NumPI November 2021

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[NOMADS] Seminar - Matthew Colbrook - Dec 1 at 17:00
by Francesco Tudisco 29 Nov '21

29 Nov '21

Dear all, We are starting this week a new series of NOMADS seminars on Numerical methods, Matrix analysis and Data Science. This week's seminar will take place on *Wednesday December 1 at 17:00* (CET) in the *GSSI Main Lecture Hall* (the "red" conference hall by the entrance of the lecture rooms building). The speaker is Matthew J Colbrook from University of Cambridge and ENS Paris, with a talk on the computation of semigroups of fractional PDEs. Title and abstract are below. In person participation is encouraged, but remote attendance will also be possible via the link: https://us02web.zoom.us/j/89668910844?pwd=TmVwQUFoNklUajAzMlEyMzB6ZVZzUT09 Further info about past and future meetings are available at the webpage: https://num-gssi.github.io/seminar/ Hope to see you all on Wednesday! And, please feel free to distribute this announcement as you see fit. Francesco and Nicola ----------- Title: Computing semigroups and time-fractional PDEs with error control Abstract: We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t > 0$, an arbitrary initial vector $u_0$ and an error tolerance $\epsilon > 0$, the algorithm computes $\exp(tA)u_0$ with error bounded by $\epsilon$. The (parallelisable) algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules and the adaptive computation of resolvents in infinite dimensions. As a particular case, we deal with semigroups on $L^2(R^d)$ that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups, we provide a quadrature rule whose error decreases like $\exp(−cN/ log(N))$ for $N$ quadrature points, that remains stable as $N \to \infty$, and which is also suitable for infinite-dimensional operators. Finally, we extend the method to time-fractional PDEs (where it avoids singularities as $t \to 0$ and large memory consumption). Numerical examples are given, including: Schrödinger and wave equations on the aperiodic Ammann–Beenker tiling and fractional beam equations arising in the modelling of small-amplitude vibration of viscoelastic materials. The spectral analysis (which is always needed for contour methods) is considerably simplified due to an infinite-dimensional “solve-then-discretise” approach. — 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_FWmWATOjsjz5v5…. For more options, visit https://groups.google.com/a/gssi.it/d/optout.

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Annuncio di seminario, 19/11 (Mariarosa Mazza)
by Leonardo Robol 18 Nov '21

18 Nov '21

Speaker: Mariarosa Mazza Affiliation: Università dell'Insubria Venue: Aula Magna, Dipartimento di Matematica Time: Venerdì, 19/11/2021, 10:00 Title: Caputo and Riemann-Liouville fractional derivatives: a matrix comparison Fractional derivatives are a mathematical tool that received much attention in the last decades because of their non-local behavior which has been demonstrated to be useful when modeling anomalous diffusion phenomena appearing, e.g., in imaging or electrophysiology. Two of the most famous definitions of fractional derivatives are the Riemann- Liouville and the Caputo ones. The two formulations are related by a well-known formula that expresses a Riemann-Liouville derivative as a Caputo one plus a term that depends on the function and its derivatives at the boundary. As a consequence of this relation, Riemann-Liouville and Caputo derivatives coincide only for sufficiently smooth functions that satisfy homogeneous conditions at the boundary. Aiming at uncovering how much this discrepancy reflects on their discretized counterparts, we focus on the relation between Riemann-Liouville and Caputo derivatives once approximated by a collocation method based on B-splines, i.e., high order finite elements with maximum regularity. We show that when the fractional order α ranges in (1,2) their difference in terms of matrices corresponds to a rank-1 correction whose spectral norm increases with the mesh-size n and is o(√n). On one hand, this implies that the spectral distribution for the B-spline collocation matrices corresponding to the Riemann-Liouville and Caputo derivatives coincide; on the other hand, the presence of the rank-1 correction makes the Caputo matrices worse conditioned for α tending to 1 due to a larger maximum singular value. Some linear algebra consequences of all this knowledge are discussed, and a selection of numerical experiments that validate our findings is provided together with a numerical study of the approximation behavior of B-spline collocation. Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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NumPI November 2021