- NumPI - Mailing list — Dipartimento di Matematica, UniPI

Math Colloquium - Lars Ruthotto - June 17 at 17:00
by Francesco Tudisco 14 Jun '21

14 Jun '21

Dear all, the next GSSI Math Colloquium will be held on *Thursday June 17 at 5pm* (please note **the time is *5pm* instead of the usual 3pm**). The speaker is Lars Ruthotto, with a lecture connecting numerical methods for differential equations and deep learning architectures. More details below. Lars Ruthotto is Associate Professor of Mathematics and Computer Science at Emory University (Atlanta, USA). To attend the talk please use to the following Zoom link: https://us02web.zoom.us/j/85179454721?pwd=TjA0V2M3L3lVTk1NNEdVcGpQcXlTdz09 Please feel free to distribute this announcement as you see fit. Looking forward to seeing you all on Thursday! Paolo Antonelli, Stefano Marchesani, Francesco Tudisco and Francesco Viola --------------------- Title: Numerical Methods for Deep Learning motivated by Partial Differential Equations Abstract: Understanding the world through data and computation has always formed the core of scientific discovery. Amid many different approaches, two common paradigms have emerged. On the one hand, primarily data-driven approaches—such as deep neural networks—have proven extremely successful in recent years. Their success is based mainly on their ability to approximate complicated functions with generic models when trained using vast amounts of data and enormous computational resources. But despite many recent triumphs, deep neural networks are difficult to analyze and thus remain mysterious. Most importantly, they lack the robustness, explainability, interpretability, efficiency, and fairness needed for high-stakes decision-making. On the other hand, increasingly realistic model-based approaches—typically derived from first principles and formulated as partial differential equations (PDEs)—are now available for various tasks. One can often calibrate these models—which enable detailed theoretical studies, analysis, and interpretation—with relatively few measurements, thus facilitating their accurate predictions of phenomena. In this talk, I will highlight recent advances and ongoing work to understand and improve deep learning by using techniques from partial differential equations. I will demonstrate how PDE techniques can yield better insight into deep learning algorithms, more robust networks, and more efficient numerical algorithms. I will also expose some of the remaining computational and numerical challenges in this area. — Francesco Tudisco Assistant Professor School of Mathematics GSSI Gran Sasso Science Institute Web: https://ftudisco.gitlab.io <https://ftudisco.gitlab.io> -- You received this message because you are subscribed to the Google Groups "nomads-list" group. To unsubscribe from this group and stop receiving emails from it, send an email to nomads-list+unsubscribe(a)gssi.it. To view this discussion on the web visit https://groups.google.com/a/gssi.it/d/msgid/nomads-list/51fd86c2-9ac2-482f-…. For more options, visit https://groups.google.com/a/gssi.it/d/optout.

1 0

Seminar on 04/06 (Margherita Porcelli)
by Leonardo Robol 04 Jun '21

04 Jun '21

Speaker: Margherita Porcelli Affiliation: Università di Bologna Time: Friday, 04/06/2021, 16:00 Title: Relaxed Interior point methods for low-rank semidefinite programming problems In this talk we will discuss a relaxed variant of interior point methods for semidefinite programming problems (SPDs). We focus on problems in which the primal variable is expected to be low-rank at optimality. Such situations are common in relaxations of combinatorial optimization problems, for example in maximum cut problems as well as in matrix completion problems. We exploit the structure of the sought solution and relax the rigid structure of IPMs for SDP. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated Newton directions can be computed. In particular, it admits the application of both the first- and the second-order methods in this context. In this talk we will focus on second-order approaches and discuss the difficulties arising in the linear algebra phase. A prototype implementation is shown as well as computational results on matrix completion problems. In particular, we will consider mildly-ill conditioned and noisy random problems as well as problems arising in diverse applications as the matrix to be recovered represents city-to- city distances, a grayscale image, game parameters in a basketball tournament. This is joint work with S. Bellavia (Unifi) and J. Gondzio (Univ. Edinburgh) https://www.dm.unipi.it/webnew/it/seminari/relaxed-interior-point-methods-l… Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

