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
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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
