the following seminar, that might be of interest to some of you. Best, -- Leonardo. -------- Messaggio originale -------- Oggetto: [Dipartimento.di] Seminar: Machine Learning & Optimization: A Fruitful Interplay - M. Sanguineti Data: 06-03-2019 16:28 Mittente: Alessio Micheli <micheli(a)di.unipi.it> Destinatario: dipartimento(a)di.unipi.it Cari tutti, con piacere annuncio il seguente seminario di Marcello Sanguineti in dipartimento. DATE/HOUR: Thursday, March 28, 2019 – 3 P.M. ROOM: Sala seminari ovest - Dip. di Informatica – Pisa TITLE: Machine Learning & Optimization: A Fruitful Interplay SPEAKER: Marcello Sanguineti, Department of Computer Science, Bioengineering,Robotics, and Systems Engineering (DIBRIS), University of Genova ABSTRACT: Since the very beginning, there has been a fruitful exchange between machine learning and optimization. While machine learning exploits optimization models and algorithms, it simultaneously poses problems which often constitute optimization challenges. This cross-fertilization is particularly evident nowadays. Many applications produce data that have to be processed and many other applications are based on data that have already been processed. In society, data-producers and data-consumers continuously exchange their roles. At the same time, scientists generate and collect huge amounts of data and need to develop methodologies for data analysis. This requires the use of ever more powerful approaches and techniques. Among these improvements, those coming from the field of optimization surely play a basic role. Not only classical optimization frameworks, such as stochastic approximation, first-order methods, and convex relaxation, are exploited in machine learning, but also more sophisticated models and techniques, such as regularized optimization, robust optimization, and functional optimization. In this talk, I provide a non-technical review of some optimization paradigms and techniques used in machine learning and I emphasize some aspects at the confluence of these two disciplines. The interplay between machine learning and optimization continues to develop, and its benefits are there for all to see. Riferimento: Alessio Micheli. -- Saluti, Alessio.

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Seminar next Monday (Vanni Noferini)
by Leonardo Robol 21 Mar '19

21 Mar '19

Dear all, next Monday we will have a seminar by Vanni Noferini; starting from the following week, we will try to go back to the original schedule of having seminars on Thursday afternoon. You are all welcome! Best wishes, -- Leonardo. Speaker: Vanni Noferini Affiliation: University of Essex / Aalto University Time: Monday, 25 March 2019, h. 10:00 Place: Sala Seminari, Dipartimento di Matematica Title: Matrices in companion rings and their Smith forms, with applications to group theory and algebraic topology In group theory, various properties of the abelianization of a cyclically presented group can be deduced by the Smith normal form of an integer circulant matrix. Motivated by this fact, we present a number of results on the Smith form of matrices that are polynomials in the companion matrix of a polynomial with coefficients in a generic elementary divisor domain. Our tools are of purely matrix theoretical nature and I will present them from the point of view of a matrix theorist. However, they enable significant advances in pure algebra. In particular, I plan to discuss how our results provide a tool to study the homology of 3-dimensional Brieskorn manifolds, which are objects of (apparently) interest for topologists. If time permits, I will mention other possible applications to group theory.

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Seminar next Thursday (Vladimir Druskin)
by Leonardo Robol 20 Mar '19

20 Mar '19

Dear all, there will be two seminars next week: in addition to the one by Carla Hernando on Monday, Vladimir Druskin will give a talk on Thursday at SNS. You find more details in the abstract below. You are all welcome to attend. Best wishes, -- Leonardo. Speaker: Vladimir Druskin Affiliation: Worcester Polytechnic Institute Time: Monday, 21 March 2019, h. 15:00 Place: Aula Mancini, SNS Title: Embedding properties of network realizations of reduced order models with applications to inverse scattering and data science Continued fractions are known since antiquity as the most compact representations of numbers. At the end of the 19th century Stieltjes connected them with physics. This connection gave rise to network syntheses in the first half of the 20th century that was at the base of modern electronics design and consecutively to model order reduction (MOR) that tremendously impacted many areas of engineering by enabling efficient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, Stieltjes continuous fractions can be used to embed the data back into the state space of the underlying dynamical system via special mechanical networks known as Stieltjes strings. Such networks can learn the underlying PDE system from the data (transfer function) via rigorously chosen hyper-parameters. Among many application of this powerful approach we discuss the following two. 1. Imaging in strongly scattering media with waves (e.g.,seismic exploration) via data-driven MOR. 2. Reduced order graph-Laplacians and efficient cluster analysis of big data sets.

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