1 2

Seminar on 14/05 (Alfredo Buttari)
by Leonardo Robol 14 May '21

14 May '21

Speaker: Alfredo Buttari Affiliation: CNRS, IRIT, Toulouse Time: Friday, 14/05/2021, 16:00 Title: Reducing the complexity of linear systems solvers through the block low-rank format Direct linear system solvers are commonly regarded as robust methods for computing the solution of linear systems of equations. Nonetheless, their complexity makes the handling of very large size problems difficult or unfeasible due to excessive execution time or memory consumption. In this talk we discuss the use of low-rank approximation techniques that allow for reducing this complexity at the price of a loss in precision which can be reliably controlled. To take advantage of these low-rank approximations, we have designed a format called block low-rank (BLR) whose objective is to achieve a favorable compromise between complexity and efficiency of operations thanks to its regular structure. We will present the basic BLR format as well as more advanced variants and the associated algorithms; we will analyze their theoretical properties and discuss the issues related to their efficient implementation on parallel computers. We will specifically focus on the use of BLR for the solution of sparse linear systems. The ombination of this format with sparse direct methods, such as the multifrontal one, leads to efficient parallel solvers with scalable complexity. These can either be used as standalone direct solvers or in combination with other techniques such as iterative or multigrid methods. We will present experimental results on real-life problems obtained by integrating the BLR format within the MUMPS parallel sparse direct solver. https://www.dm.unipi.it/webnew/it/seminari/reducing-complexity-linear-syste… Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

1 1

Seminar on 07/05 (Stefano Pozza)
by Leonardo Robol 07 May '21

07 May '21

Speaker: Stefano Pozza Affiliation: Charles University, Prague Time: Friday, 07/05/2021, 16:00 Title: The Short-term Rational Lanczos Method and Applications Rational Krylov subspaces have become a reference tool in dimension reduction procedures for several application problems. When data matrices are symmetric, a short-term recurrence can be used to generate an associated orthonormal basis. In the past this procedure was abandoned because it requires twice the number of linear system solves per iteration than with the classical long-term method. We propose an implementation that allows one to obtain key rational subspace matrices without explicitly storing the whole orthonormal basis, with a moderate computational overhead associated with sparse system solves. Several applications are discussed to illustrate the advantages of the proposed procedure. https://www.dm.unipi.it/webnew/it/seminari/short-term-rational-lanczos-meth… Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

1 1

Seminar on 23/04 (Philipp Birken)
by Leonardo Robol 23 Apr '21

23 Apr '21

Speaker: Philipp Birken Affiliation: Lund University Time: Friday, 23/04/2021, 16:00 Title: Conservative iterative solvers in computational fluid dynamics The governing equations in computational fluid dynamics such as the Navier-Stokes- or Euler equations are conservation laws. Finite volume methods are designed to respect this and the theorem of Lax-Wendroff underscores the importance of it. It roughly states that for a nonlinear (!) scalar conservation law in 1D , if the numerical method with explicit Euler time integration is consistent and (locally) conservative, then in case of convergence, the numerical method converges to a weak solution. When using implicit time integration, the widespread believe in the community is that conservation is lost. This is however, not necessarily due to the time integration, but due to the use of iterative solvers. We first present a catalogue of iterative solvers that preserve the weaker property of global conservation to identify candidates of solvers that preserve local conservation as used in the Lax-Wendroff theorem. We then proceed to prove an extension of the Lax-Wendroff theorem for the situation that we perform a fixed number of steps of a so called pseudo time iteration per time step. It turns out that in this case, the numerical method converges to a weak solution of the conservation law with a modified propagation speed. This can be exploited to improve performance of the iterative method. https://www.dm.unipi.it/webnew/it/seminari/conservative-iterative-solvers-c… Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

1 1

Seminar on 09/04 (Jie Meng)
by Leonardo Robol 09 Apr '21

09 Apr '21

Speaker: Jie Meng Affiliation: Università di Pisa Time: Friday, 09/04/2021, 16:00 Title: Geometric means of quasi-Toeplitz matrices We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices A = (a_{i,j}) i,j=1,2,... of the form A = T(a) + E, where E represents a compact operator, and T(a) is a semi-infinite Toeplitz matrix associated with the function a, with Fourier series \sum_{l} a_l e^{ilt} , in the sense that (T(a))_{i,j} = a_{j-i}. If a is real valued and essentially bounded, then these matrices represent bounded self-adjoint operators on l^2. We consider the case where a is a continuous function, where quasi-Toeplitz matrices coincide with a classical Toeplitz algebra, and the case where a is in the Wiener algebra, that is, has absolutely convergent Fourier series. We prove that if a_1, ... , a_p are continuous and positive functions, or are in the Wiener algebra with some further conditions, then means of geometric type, such as the ALM, the NBMP and the Karcher mean of quasi-Toeplitz positive definite matrices associated with a_1, ..., a_p , are quasi-Toeplitz matrices associated with the geometric mean (a_1 ... a_p)^{1/p}, which differ only by the compact correction. We show by numerical tests that these operator means can be practically approximated. https://www.dm.unipi.it/webnew/it/seminari/geometric-means-quasi-toeplitz-m… Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

