NumPI March 2021

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[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.

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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

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[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.

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[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.

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Seminar on 05/03 (Davide Bianchi)
by Leonardo Robol 05 Mar '21

05 Mar '21

Speaker: Davide Bianchi Affiliation: University of Insubria Time: Friday, 05/03/2021, 16:00 Title: Compatibility, embedding and regularization of non-local random walks on graphs Several variants of the graph Laplacian have been introduced to model non-local diffusion pro- cesses, which allow a random walker to “jump” to non-neighborhood nodes, most notably the path graph Laplacians and the fractional graph Laplacian, see [2, 3]. From a rigorous point of view, this new dynamics is made possible by having replaced the original graph G with a weighted complete graph G 0 on the same node-set, that depends on G and wherein the presence of new edges allows a direct passage between nodes that were not neighbors in G. A natural question arises: are the dynamics on the “old” walks along the edges of G compatible with the new dynamics? Indeed, it would be desirable to introduce long-range jumps but preserving at the same time the original dynamics if we move along the edges of G. In other words, for any time-interval where does not take place any long-range jump, a random walk on G 0 should be indistinguishable from the original random walk on G. One can easily figure this by a simple but clarifying example: let us suppose that our random walker is surfing the Net (the original graph G), and just for the sake of simplicity let us suppose that the Net is undirected. The walker then can move towards linked web-pages with a probability that can be both uniforms on the number of total links or dependent on some other parameters. Suppose now that we allow the walker to jump from one web-page to non-linked web-pages by just typing an URL address in the navigation bar so that he can virtually reach directly any possible web-pages on the Net (the induced graph G 0 ). If in any moment, for any reason, the walker is forced again to surf the Net by just following the links, then we should see him moving exactly as he used to do, namely, the probability he moves to the next linked web-page has to be the same as before. Unfortunately, in general, the induced complete graph G 0 , defined accordingly to the proposal in the literature, breaks that compatibility and the new models cease to be expressions of the original model G. In this talk, we will present some of the main results obtained in [1]. We will first introduce a rigorous definition of compatibility and embedding, which stem from a probabilistic and purely an- alytical point of view, respectively. Secondly, we will propose a regularization method to guarantee such compatibility and preserving at the same time all the nice properties granted by G 0 . Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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