1 1

[NOMADS] Seminar - Ivan Markovsky - March 30 at 18:00
by Francesco Tudisco 29 Mar '21

29 Mar '21

Dear all, You are all invited to this week's NOMADS seminar at GSSI. The seminar will take place tomorrow *March 30 at 18:00 (CET)*. The speaker is Ivan Markovsky from Vrije Universiteit Brussel (Belgium) who will give a talk on Data-driven dynamic interpolation and approximation. Abstract and more info below. The seminar will be given via Zoom. To attend the seminar please use the following link: https://us02web.zoom.us/j/85393475759?pwd=ckNDOGNGY0d0bTBZVXBmd1FibXJVUT09 <https://us02web.zoom.us/j/85393475759?pwd=ckNDOGNGY0d0bTBZVXBmd1FibXJVUT09> Further info about past and future meetings are available at the webpage: https://num-gssi.github.io/seminar/ Please feel free to distribute this announcement as you see fit. Hope to see you all Tomorrow! Francesco and Nicola ------------------------------------------------- Data-driven dynamic interpolation and approximation The behavioral system theory give theoretical foundation for nonparameteric representations of linear time-invariant systems based on Hankel matrices constructed from data. These data-driven representations led in turn to new system identification, signal processing, and control methods. In particular, data-driven simulation and linear quadratic tracking control problems were solved using the new approach [1,2]. This talk shows how the approach can be used further on for solving data-driven interpolation and approximation problems (missing data estimation) and how it can be generalized to some classes of nonlinear systems. The theory leads to algorithms that are both general (can deal simultaneously with missing, exact, and noisy data of multivariable systems) and simple (require existing numerical linear algebra methods only). This opens a practical computational way of doing system theory and signal processing directly from data without identification of a transfer function or a state space representation and doing model-based design. References: [1] I. Markovsky and P. Rapisarda. “Data-driven simulation and control”. Int. J. Control 81.12 (2008), pp. 1946--1959. [2] I. Markovsky. A missing data approach to data-driven filtering and control. IEEE Trans. Automat. Contr., 62:1972--1978, April 2017. [3] I. Markovsky and F. Dörfler. Data-driven dynamic interpolation and approximation. Technical report, Vrije Universiteit Brussel, 2021. Available from http://homepages.vub.ac.be/~imarkovs/publications/ddint.pdf — Francesco Tudisco Assistant Professor School of Mathematics GSSI Gran Sasso Science Institute Web: https://ftudisco.gitlab.io <https://ftudisco.gitlab.io> -- You received this message because you are subscribed to the Google Groups "nomads-list" group. To unsubscribe from this group and stop receiving emails from it, send an email to nomads-list+unsubscribe(a)gssi.it. To view this discussion on the web visit https://groups.google.com/a/gssi.it/d/msgid/nomads-list/51aafd85-e7f6-58a0-…. For more options, visit https://groups.google.com/a/gssi.it/d/optout.

1 0

Seminar on 19/03 (Alice Cortinovis)
by Leonardo Robol 19 Mar '21

19 Mar '21

Speaker: Alice Cortinovis Affiliation: EPFL Time: Friday, 19/03/2021, 16:00 Title: Randomized trace estimates for indefinite matrices with an application to determinants Randomized trace estimation is a popular technique to approximate the trace of a large-scale matrix A by computing the average of quadratic forms x^T * A * x for many samples of a random vector X. We show new tail bounds for randomized trace estimates in the case of Rademacher and Gaussian random vectors, which significantly improve existing results for indefinite matrices. Then we focus on the approximation of the determinant of a symmetric positive definite matrix B, which can be done via the relation log(det(B)) = trace(log(B)), where the matrix log(B) is usually indefinite. We analyze the convergence of the Lanczos method to approximate quadratic forms x^T * log(B) * x by exploiting its connection to Gauss quadrature. Finally, we combine our tail bounds on randomized trace estimates with the analysis of the Lanczos method to improve and extend an existing result on log determinant approximation to not only cover Rademacher but also Gaussian random vectors. https://www.dm.unipi.it/webnew/it/seminari/randomized-trace-estimates-indef… Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

1 1

[NOMADS] Seminar - Eugene Tyrtyshnikov - Today at 18:00
by Francesco Tudisco 09 Mar '21

09 Mar '21

Good morning everyone, This is just a gentle reminder about *today's seminar by Eugene Tyrtyshnikov* (MSU and INM-RAS). The seminar will take place *at 18:00* via the Zoom meeting: https://us02web.zoom.us/j/84101660726?pwd=TDhrWlFKdnhQVnBTZFdMWmw3Q3J4QT09 Hope to see you all there! Francesco and Nicola ---------------- Tikhonov's solution to a class of linear systems equivalent within perturbations A standard approach to incorrect problems suggests that a problem of interest is reformulated with the knowledge of some additional a-priori information. This can be done by several well-known regularization techniques. Many practical problems are successfully solved on this way. What does not still look as completely satisfactory is that the new reset problem seems to appear rather implicitly in the very process of its solution. In 1980, A. N. Tikhonov proposed a reformulation [1] that arises explicitly before the discussion of the solution methods. He suggested a notion of normal solution to a family of linear algebraic systems described by a given individual system and its vicinity comprising perturbed systems, under the assumption that there are compatible systems in the class notwithstanding the compatibility property of the given individual system. Tikhovov proved that the normal solution exists and is unique. However, a natural question about the correctness of the reset problem was not answered. In this talk we address a question of correctness of the reformulated incorrect problems that seems to have been missed in all previous considerations. The main result is the proof of correctness for Tikhonov's normal solution. Possible generalizations and difficulties will be also discussed. [1] A. N. Tikhonov, Approximate systems of linear algebraic equations, USSR Computational Mathematics and Mathematical Physics, vol. 20, issue 6 (1980) — Francesco Tudisco Assistant Professor School of Mathematics GSSI Gran Sasso Science Institute Web: https://ftudisco.gitlab.io -- You received this message because you are subscribed to the Google Groups "nomads-list" group. To unsubscribe from this group and stop receiving emails from it, send an email to nomads-list+unsubscribe(a)gssi.it. To view this discussion on the web visit https://groups.google.com/a/gssi.it/d/msgid/nomads-list/2fe6fc1f-c30c-9ee8-…. For more options, visit https://groups.google.com/a/gssi.it/d/optout.

1 0

[NOMADS] Seminar - Eugene Tyrtyshnikov - March 9 at 18:00
by Francesco Tudisco 08 Mar '21

08 Mar '21

Dear all, You are all invited to this week's NOMADS seminar at GSSI. *The seminar schedule is changed and will now run on Tuesdays @ 18:00 (CET)* most of the times. This week's seminar is on *March 09 at 18:00 (CET).* The speaker is Eugene Tyrtyshnikov from Moscow University and INM-RAS (Russia). The talk will be focused on the Tikhonov solution to a class of linear system problems. Please find abstract and title below. The seminar will be given via Zoom. To attend the seminar please use the following link: https://us02web.zoom.us/j/84101660726?pwd=TDhrWlFKdnhQVnBTZFdMWmw3Q3J4QT09 Further info about past and future meetings are available at the webpage: https://num-gssi.github.io/seminar/ Please feel free to distribute this announcement as you see fit. Hope to see you all on Tuesday! Francesco and Nicola ------ Tikhonov's solution to a class of linear systems equivalent within perturbations A standard approach to incorrect problems suggests that a problem of interest is reformulated with the knowledge of some additional a-priori information. This can be done by several well-known regularization techniques. Many practical problems are successfully solved on this way. What does not still look as completely satisfactory is that the new reset problem seems to appear rather implicitly in the very process of its solution. In 1980, A. N. Tikhonov proposed a reformulation [1] that arises explicitly before the discussion of the solution methods. He suggested a notion of normal solution to a family of linear algebraic systems described by a given individual system and its vicinity comprising perturbed systems, under the assumption that there are compatible systems in the class notwithstanding the compatibility property of the given individual system. Tikhovov proved that the normal solution exists and is unique. However, a natural question about the correctness of the reset problem was not answered. In this talk we address a question of correctness of the reformulated incorrect problems that seems to have been missed in all previous considerations. The main result is the proof of correctness for Tikhonov's normal solution. Possible generalizations and difficulties will be also discussed. [1] A. N. Tikhonov, Approximate systems of linear algebraic equations, USSR Computational Mathematics and Mathematical Physics, vol. 20, issue 6 (1980) — Francesco Tudisco Assistant Professor School of Mathematics GSSI Gran Sasso Science Institute Web: https://ftudisco.gitlab.io -- You received this message because you are subscribed to the Google Groups "nomads-list" group. To unsubscribe from this group and stop receiving emails from it, send an email to nomads-list+unsubscribe(a)gssi.it. To view this discussion on the web visit https://groups.google.com/a/gssi.it/d/msgid/nomads-list/CAO2_FWkAyR1tebRPom…. For more options, visit https://groups.google.com/a/gssi.it/d/optout.

1 0